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Stability Properties of Linear File-Sharing Networks

L. Leskelä, Philippe Robert, Florian Simatos

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L. Leskelä, Philippe Robert, Florian Simatos. Stability Properties of Linear File-Sharing Networks. 2009. inria-00401104

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Preprint submitted on 2 Jul 2009

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STABILITY PROPERTIES OF LINEAR FILE-SHARING NETWORKS

LASSE LESKELÄ, PHILIPPE ROBERT, AND FLORIAN SIMATOS

ABSTRACT. File-sharing networks are distributed systems used to disseminate files among a subset of the nodes of the Internet. A file is split into several pieces called chunks, the general simple principle is that once a node of the system has retrieved a chunk, it may become a server for this chunk. A stochastic model is considered for arrival times and durations of time to download chunks. One investigates the maximal arrival rate that such a network can accommodate, i.e., the conditions under which the Markov process describing this network is ergodic. Technical estimates related to the survival of interacting branching processes are key ingredients to establish the stability of these systems. Several cases are considered: networks with one and two chunks where a complete classification is obtained and several cases of a network with n chunks.

CONTENTS

1. Introduction 1

2. Analysis of the Single-Chunk Network 4

3. Yule Processes with Deletions 9

4. Analysis of the Multi-Chunk Network 15

Appendix A. Proof of Proposition 3.3 21

References 24

1. INTRODUCTION

File-sharing networks are distributed systems used to disseminate information among a subset of the nodes of the Internet (overlay network). The general simple principle is the following: once a node of the system has retrieved a file it becomes a server for this file. The advantage of this scheme is that it disseminates information in a very efficient way as long as the number of servers is growing rapidly. The growth of the number of servers is not necessarily without bounds since a node having this file may stop being a server after some time. These schemes have been used for some time now in peer-to-peer systems such as BitTorrent or Emule, for example to distribute large files over the Internet.

An improved version of this principle consists in splitting the original file into several pieces (called “chunks”) so that a given node can retrieve simultaneously several chunks of the same file from different servers. In this case, the rate to get a given file may thus increase significantly. At the same time, the global capacity of

Date: July 2, 2009.

Key words and phrases. Peer-to-Peer Algorithms; Killed Branching Processes;

Work partially supported by SCALP Project funded by EEC Network of Excellence Euro-FGI, and the Academy of Finland. ---PAGE_BREAK---

the file-sharing system is also increased since a node becomes a server of a chunk as soon as it has retrieved it and not only when it has the whole file. This improvement has interesting algorithmic implications since each node has to establish a matching between chunks and servers. Strategies to maximize the global efficiency of the file sharing systems have to be devised. See for instance Massoulié and Vojnović [12], Bonald et al. [4] and Massoulié and Twigg [11].

The efficiency of these systems can be considered from different points of view.

Transient behavior: A new file is owned by one node, given there are potentially N other nodes interested by it, how long does it take so that a given node retrieves it ? significant fraction $\alpha \in (0, 1]$ of the N nodes retrieve it ? See Yang and de Veciana [26] and Simatos et al. [22]. See also Robert and Simatos [19].

Stationary behavior: A constant flow of requests enters, is the capacity of the file-sharing system sufficient to cope with this flow ?

In this paper, the stationary behavior is investigated in a stochastic context: arrival times are random as well as chunk transmission times. In this setting mathematical studies are quite scarce, see Qiu and Srikant [17], Simatos et al. [22], Susitaival et al. [24] and references therein. A simple strategy to disseminate chunks is considered: chunks are retrieved sequentially and a given node can be the server of only the last chunk it got. See Massoulié and Vojnović [12] and Parvez et al. [16] for a detailed motivation of this situation.

In this paper, the sequential scheme for disseminating a file that is divided into n chunks is analyzed. New requests arrive according to a Poisson process at rate $\lambda$, and become downloaders of chunk 1. Users who have obtained chunks 1,...,k act simultaneously as uploaders of chunk k and downloaders of chunk k + 1, and the users who have all the chunks leave the network at rate $\nu$. The transmission rate of chunk k is denoted by $\mu_k$, and $x_k$ is the number of users having obtained chunks 1,...,k. In this way, the total transmission rate of chunk k in the network is $\mu_k x_k$. The flow of users can be modeled as the linear network depicted in Figure 1.

FIGURE 1. Transition rates of the linear network outside boundaries.

The main problem analyzed in the paper is the determination of a constant $\lambda^*$ such that if $\lambda < \lambda^*$ [resp. $\lambda > \lambda^*$], then the associated Markov process is ergodic [resp. transient]. As it will be seen, the constant $\lambda^*$ may be infinite in some cases so that the file-sharing network is always stable independently of the value of $\lambda$. The main technical difficulty to prove stability/instability results for this class of stochastic networks is that, except for the input, the Markov process has unbounded jump rates, in fact proportional to one of the coordinates of the current state. Note that loss networks have also this characteristic but in this case, the stability problem is trivial since the state space is finite. See Kelly [8].

Fluid Limits for File-Sharing Networks. Classically, to analyze the stability properties of stochastic networks, one can use the limits of a scaling of the Markov ---PAGE_BREAK---

process, the so-called fluid limits. The scaling consists in speeding up time by the norm $|x|$ of the initial state $x$, by scaling the state vector by $1/|x|$ and by letting $|x|$ go to infinity. See Bramson [5], Chen and Yao [6] and Robert [18] for example. This scaling is, however, better suited to "locally additive" processes, that is, Markov processes that behave locally as random walks. Since the transition rates are unbounded, it may occur that the corresponding fluid limits have discontinuities; this complicates a lot the analysis of a possible limiting dynamical system. Roughly speaking, this is due to the fact that, because of the unbounded transition rates, events occur on the time scale $t \mapsto t \log |x|$ instead of $t \mapsto |x|t$. See the case of the $M/M/\infty$ queue in Chapter 9 of Robert [18], and Simatos and Tibi [23] for a discussion of this phenomenon in a related context.

A "fluid scaling" is nevertheless available for file-sharing networks. A possible description for a possible candidate $(x_i(t))$ for this limiting picture would satisfy the following differential equations,

For the sake of simplicity the behavior at the boundaries {$x : x_i = 0$}, $i \ge 1$ is ignored in the above equations. This has been, up to now, one of the main tools to investigate mathematical models of file-sharing networks. See Qiu and Srikant [17], Núñez-Queija and Prabhu [15] for example. In the context of loss networks, an analogous limiting picture can be rigorously justified when the input rates and buffer sizes are scaled by some $N$ and the state variable by $1/N$. This scaling is not useful here, since the problem is precisely of determining the values of $\lambda$ for which the associated Markov is ergodic whereas in the above scaling $\lambda$ is scaled. From this point of view Equations (1) are therefore quite informal. They can nevertheless give some insight into the qualitative behavior of these networks but they cannot apparently be used to prove stability results. Their interpretation near boundaries is in particular not clear.

Interacting Branching Processes. Since scaling techniques do not apply here, one needs to resort to different techniques to study stability: coupling the linear file-sharing network with interacting branching processes is a key idea. For $i \ge 1$, without the departures the process $(X_i(t))$ would be a branching process where individuals give birth to one child at rate $\mu_i$. This description of such a file-sharing system as a branching process is quite natural. It has been used to analyze the transient behavior of these systems. See Yang and de Veciana [26], Dang et al. [7] and Simatos et al. [22]. A departure for $(X_i(t))$ can be seen as a death of an individual of class i and at the same time as a birth of an individual of class i+1. The file-sharing network can thus be described as a system of interacting branching processes with a constant input rate $\lambda$.

To tackle the general problem of stability, several key ingredients are used in this paper: Lyapunov functions, coupling arguments and precise estimations of the growth of a branching process killed by another branching process. As it will be seen, several results used come from the branching process formulation of the stochastic model. In particular Section 3 is devoted to the derivation of results concerning killed branching processes. The stability properties of networks with ---PAGE_BREAK---

a single-chunk file are analyzed in detail in Section 2. In Section 4, file-sharing networks with $n$ chunks are studied and the case $n = 2$ is investigated thoroughly.

Acknowledgements.

This paper has benefited from various interesting discussions with S. Borst, I. Norros, R. Núñez-Queija, B.J. Prabhu, and H. Reittu.

2. ANALYSIS OF THE SINGLE-CHUNK NETWORK

This section is devoted to the study of a class of two-dimensional Markov jump processes $(X_0(t), X_1(t))$, the corresponding Q-matrix $\Omega_r$ is given, for $x = (x_0, x_1) \in \mathbb{N}^2$, by

$$(2)\{\begin{array}{l}{\mathrm{\Omega }}_r[(x_0,x_1),(x_0+1,x_1)]=\lambda ,\\ {\mathrm{\Omega }}_r[(x_0,x_1),(x_0-1,x_1+1)]=\mu r(x_0,x_1)(x_1\vee 1){\mathbf{1}}_{\{x_0>0\}},\\ {\mathrm{\Omega }}_r[(x_0,x_1),(x_0,x_1-1)]=\nu x_1,\end{array}(2)\; \backslash quad\; \backslash begin\{cases\}\; \backslash Omega\_r[(x\_0,\; x\_1),\; (x\_0\; +\; 1,\; x\_1)]\; =\; \backslash lambda,\; \backslash \backslash \; \backslash Omega\_r[(x\_0,\; x\_1),\; (x\_0\; -\; 1,\; x\_1\; +\; 1)]\; =\; \backslash mu\; r(x\_0,\; x\_1)\; (x\_1\; \backslash lor\; 1)\; \backslash mathbf\{1\}\_\{\backslash \{x\_0>0\backslash \}\},\; \backslash \backslash \; \backslash Omega\_r[(x\_0,\; x\_1),\; (x\_0,\; x\_1\; -\; 1)]\; =\; \backslash nu\; x\_1,\; \backslash end\{cases\}$$

where $x \mapsto r(x)$, referred to as the rate function, is some fixed function on $\mathbb{N}^2$ with values in $[0, 1]$ and $n \lor m$ denotes $\max(n, m)$ for $n, m \in \mathbb{N}^2$. This corresponds to a more general model than the linear file-sharing network of Figure 1 in the case $n=1$, where for the sake of simplicity $\mu_1$ is noted $\mu$ in this section.

From a modeling perspective, this Markov process describes the following system. Requests for a single file arrive with rate $\lambda$, the first component $X_0(t)$ is the number of requests which did not get the file, whereas the second component is the number of requests having the file and acting as servers until they leave the file-sharing network. The constant $\mu$ can be viewed as the file transmission rate, and $\nu$ as the rate at which servers having all chunks leave. The term $r(x_0, x_1)$ describes the interaction of downloaders and uploaders in the system. The term $x_1 \lor 1$ can be interpreted so that there is one server permanent server in the network, which is contacted if there are no other上传er nodes in the system. A related system where there is always one permanent server for the file can be modeled by replacing the term $x_1 \lor 1$ by $x_1 + 1$. See the remark at the end of this section.

Several related examples of this class of models have been recently investigated. The case

$$r(x_0,x_1)=\frac{x_0}{x_0+x_1}r(x\_0,\; x\_1)\; =\; \backslash frac\{x\_0\}\{x\_0\; +\; x\_1\}$$

is considered in Núñez-Queija and Prabhu [15] and Massoulié and Vojnović [12]; in this case the downloading time of the file is neglected. Susitaival et al. [24] analyzes the rate function $r(x)$

$$r(x_0,x_1)=1\wedge \left(\alpha \frac{x_0}{x_1}\right)r(x\_0,\; x\_1)\; =\; 1\; \backslash wedge\; \backslash left(\; \backslash alpha\; \backslash frac\{x\_0\}\{x\_1\}\; \backslash right)$$

with $\alpha > 0$ and $a \land b$ denotes $\min(a, b)$ for $a, b \in \mathbb{R}$. This model allows to take into account that a request cannot be served by more than one server. See also Qiu and Srikant [17].

With a slight abuse of notation, for $0 < \delta \le 1$, the matrix $\Omega_\delta$ will refer to the case when the function $r$ is identically equal to $\delta$. Note that the boundary condition $x_1 \lor 1$ for departures from the first queue prevents the second coordinate from ending up in the absorbing state 0. Other possibilities are discussed at the end of this section. In the following $(X^r(t)) = (X_0^r(t), X_1^r(t))$ [resp. $(X^\delta(t))$] will denote a Markov process with Q-matrix $\Omega_r$ [resp. $\Omega_\delta$]. ---PAGE_BREAK---

Free Process. For $\delta > 0$, $Q_\delta$ denotes the following $Q$-matrix

$$(3)\{\begin{array}{l}{Q}_{\delta }[(y_0,y_1),(y_0+1,y_1)]=\lambda ,\\ Q_{\delta }[(y_0,y_1),(y_0-1,y_1+1)]=\mu \delta (y_1\vee 1),\\ Q_{\delta }[(y_0,y_1),(y_0,y_1-1)]=\nu y_1\mathrm{.}\end{array}(3)\; \backslash qquad\; \backslash begin\{cases\}\; Q\_\backslash delta[(y\_0,\; y\_1),\; (y\_0\; +\; 1,\; y\_1)]\; =\; \backslash lambda,\; \backslash \backslash \; Q\_\backslash delta[(y\_0,\; y\_1),\; (y\_0\; -\; 1,\; y\_1\; +\; 1)]\; =\; \backslash mu\backslash delta(y\_1\; \backslash vee\; 1),\; \backslash \backslash \; Q\_\backslash delta[(y\_0,\; y\_1),\; (y\_0,\; y\_1\; -\; 1)]\; =\; \backslash nu\; y\_1.\; \backslash end\{cases\}$$

The process $(Y^\delta(t)) = (Y_0^\delta(t), Y_1^\delta(t))$, referred to as the free process, will denote a Markov process with $Q$-matrix $Q_\delta$. Note that the first coordinate $Y_0^\delta$ may become negative. The second coordinate $(Y_1^\delta(t))$ of the free process is a classical birth-and-death process. It is easily checked that if $\rho_\delta$ defined as $\delta\mu/\nu$ is such that $\rho_\delta < 1$, then $(Y_1^\delta(t))$ is an ergodic Markov process converging in distribution to $Y_1^\delta(\infty)$ and that

$$(4){\lambda }^{\ast }(\delta )\nu \mathbb{E}(Y_1^{\delta }(\mathrm{\infty }))=\mu \mathbb{E}(Y_1^{\delta }(\mathrm{\infty })\vee 1)=\frac{\delta \mu }{(1-{\rho }_{\delta })(1-\log (1-{\rho }_{\delta }))}\mathrm{.}(4)\; \backslash quad\; \backslash lambda^*(\backslash delta)\; \backslash stackrel\{\backslash text\{def.\}\}\{=\}\; \backslash nu\; \backslash mathbb\{E\}(Y\_1^\backslash delta(\backslash infty))\; =\; \backslash mu\; \backslash mathbb\{E\}(Y\_1^\backslash delta(\backslash infty)\; \backslash vee\; 1)\; =\; \backslash frac\{\backslash delta\; \backslash mu\}\{(1\; -\; \backslash rho\_\backslash delta)(1\; -\; \backslash log(1\; -\; \backslash rho\_\backslash delta))\}.$$

When $\rho_\delta > 1$, then the process $(Y^\delta(t))$ converges almost surely to infinity. In the sequel $\lambda^*(1)$ is simply denoted $\lambda^*$.

In the following it will be assumed, Condition (C) below, that the rate function $r$ converges to 1 as the first coordinate goes to infinity; as will be seen, the special case $r \equiv 1$ then plays a special role, and so before analyzing the stability properties of $(X^r(t))$, one begins with an informal discussion when the rate function $r$ is identically equal to 1. Since the departure rate from the system is proportional to the number of requests/servers in the second queue, a large number of servers in the second queue gives a high departure rate, irrespectively of the state of the first queue. The input rate of new requests being constant, the real bottleneck with respect to stability is therefore when the first queue is large. The interaction of the two processes $(X_0^1(t))$ and $(X_1^1(t))$ is expressed through the indicator function of the set ${X_0^1(t) > 0}$. The second queue $(X_1^1(t))$ locally behaves like the birth-and-death process $(Y_1^1(t))$ as long as $(X_0^1(t))$ is away from 0. The two cases $\rho_1 > 1$ and $\rho_1 < 1$ are considered.

If $\rho_1 > 1$, i.e., $\mu > \nu$, the process $(X_1^1(t))$ is a transient process as long as the first coordinate is non-zero. Consequently, departures from the second queue occur faster and faster. Since, on the other hand, arrivals occur at a steady rate, departures eventually outpace arrivals. The fact that the second queue grows when $(X_0(t))$ is away from 0 stabilizes the system independently of the value of $\lambda$, and so the system should be stable for any $\lambda > 0$.

If $\rho_1 < 1$, and as long as $(X_0(t))$ is away from 0, the coordinate $(X_1^1(t))$ locally behaves like the ergodic Markov process $(Y_1^1(t))$. Hence if $(X_0^1(t))$ is non-zero for long enough, the requests in the first queue see in average $\mathbb{E}(Y_1^1(\infty) \vee 1)$ servers which work at rate $\mu$. Therefore, the stability condition for the first queue should be

where $\lambda^* = \lambda^*(1)$ is defined by Equation (4). Otherwise if $\lambda > \lambda^*$, the system should be unstable.

Markovian Notations. In the following, one will use the following convention, if $(U(t))$ is a Markov process, the index $u$ of $\mathbb{P}_u((U(t)) \in \cdot)$ will refer to the initial condition of this Markov process. ---PAGE_BREAK---

Transience and Recurrence Criteria for $(X^r(t))$.

Proposition 2.1 (Coupling). If $X^r(0) = Y^1(0) \in \mathbb{N}^2$, there exists a coupling of the processes $(X^r(t))$ and $(Y^1(t))$ such that the relation

$$(5)X_0^r(t)\ge Y_0^1(t)\text{ and }{X}_1^r(t)\le Y_1^1(t),(5)\; \backslash qquad\; X\_0^r(t)\; \backslash ge\; Y\_0^1(t)\; \backslash text\{\; and\; \}\; X\_1^r(t)\; \backslash le\; Y\_1^1(t),$$

holds for all $t \ge 0$ and for any sample path.

For any $0 \le \delta \le 1$, if

$${\tau }_{\delta }=\inf \{t\ge 0:r(X^r(t))\le \delta \}\text{ and }\sigma =\inf \{t\ge 0:X_0^r(t)=0\},\backslash tau\_\{\backslash delta\}\; =\; \backslash inf\backslash \{t\; \backslash ge\; 0\; :\; r(X^r(t))\; \backslash le\; \backslash delta\backslash \}\; \backslash text\{\; and\; \}\; \backslash sigma\; =\; \backslash inf\backslash \{t\; \backslash ge\; 0\; :\; X\_0^r(t)\; =\; 0\backslash \},$$

and if $X^1(0) = Y^\delta(0) \in \mathbb{N}^2$ then there exists a coupling of the processes $(X^r(t))$ and $(Y^\delta(t))$ such that, for any sample path, the relation

$$(6)X_0^r(t)\le Y_0^{\delta }(t)\text{ and }{X}_1^r(t)\ge Y_1^{\delta }(t)(6)\; \backslash qquad\; X\_0^r(t)\; \backslash le\; Y\_0^\backslash delta(t)\; \backslash text\{\; and\; \}\; X\_1^r(t)\; \backslash ge\; Y\_1^\backslash delta(t)$$

holds for all $t \le \tau_\delta \wedge \sigma$.

Proof. Let $X^r(0) = (x_0, x_1)$ and $Y^1(0) = (y_0, y_1)$ be such that $x_0 \ge y_0$ and $x_1 \le y_1$, one has to prove that the processes $(X^r(t))$ and $(Y^1(t))$ can be constructed such that Relation (5) holds at the time of the next jump of one of them. See Leskelä [10] for the existence of couplings using analytical, nonconstructive techniques.

The arrival rates in the first queue are the same for both processes. If $x_1 < y_1$, a departure from the second queue for $(Y^1(t))$ or $(X^r(t))$ preserves the order relation (5) and if $x_1 = y_1$, this departure occurs at the same rate for both processes and thus the corresponding instant can be chosen at the same (exponential) time. For the departures from the first to the second queue, the departure rate for $(X^r(t))$ is $\mu r(x_0, x_1)(x_1 \vee 1)\mathbb{I}_{{x_0>0}} \le \mu(y_1 \vee 1)$ which is the departure rate for $(Y^1(t))$, hence the corresponding departure instants can be taken in the reverse order so that Relation (5) also holds at the next jump instant. The first part of the proposition is proved.

The rest of the proof is done in a similar way: The initial states $X^r(0) = (x_0, x_1)$ and $Y^\delta(0) = (y_0, y_1)$ are such that $x_0 \le y_0$ and $x_1 \ge y_1$. With the killing of the processes at time $\tau_\delta \wedge \sigma$ one can assume additionally that $x_0 \neq 0$ and that the relation $r(x_0, x_1) \ge \delta$ holds; Under these assumptions one can check by inspecting the next transition that (6) holds. The proposition is proved. $\square$

Proposition 2.2. Under the condition $\mu < \nu$, the relation

$$\underset{t\to +\mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{X_0^r(t)}{t}\ge \lambda -{\lambda }^{\ast }\backslash liminf\_\{t\; \backslash to\; +\backslash infty\}\; \backslash frac\{X\_0^r(t)\}\{t\}\; \backslash geq\; \backslash lambda\; -\; \backslash lambda^*$$

holds almost surely. In particular, if $\mu < \nu$ and $\lambda > \lambda^*$, then the process $(X^r(t))$ is transient.

Proof. By Proposition 2.1, one can assume that there exists a version of $(Y^1(t))$ such that $X_0^r(0) = Y_0^1(0)$ and the relation $X_0^r(t) \ge Y_0^1(t)$ holds for any $t \ge 0$. From Definition (3) of the Q-matrix of $(Y^1(t))$, one has, for $t \ge 0$,

$$Y^1(t)=Y^1(0)+N_{\lambda }(t)-A(t),Y^\{1\}(t)\; =\; Y^\{1\}(0)\; +\; N\_\{\backslash lambda\}(t)\; -\; A(t),$$

where $(N_\lambda(t))$ is a Poisson process with parameter $\lambda$ and $(A(t))$ is the number of arrivals (jumps of size 1) for the second coordinate $(Y_1^1(t))$: in particular

$$\mathbb{E}(A(t))=\mu \mathbb{E}({\int }_0^t{Y}_1^1(s)\vee 1ds)\mathrm{.}\backslash mathbb\{E\}(A(t))\; =\; \backslash mu\; \backslash mathbb\{E\}\; \backslash left(\; \backslash int\_\{0\}^\{t\}\; Y\_\{1\}^\{1\}(s)\; \backslash vee\; 1\; ds\; \backslash right).$$ ---PAGE_BREAK---

Since $(Y_1^1(t))$ is an ergodic Markov process under the condition $\mu < \nu$, the ergodic theorem in this setting gives that

$$\underset{t\to +\mathrm{\infty }}{\lim }\frac{1}{t}A(t)=\underset{t\to +\mathrm{\infty }}{\lim }\frac{1}{t}\mathbb{E}(A(t))=\mu \mathbb{E}(Y_1^1(\mathrm{\infty })\vee 1)={\lambda }^{\ast },\backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; \backslash frac\{1\}\{t\}\; A(t)\; =\; \backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; \backslash frac\{1\}\{t\}\; \backslash mathbb\{E\}(A(t))\; =\; \backslash mu\; \backslash mathbb\{E\}\; (Y\_1^1(\backslash infty)\; \backslash lor\; 1)\; =\; \backslash lambda^*,$$

by Equation (4), hence $(Y_0^1(t)/t)$ converges almost surely to $\lambda - \lambda^*$. The proposition is proved. $\square$

The next result establishes the ergodicity result of this section.

Proposition 2.3. If the rate function $r$ is such that, for any $x_1 \in \mathbb{N}$,

(C)

$$\underset{x_0\to +\mathrm{\infty }}{\lim }r(x_0,x_1)=1,\backslash lim\_\{x\_0\; \backslash to\; +\backslash infty\}\; r(x\_0,\; x\_1)\; =\; 1,$$

and if $\mu \ge \nu$, or if $\mu < \nu$ and $\lambda < \lambda^*$ with

$${\lambda }^{\ast }=\frac{\mu }{(1-\rho )(1-\log (1-\rho ))},\backslash lambda^*\; =\; \backslash frac\{\backslash mu\}\{(1-\backslash rho)(1-\backslash log(1-\backslash rho))\},$$

and $\rho = \mu/\nu$, then $(X^r(t))$ is an ergodic Markov process.

Note that Condition (C) is satisfied for the functions $r$ considered in the models considered by Núñez-Queija and Prabhu [15] and in Susitaival et al. [24]. See above.

Proof. If $x = (x_0, x_1) \in \mathbb{R}^2$, $|x|$ denotes the norm of $x$, $|x| = |x_0| + |x_1|$. The proof uses Foster's criterion as stated in Robert [18, Theorem 9.7]. If there exist constants $K_0, K_1, t_0, t_1$ and $\eta > 0$ such that, for $x = (x_0, x_1) \in \mathbb{N}^2$,

(8)

$${\mathbb{E}}_{(x_0,x_1)}(\mathrm{\mid }{X}^r(t_1)\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })\le -t_1,\text{ if }{x}_1\ge K_1,\backslash mathbb\{E\}\_\{(x\_0,x\_1)\}(|X^r(t\_1)|\; -\; |x|)\; \backslash leq\; -t\_1,\; \backslash text\{\; if\; \}\; x\_1\; \backslash geq\; K\_1,$$

(9)

then the Markov process $(X^r(t))$ is ergodic.

Relation (8) is straightforward to establish: if $x_1 \ge K_1$, one gets, by considering only $K_1$ of the $x_1$ initial servers in the second queue and the Poisson arrivals, that

$${\mathbb{E}}_{(x_0,x_1)}(\mathrm{\mid }{X}^r(1)\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })\le \lambda -K_1(1-e^{-\nu }),\backslash mathbb\{E\}\_\{(x\_0,x\_1)\}(|X^r(1)|\; -\; |x|)\; \backslash leq\; \backslash lambda\; -\; K\_1(1\; -\; e^\{-\backslash nu\}),$$

hence it is enough to take $t_1 = 1$ and $K_1 = (\lambda+1)/(1-e^{-\nu})$ to have Relation (8).

One has therefore to establish Inequality (9). Let $\tau_\delta$ and $\sigma$ be the stopping times introduced in Proposition 2.1, one first proves an intermediate result: for any $t > 0$ and any $x_1 \in \mathbb{N}$,

$$(10)\underset{x_0\to +\mathrm{\infty }}{\lim }{\mathbb{P}}_{(x_0,x_1)}(\sigma \wedge {\tau }_{\delta }\le t)=0.(10)\; \backslash quad\; \backslash lim\_\{x\_0\; \backslash to\; +\backslash infty\}\; \backslash mathbb\{P\}\_\{(x\_0,x\_1)\}(\backslash sigma\; \backslash wedge\; \backslash tau\_\backslash delta\; \backslash le\; t)\; =\; 0.$$

Fix $x_1 \in \mathbb{N}$ and $t \ge 0$: for $\varepsilon > 0$, there exists $D_1$ such that

$${\mathbb{P}}_{x_1}(\underset{0\le s\le t}{\sup }{Y}_1^1(s)\ge D_1)\le \epsilon ,\backslash mathbb\{P\}\_\{x\_1\}\; \backslash left(\; \backslash sup\_\{0\; \backslash le\; s\; \backslash le\; t\}\; Y\_1^1(s)\; \backslash ge\; D\_1\; \backslash right)\; \backslash le\; \backslash varepsilon,$$

from Proposition 2.1, this gives the relation valid for all $x_0 \ge 0$,

$${\mathbb{P}}_{(x_0,x_1)}(\underset{0\le s\le t}{\sup }{X}_1^r(s)\ge D_1)\le \epsilon \mathrm{.}\backslash mathbb\{P\}\_\{(x\_0,x\_1)\}\; \backslash left(\; \backslash sup\_\{0\; \backslash le\; s\; \backslash le\; t\}\; X\_1^r(s)\; \backslash ge\; D\_1\; \backslash right)\; \backslash le\; \backslash varepsilon.$$

By Condition (C), there exists $\gamma \ge 0$ (that depends on $x_1$) such that $r(x_0, x_1) \ge \delta$ when $x_0 \ge \gamma$. As long as $(X^r(t))$ stays in the subset ${(y_0, y_1) : y_1 \le D_1}$, the transition rates of the first component $(X_0^r(t))$ are uniformly bounded. Consequently, ---PAGE_BREAK---

there exists $K$ such that, for $x_0 \ge K$,

$${\mathbb{P}}_{(x_0,x_1)}[\underset{s\le t}{\sup }{X}_0^r(s)\le \gamma ,\underset{s\le t}{\sup }{X}_1^r(s)\le D_1]\le \epsilon \mathrm{.}\backslash mathbb\{P\}\_\{(x\_0,x\_1)\}\; \backslash left[\; \backslash sup\_\{s\; \backslash le\; t\}\; X\_0^r(s)\; \backslash le\; \backslash gamma,\; \backslash sup\_\{s\; \backslash le\; t\}\; X\_1^r(s)\; \backslash le\; D\_1\; \backslash right]\; \backslash le\; \backslash varepsilon.$$

Relation (10) follows from the last two inequalities and the identity

$${\mathbb{P}}_{(x_0,x_1)}(\sigma \wedge {\tau }_{\delta }\le t)\le {\mathbb{P}}_{(x_0,x_1)}(\underset{s\le t}{\sup }{X}_0^r(s)\le \gamma )\mathrm{.}\backslash mathbb\{P\}\_\{(x\_0,x\_1)\}(\backslash sigma\; \backslash wedge\; \backslash tau\_\backslash delta\; \backslash le\; t)\; \backslash le\; \backslash mathbb\{P\}\_\{(x\_0,x\_1)\}\backslash left(\backslash sup\_\{s\; \backslash le\; t\}\; X\_0^r(s)\; \backslash le\; \backslash gamma\backslash right).$$

One returns to the proof of Inequality (9). By definition of the Q-matrix of the process $(X^r(t))$,

$${\mathbb{E}}_{(x_0,x_1)}(\mathrm{\mid }{X}^r(t)\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })=\lambda t-\nu {\int }_0^t{\mathbb{E}}_{(x_0,x_1)}(X_1^r(u))du,x\in {\mathbb{N}}^2,t\ge 0.\backslash mathbb\{E\}\_\{(x\_0,x\_1)\}(|X^r(t)|\; -\; |x|)\; =\; \backslash lambda\; t\; -\; \backslash nu\; \backslash int\_0^t\; \backslash mathbb\{E\}\_\{(x\_0,x\_1)\}(X\_1^r(u))\; du,\; x\; \backslash in\; \backslash mathbb\{N\}^2,\; t\; \backslash ge\; 0.$$

For any $x \in \mathbb{N}^2$, there exists a version of $(Y^\delta(t))$ with initial condition $Y^\delta(0) = X^r(0) = x$, and such that Relation (6) holds for $t < \tau_\delta \wedge \sigma$, in particular

Cauchy-Schwarz inequality shows that for any $t \ge 0$ and $x \in \mathbb{N}^2$

by gathering these inequalities, and by using the fact that the process $(Y_1^\delta(t))$ depends only on $x_1$ and not $x_0$, one finally gets the relation

$$(11)\frac{1}{t}{\mathbb{E}}_x(\mathrm{\mid }X(t)\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })\le \lambda -\frac{\nu }{t}{\int }_0^t{\mathbb{E}}_{x_1}(Y_1^{\delta }(u))du+c(x_1,t)\sqrt{{\mathbb{P}}_x({\tau }_{\delta }\wedge \sigma \le t)}(11)\; \backslash quad\; \backslash frac\{1\}\{t\}\; \backslash mathbb\{E\}\_x(|X(t)|\; -\; |x|)\; \backslash leq\; \backslash lambda\; -\; \backslash frac\{\backslash nu\}\{t\}\; \backslash int\_0^t\; \backslash mathbb\{E\}\_\{x\_1\}(Y\_1^\backslash delta(u))\; du\; +\; c(x\_1,\; t)\; \backslash sqrt\{\backslash mathbb\{P\}\_x(\backslash tau\_\backslash delta\; \backslash wedge\; \backslash sigma\; \backslash le\; t)\}$$

with

$$c(x_1,t)=\frac{\nu }{t}{\int }_0^t\sqrt{{\mathbb{E}}_{x_1}[(Y_1^{\delta }(u){)}^2]}du\mathrm{.}c(x\_1,\; t)\; =\; \backslash frac\{\backslash nu\}\{t\}\; \backslash int\_\{0\}^\{t\}\; \backslash sqrt\{\backslash mathbb\{E\}\_\{x\_1\}\; [\; (Y\_1^\{\backslash delta\}(u))^2\; ]\}\; du.$$

Two cases are considered.

(1) If $\mu > \nu$, if $\delta < 1$ is such that $\delta\mu > \nu$, the process $(Y_1^\delta(t))$ is transient, so that

$$\underset{t\to +\mathrm{\infty }}{\lim }\frac{1}{t}{\int }_0^t{\mathbb{E}}_{x_1}(Y_1^{\delta }(u))du=+\mathrm{\infty },\backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; \backslash frac\{1\}\{t\}\; \backslash int\_0^t\; \backslash mathbb\{E\}\_\{x\_1\}(Y\_1^\backslash delta(u))\; du\; =\; +\backslash infty,$$

for each $x_1 \ge 0$.

(2) If $\mu < \nu$, one takes $\delta = 1$, or if $\mu = \nu$, one takes $\delta < 1$ close enough to 1 so that $\lambda < \lambda^*(\delta)$. In both cases, $\lambda < \lambda^*(\delta)$ and the process $(Y_1^\delta(t))$ converges in distribution, hence

$$\underset{t\to +\mathrm{\infty }}{\lim }\frac{1}{t}{\int }_0^t{\mathbb{E}}_{x_1}(Y_1^{\delta }(u))du=\nu E(Y_1^{\delta }(\mathrm{\infty }))={\lambda }^{\ast }(\delta )>\lambda \backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; \backslash frac\{1\}\{t\}\; \backslash int\_0^t\; \backslash mathbb\{E\}\_\{x\_1\}(Y\_1^\backslash delta(u))\; du\; =\; \backslash nu\; E(Y\_1^\backslash delta(\backslash infty))\; =\; \backslash lambda^*(\backslash delta)\; >\; \backslash lambda$$

for each $x_1 \ge 0$. ---PAGE_BREAK---

Consequently in both cases, there exist constants $\eta > 0$, $\delta < 1$ and $t_0 > 0$ such that for any $x_1 \le K_1$,

$$(12)\lambda -\nu \frac{1}{t_0}{\int }_0^{t_0}{\mathbb{E}}_{x_1}(Y_1^{\delta }(u))du\le -\eta ,(12)\; \backslash qquad\; \backslash lambda\; -\; \backslash nu\; \backslash frac\{1\}\{t\_0\}\; \backslash int\_0^\{t\_0\}\; \backslash mathbb\{E\}\_\{x\_1\}(Y\_1^\backslash delta(u))\; du\; \backslash le\; -\backslash eta,$$

with Relation (11), one gets that if $x_1 \le K_1$ then

$$\frac{1}{t_0}{\mathbb{E}}_x(\mathrm{\mid }X(t_0)\mathrm{\mid }-\mathrm{\mid }x\mathrm{\mid })\le -\eta +c^{\ast }\sqrt{{\mathbb{P}}_x({\tau }_{\delta }\wedge \sigma \le t_0)},\backslash frac\{1\}\{t\_0\}\; \backslash mathbb\{E\}\_x(|X(t\_0)|\; -\; |x|)\; \backslash le\; -\backslash eta\; +\; c^*\; \backslash sqrt\{\backslash mathbb\{P\}\_x(\backslash tau\_\backslash delta\; \backslash wedge\; \backslash sigma\; \backslash le\; t\_0)\},$$

where $c^* = \max(c(n, t_0), 0 \le n \le K_1)$. By Identity (10), there exists $K_0$ such that, for all $x_0 \ge K_0$ and $x_1 \le K_1$, the relation

$$c^{\ast }\sqrt{{\mathbb{P}}_{(x_0,x_1)}({\tau }_{\delta }\wedge \sigma \le t_0)}\le \frac{\eta }{2}c^*\; \backslash sqrt\{\backslash mathbb\{P\}\_\{(x\_0,x\_1)\}(\backslash tau\_\{\backslash delta\}\; \backslash wedge\; \backslash sigma\; \backslash le\; t\_0)\}\; \backslash le\; \backslash frac\{\backslash eta\}\{2\}$$

holds. This relation and the inequalities (12) and (11) give Inequality (9). The proposition is proved. $\square$

Another Boundary Condition. The boundary condition $x_1 \lor 1$ in the transition rates of $(X(t))$, Equation (2), prevents the second coordinate from ending up in the absorbing state 0. It amounts to suppose that a permanent server gets activated when no node may offer the file. Another way to avoid this absorbing state is to suppose that a permanent node is always active, which gives transition rates with $x_1+1$ instead. This choice was for instance made in Núñez-Queija and Prabhu [15]. All our results apply for this other boundary condition: the only difference that is when $\nu > \mu$, the value of the threshold $\lambda^*$ of Equation (4) is given by the quantity $\lambda^* = \mu\nu/(\nu - \mu)$.

3. YULE PROCESSES WITH DELETIONS

This section introduces the tools which are necessary in order to generalize the results of the previous section to the multi-chunk case $n \ge 2$. A Yule process $(Y(t))$ with rate $\mu > 0$ is a Markovian branching process with Q-matrix

$$(13)q_Y(x,x+1)=\mu x,\mathrm{\forall }x\ge 0.(13)\; \backslash qquad\; q\_Y(x,\; x+1)\; =\; \backslash mu\; x,\; \backslash quad\; \backslash forall\; x\; \backslash ge\; 0.$$

An individual gives birth to a child, or equivalently splits into two particles, with rate $\mu$. Let $(\sigma_n)$ be the split times of a Yule process started with one particle, it is not difficult to check that, for $n \ge 1$,

$${\sigma }_n\sum _{\mathrm{\ell }=1}^n\frac{E_{\mathrm{\ell }}^{\mu }}{\mathrm{\ell }}\max (E_1^{\mu },\dots ,E_n^{\mu }),\backslash sigma\_n\; \backslash stackrel\{\backslash text\{dist.\}\}\{=\}\; \backslash sum\_\{\backslash ell=1\}^\{n\}\; \backslash frac\{E\_\{\backslash ell\}^\{\backslash mu\}\}\{\backslash ell\}\; \backslash stackrel\{\backslash text\{dist.\}\}\{=\}\; \backslash max(E\_1^\{\backslash mu\},\; \backslash dots,\; E\_n^\{\backslash mu\}),$$

where $(E_{\ell}^{\mu})$ are i.i.d. exponential random variables with parameter $\mu$. If $\lambda > \mu$ then, by using Fubini's Theorem,

In this section one considers some specific results on variants of this stochastic model when some individuals are killed. In terms of branching processes, this amounts to prune the tree, i.e., to cut some edges of the tree, and the subtree attached to ---PAGE_BREAK---

it. This procedure is fairly common for branching processes, in the Crump-Mode- Jagers model for example, see Kingman [9]. See also Neveu [14] or Aldous and Pitman [1]. Two situations are considered: the first one when the deletions are part of the internal dynamics, so that each individual dies out after an exponential time, and the other when killings are given by an exogenous process and occur at fixed (random or deterministic) epochs.

Constant Death Rate and Regeneration. Let $(Z(t))$ be the birth-and-death process whose $Q$-matrix $Q_Z$ is given by, for $\mu_Z > 0$ and $\nu > 0$,

$$(15)q_Z(z,z+1)={\mu }_Z(z\vee 1)\text{ and }{q}_Z(z,z-1)=\nu z\mathrm{.}(15)\; \backslash qquad\; q\_Z(z,\; z+1)\; =\; \backslash mu\_Z(z\; \backslash lor\; 1)\; \backslash text\{\; and\; \}\; q\_Z(z,\; z-1)\; =\; \backslash nu\; z.$$

The lifetime of an individual is exponentially distributed with parameter $v$, and the process restarts with one individual after some time when it hits 0. This process can be described equivalently as a time-changed $M/M/1$ queue or as a sequence of independent branching processes. As it will be seen these two viewpoints are complementary.

In the rest of this part, $\mu_Z$ and $\nu$ are fixed, $(Z(t))$ is the Markov process with $Q$-matrix $Q_Z$, $(\sigma_n)$ is the sequence of times of its positive jumps, the birth instants, and $(B_\sigma(t))$ is the corresponding counting process of $(\sigma_n)$, for $t \ge 0$,

$$B_{\sigma }(t)=\sum _{i\ge 1}{1}_{\{{\sigma }_i\le t\}}\mathrm{.}B\_\{\backslash sigma\}(t)\; =\; \backslash sum\_\{i\; \backslash ge\; 1\}\; 1\_\{\backslash \{\backslash sigma\_i\; \backslash le\; t\backslash \}\}.$$

Proposition 3.1 (Queueing Representation). If $Z(0) = z \in \mathbb{N}$, then

$$(16)(Z(t),t\ge 0)(L(C(t)),t\ge 0),(16)\; \backslash qquad\; (Z(t),\; t\; \backslash ge\; 0)\; \backslash stackrel\{\backslash text\{dist.\}\}\{=\}\; (L(C(t)),\; t\; \backslash ge\; 0),$$

where $(L(t))$ is the process of the number of jobs of an $M/M/1$ queue with input rate $\mu_Z$ and service rate $\nu$ and with $L(0) = z$ and $C(t) = \inf{s > 0 : A(s) > t}$, where

$$A(t)={\int }_0^t\frac{1}{1\vee L(u)}du\mathrm{.}A(t)\; =\; \backslash int\_\{0\}^\{t\}\; \backslash frac\{1\}\{1\; \backslash vee\; L(u)\}\; du.$$

Proof. It is not difficult to check that the process $(M(t)) \stackrel{\text{def.}}{=} (L(C(t)))$ has the Markov property. Let $Q_M$ be its $Q$-matrix. For $z \ge 0$,

$$\mathrm{\P }(L(C(h))=z+1\mathrm{\mid }L(0)=z)={\mu }_Z\mathbb{E}(C(h))+o(h)={\mu }_Z(z\vee 1)h+o(h),\backslash P(L(C(h))\; =\; z\; +\; 1\; |\; L(0)\; =\; z)\; =\; \backslash mu\_Z\; \backslash mathbb\{E\}(C(h))\; +\; o(h)\; =\; \backslash mu\_Z\; (z\; \backslash vee\; 1)h\; +\; o(h),$$

hence $q_M(z, z + 1) = \mu_Z(z \vee 1)$. Similarly $q_M(z, z - 1) = \nu z$. The proposition is proved. $\square$

Corollary 3.1. For any $\gamma > (\mu_Z - \nu) \lor 0$ and $z = Z(0) \in \mathbb{N}$,

Proof. Proposition 3.1 shows that, in particular, the sequences of positive jumps of $(Z(t))$ and of $(L(C(t)))$ have the same distribution. Hence, if $N_{\mu_Z} = (t_n)$ is the arrival process of the $M/M/1$ queue, a Poisson process with parameter $\mu_Z$, then, with the notations of the above proposition, the relation

$$({\sigma }_n)(A(t_n))(\backslash sigma\_n)\; \backslash stackrel\{\backslash text\{dist.\}\}\{=\}\; (A(t\_n))$$ ---PAGE_BREAK---

holds. By using standard martingale properties of stochastic integrals with respect to Poisson processes, see Rogers and Williams [20], one gets for $t \ge 0$,

where Relation (16) has been used for the last equality. Kolmogorov's equation for the process $(Z(t))$ gives that

therefore, by Gronwall's Lemma,

$$\varphi (t)\le \varphi (0)+{\mu }_Z{\int }_0^{t}u{e}^{({\mu }_Z-\nu )u}du\le z+\frac{{\mu }_Z}{{\mu }_Z-\nu }t{e}^{({\mu }_Z-\nu )t}\mathrm{.}\backslash phi(t)\; \backslash le\; \backslash phi(0)\; +\; \backslash mu\_Z\; \backslash int\_0^t\; ue^\{(\backslash mu\_Z\; -\; \backslash nu)u\}\; du\; \backslash le\; z\; +\; \backslash frac\{\backslash mu\_Z\}\{\backslash mu\_Z\; -\; \backslash nu\}\; te^\{(\backslash mu\_Z\; -\; \backslash nu)t\}.$$

From Equation (18), one concludes that

The proposition is proved. $\square$

A Branching Process. Before hitting 0, the Markov process $(Z(t))$ whose Q-matrix is given by Relation (15) can be seen a Bellman-Harris branching process. Its Malthusian parameter is given by $\alpha = \mu_Z - \nu$. See Athreya and Ney [3]. In this setting, it describes the evolution of a population of independent particles, at rate $\lambda \stackrel{\text{def.}}{=} \mu_Z + \nu$ each of these particles either splits into two particles with probability $p \stackrel{\text{def.}}{=} \mu_Z / (\mu_Z + \nu)$ or dies. These processes will be referred to as $(p, \lambda)$-branching processes in the sequel.

A $(p, \lambda)$-branching process survives with positive probability only when $p > 1/2$, in which case the probability of extinction $q$ is equal to $q = (1-p)/p = \nu/\mu_Z$. The main (and only) difference with a branching process is that $Z$ regenerates after hitting 0. When it regenerates, it again behaves as a $(p, \lambda)$-branching process (started with one particle), until it hits 0 again.

Proposition 3.2 (Branching Representation). If $Z(0) = z \in \mathbb{N}$ and $(\tilde{Z}(t))$ is a $(p, \lambda)$-branching process started with $z \in \mathbb{N}$ particles and $\tilde{T}$ its extinction time, then

$$(Z(t),0\le t\le T)(\tilde{Z}(t),0\le t\le \tilde{T}),(Z(t),\; 0\; \backslash le\; t\; \backslash le\; T)\; \backslash stackrel\{\backslash text\{dist.\}\}\{=\}\; (\backslash tilde\{Z\}(t),\; 0\; \backslash le\; t\; \backslash le\; \backslash tilde\{T\}),$$

where $T = \inf{t \ge 0 : Z(t) = 0}$ is the hitting time of 0 by $(Z(t))$.

Corollary 3.2. Suppose that $\mu_Z > \nu$. Then $\mathbb{P}_z$-almost surely for any $z \ge 0$, there exists a finite random variable $Z(\infty)$ such that,

$$\underset{t\to +\mathrm{\infty }}{\lim }{e}^{-({\mu }_Z-\nu )t}Z(t)=Z(\mathrm{\infty })\text{and}Z(\mathrm{\infty })>0.\backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; e^\{-(\backslash mu\_Z\; -\; \backslash nu)t\}\; Z(t)\; =\; Z(\backslash infty)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; Z(\backslash infty)\; >\; 0.$$ ---PAGE_BREAK---

Proof. When $\mu_Z > \nu$, the process $(Z(t))$ couples in finite time with a supercritical $(p, \lambda)$-branching process $(\tilde{Z}(t))$ conditioned on non-extinction; this follows readily from Proposition 3.2 (or see the Appendix for details). Since for any supercritical $(p, \lambda)$-branching process, $(\exp(-(\mu_Z - \nu)t)\tilde{Z}(t))$ converges almost surely to a finite random variable $\tilde{Z}(\infty)$, positive on the event of non-extinction (see Nerman [13]), one gets the desired result. $\square$

Due to its technicality, the proof of the following result is postponed to the Appendix; this result is used in the proof of Proposition 3.5.

Proposition 3.3. Suppose that $\mu_Z > \nu$, if

then for any $0 < \eta < \eta^*(\nu/\mu_Z)$,

A Yule Process Killed at Fixed Instants. In this part, it is assumed that, provided that it is non-empty, at epochs $\sigma_n$, $n \ge 1$, an individual is removed from the population of an ordinary Yule process ($Y(t)$) with rate $\mu_W$ starting with $Y(0) = w \in \mathbb{N}$ individuals. It is assumed that $(\sigma_n)$ is some fixed non-decreasing sequence. It will be shown that the process $(W(t))$ obtained by killing one individual of $Y(t)$) at each of the successive instants $(\sigma_n)$ survives with positive probability when the series with general term $(\exp(-\mu_W\sigma_n))$ converges.

In the following, a related result will be considered in the case where the sequence $(\sigma_n)$ is given by the sequence of birth times of the process $(Z(t))$ introduced above. See Alsmeyer [2] and the references therein for related models.

One denotes

$$\kappa =\inf \{n\ge 1:W({\sigma }_n)=0\}\mathrm{.}\backslash kappa\; =\; \backslash inf\backslash \{n\; \backslash ge\; 1\; :\; W(\backslash sigma\_n)\; =\; 0\backslash \}.$$

The process $(W(t))$ can be represented in the following way:

$$(20)W(t)=Y(t)-\sum _{i=1}^{\kappa }{X}_i(t)1_{\{{\sigma }_i\le t\}},(20)\; \backslash qquad\; W(t)\; =\; Y(t)\; -\; \backslash sum\_\{i=1\}^\{\backslash kappa\}\; X\_i(t)\; 1\_\{\backslash \{\backslash sigma\_i\; \backslash le\; t\backslash \}\},$$

where, for $1 \le i \le \kappa$ and $t \ge \sigma_i$, $X_i(t)$ is the total number of children at time $t$ in the original Yule process of the $i$th individual killed at time $\sigma_i$. In terms of trees, $(W(t))$ can be seen as a subtree of $(Y(t))$: for $1 \le i \le \kappa$, $(X_i(t))$ is the subtree of $(Y(t))$ associated with the $i$th particle killed at time $\sigma_i$.

It is easily checked that $(X_i(t - \sigma_i), t \ge \sigma_i)$ is a Yule process starting with one individual and, since a killed individual cannot have one of his descendants killed, that the processes

$$({\tilde{X}}_i(t))=(X_i(t+{\sigma }_i),t\ge 0),1\le i\le \kappa ,(\backslash tilde\{X\}\_i(t))\; =\; (X\_i(t\; +\; \backslash sigma\_i),\; t\; \backslash ge\; 0),\; \backslash quad\; 1\; \backslash le\; i\; \backslash le\; \backslash kappa,$$

are independent Yule processes.

For any process $(U(t))$, one denotes:

$$(21)(M_U(t))(e^{-{\mu }_{W}t}U(t))\mathrm{.}(21)\; \backslash qquad\; (M\_U(t))\; \backslash stackrel\{\backslash text\{def.\}\}\{=\}\; (e^\{-\backslash mu\_W\; t\}\; U(t)).$$ ---PAGE_BREAK---

If $(\tilde{X}(t))$ is a Yule process with rate $\mu_W$, the martingale $(M_{\tilde{X}}(t))$ converges almost surely and in $L_2$ to a random variable $M_{\tilde{X}}(\infty)$ with an exponential distribution with mean $\tilde{X}(0)$, and by Doob's Inequality

See Athreya and Ney [3]. Consequently

$$e^{-{\mu }_{W}t}W(t)=M_Y(t)-\sum _{i=1}^{\kappa }{e}^{-{\mu }_W{\sigma }_i}{M}_{{\tilde{X}}_i}(t-{\sigma }_i)1_{\{{\sigma }_i\le t\}},e^\{-\backslash mu\_W\; t\}\; W(t)\; =\; M\_Y(t)\; -\; \backslash sum\_\{i=1\}^\{\backslash kappa\}\; e^\{-\backslash mu\_W\; \backslash sigma\_i\}\; M\_\{\backslash tilde\{X\}\_i\}(t\; -\; \backslash sigma\_i)\; 1\_\{\backslash \{\backslash sigma\_i\; \backslash le\; t\backslash \}\},$$

and for any $t \ge 0$,

$$\sum _{i=1}^{\kappa }{e}^{-{\mu }_W{\sigma }_i}{M}_{{\tilde{X}}_i}(t-{\sigma }_i)1_{\{{\sigma }_i\le t\}}\le \sum _{i=1}^{\kappa }{e}^{-{\mu }_W{\sigma }_i}\underset{s\ge 0}{\sup }{M}_{{\tilde{X}}_i}(s)\mathrm{.}\backslash sum\_\{i=1\}^\{\backslash kappa\}\; e^\{-\backslash mu\_W\; \backslash sigma\_i\}\; M\_\{\backslash tilde\{X\}\_i\}(t-\backslash sigma\_i)\; 1\_\{\backslash \{\backslash sigma\_i\; \backslash le\; t\backslash \}\}\; \backslash le\; \backslash sum\_\{i=1\}^\{\backslash kappa\}\; e^\{-\backslash mu\_W\; \backslash sigma\_i\}\; \backslash sup\_\{s\; \backslash ge\; 0\}\; M\_\{\backslash tilde\{X\}\_i\}(s).$$

Assume now that $\sum_{i \ge 1} e^{-\mu_W \sigma_i} < +\infty$: then the last expression is integrable, and Lebesgue's Theorem implies that $(M_W(t)) = (\exp(-\mu_W t)W(t))$ converges almost surely and in $L_2$ to

$$M_W(\mathrm{\infty })=M_Y(\mathrm{\infty })-\sum _{i=1}^{\kappa }{e}^{-{\mu }_W{\sigma }_i}{M}_{{\tilde{X}}_i}(\mathrm{\infty })\mathrm{.}M\_W(\backslash infty)\; =\; M\_Y(\backslash infty)\; -\; \backslash sum\_\{i=1\}^\{\backslash kappa\}\; e^\{-\backslash mu\_W\; \backslash sigma\_i\}\; M\_\{\backslash tilde\{X\}\_i\}(\backslash infty).$$

Clearly, for some $w^*$ large enough and then for any $w \ge w^*$, one has

$${\mathbb{E}}_w(M_W(\mathrm{\infty }))\ge w-\sum _{i=1}^{+\mathrm{\infty }}{e}^{-{\mu }_W{\sigma }_i}>0,\backslash mathbb\{E\}\_w(M\_W(\backslash infty))\; \backslash ge\; w\; -\; \backslash sum\_\{i=1\}^\{+\backslash infty\}\; e^\{-\backslash mu\_W\; \backslash sigma\_i\}\; >\; 0,$$

in particular $\mathbb{P}_w(M_W(\infty) > 0) > 0$ and $\mathbb{P}_w(W(t) \ge 1, \forall t \ge 0) > 0$. If $Y(0) = w < w^*$ and $\sigma_1 > 0$, then $\mathbb{P}_w(Y(\sigma_1) \ge w^* + 1) > 0$ and therefore, by translation at time $\sigma_1$, the same conclusion holds when the sequence $(\exp(-\mu_W \sigma_i))$ has a finite sum. The following proposition has thus been proved.

Proposition 3.4. Let $(W(t))$ be a process growing as a Yule process with rate $\mu_W$ and for which individuals are killed at non-decreasing instants $(\sigma_n)$ with $\sigma_1 > 0$. If

then as $t$ gets large, and for any $w \ge 1$, the variable $(\exp(-\mu_W t)W(t))$ converges $\mathbb{P}_w$-almost surely and in $L_2$ to a finite random variable $M_W(\infty)$ such that $\mathbb{P}_w(M_W(\infty) > 0) > 0$.

The previous proposition establishes the minimal results needed in Section 4. However, Kolmogorov's Three-Series, see Williams [25], can be used in conjunction with Fatou's Lemma to show that $(W(t))$ dies out almost surely when the series with general term $(\exp(-\mu_W \sigma_n))$ diverges.

A Yule Process Killed at the Birth Instants of a Bellman-Harris Process.

In this subsection, one considers a Yule process $(Y(t))$ with parameter $\mu_W$ with Q-matrix defined by Relation (13) and an independent Markov process $(Z(t))$ with Q-matrix defined by Relation (15). In particular $\mu_Z - \nu$ is the Malthusian parameter of $(Z(t))$. A process $(W(t))$ is defined by killing one individual of $(Y(t))$ at each of ---PAGE_BREAK---

the birth instants $(\sigma_n)$ of $(Z(t))$. As before $(B_\sigma(t))$ denotes the counting process association to the non-decreasing sequence $(\sigma_n)$,

$$B_{\sigma }(t)=\sum _{i\ge 1}{1}_{\{{\sigma }_i\le t\}}\mathrm{.}B\_\{\backslash sigma\}(t)\; =\; \backslash sum\_\{i\; \backslash ge\; 1\}\; 1\_\{\backslash \{\backslash sigma\_i\; \backslash le\; t\backslash \}\}.$$

Proposition 3.5. Assume that $\mu_Z - \nu > \mu_W$, and let $H_0$ be the extinction time of $(W(t))$, i.e.,

$$H_0=\inf \{t\ge 0:W(t)=0\},H\_0\; =\; \backslash inf\backslash \{t\; \backslash ge\; 0\; :\; W(t)\; =\; 0\backslash \},$$

then the random variable $H_0$ is almost surely finite and:

(i) $Z(H_0) - Z(0) \le e^{\mu_W H_0} M_Y^*$ where $$M_Y^{\ast }=\underset{t\ge 0}{\sup }{e}^{-{\mu }_{W}t}Y(t)\mathrm{.}M\_Y^*\; =\; \backslash sup\_\{t\; \backslash ge\; 0\}\; e^\{-\backslash mu\_W\; t\}\; Y(t).$$

(ii) There exists a finite constant $C$ such that for any $z \ge 0$ and $w \ge 1$,

$$(22){\mathbb{E}}_{(w,z)}(H_0)\le C(\log (w)+1)\mathrm{.}(22)\; \backslash qquad\; \backslash mathbb\{E\}\_\{(w,z)\}(H\_0)\; \backslash le\; C\; (\backslash log(w)\; +\; 1).$$

Note that the subscript $(w, z)$ refers to the initial state of the Markov process $(W(t), Z(t))$.

Proof. Define $\alpha = \mu_Z - \nu$. Concerning the almost sure finiteness of $H_0$, note that Equation (20) entails that $W(t) \le Y(t) - B_\sigma(t)$ for all $t \ge 0$ on the event ${H_0 = +\infty}$. As $t$ goes to infinity, both $\exp(-\mu_W t)Y(t)$ and $\exp(-\alpha t)B_\sigma(t)$ converge almost surely to positive and finite random variables (see Nerman [13]), which implies, when $\alpha = \mu_Z - \nu > \mu_W$, that $W(t)$ converges to $-\infty$ on ${H_0 = +\infty}$, and so this event is necessarily of probability zero.

The first point (i) of the proposition comes from Identity (20) at $t = H_0$:

$$(23)Z(H_0)-Z(0)\le B_{\sigma }(H_0)\le Y(H_0)\le e^{{\mu }_W{H}_0}{M}_Y^{\ast }\mathrm{.}(23)\; \backslash qquad\; Z(H\_0)\; -\; Z(0)\; \backslash le\; B\_\backslash sigma(H\_0)\; \backslash le\; Y(H\_0)\; \backslash le\; e^\{\backslash mu\_W\; H\_0\}\; M\_Y^*.$$

By using the relation $\exp(x) \ge x$, Equation (22) follows from the following bound: for any $\eta < \eta^*(\nu/\mu_Z)$ (recall that $\eta^*$ is given by Equation (19)),

So all is left to prove is this bound. Under $\mathbb{P}{(w,z)}$, $(Y(t))$ can be represented as the sum of $w$ i.i.d. Yule processes, and so $M_Y^* \le M{Y,1}^* + \cdots + M_{Y,w}^*$ with $(M_{Y,i}^*)$ i.i.d. distributed like $M_Y^*$ under $\mathbb{P}_{(1,z)}$; Inequality (23) then entails that

$$e^{(\alpha -{\mu }_W)H_0}\le \left(\sum _{i=1}^w{M}_{Y,i}^{\ast }\right)\times \underset{t\ge {\sigma }_1}{\sup }(e^{\alpha t}\mathrm{/}{B}_{\sigma }(t))\mathrm{.}e^\{(\backslash alpha\; -\; \backslash mu\_W)H\_0\}\; \backslash le\; \backslash left(\; \backslash sum\_\{i=1\}^\{w\}\; M\_\{Y,i\}^\{*\}\; \backslash right)\; \backslash times\; \backslash sup\_\{t\; \backslash ge\; \backslash sigma\_1\}\; \backslash left(\; e^\{\backslash alpha\; t\}\; /\; B\_\{\backslash sigma\}(t)\; \backslash right).$$

By independence of $(M_{Y,i}^*)$ and $(B_\sigma(t))$, Jensen's inequality gives for any $\eta < 1$:

$${\mathbb{E}}_{(w,z)}(e^{\eta (\alpha -{\mu }_W)H_0})\le w^{\eta }(\mathbb{E}(M_{Y,1}^{\ast }){)}^{\eta }{\mathbb{E}}_z(\underset{t\ge {\sigma }_1}{\sup }(e^{\eta \alpha t}{B}_{\sigma }(t{)}^{-\eta })),\backslash mathbb\{E\}\_\{(w,z)\}\; (e^\{\backslash eta(\backslash alpha\; -\; \backslash mu\_W)H\_0\})\; \backslash le\; w^\backslash eta\; (\backslash mathbb\{E\}(M\_\{Y,1\}^*))^\backslash eta\; \backslash mathbb\{E\}\_z\; \backslash left(\; \backslash sup\_\{t\; \backslash ge\; \backslash sigma\_1\}\; (e^\{\backslash eta\backslash alpha\; t\}\; B\_\backslash sigma(t)^\{-\backslash eta\})\; \backslash right),$$

hence the bound (24) follows from Proposition 3.3. $\square$

One concludes this section with a Markov chain which will be used in Section 4. Define recursively the sequence $(V_n)$ by, $V_0 = v$ and

$$(25)V_{n+1}=\sum _{k=1}^{A_n(V_n)}{I}_k,n\ge 0,(25)\; \backslash qquad\; V\_\{n+1\}\; =\; \backslash sum\_\{k=1\}^\{A\_n(V\_n)\}\; I\_k,\; n\; \backslash ge\; 0,$$ ---PAGE_BREAK---

where $(I_k)$ are identically distributed integer valued random variables independent of $V_n$ and $A_n(V_n)$, and such that $\mathbb{E}(I_1) = p$ for some $p \in (0, 1)$. For $v > 0$, $A_n(v)$ is an independent random variable with the same distribution as $Z(H_0)$ under $\mathbb{P}_{(1,v)}$, i.e., with the initial condition $(W(0), Z(0)) = (1, v)$.

The above equation (25) can be interpreted as a branching process with immi- gration, see Seneta [21], or also as an autoregressive model.

Proposition 3.6. Under the condition $\mu_Z - \nu > \mu_W$, if $(V_n)$ is the Markov chain defined by Equation (25) and, for $K \ge 0$,

$$N_K=\inf \{n\ge 0:V_n\le K\},N\_K\; =\; \backslash inf\backslash \{n\; \backslash ge\; 0\; :\; V\_n\; \backslash le\; K\backslash \},$$

then there exist $\gamma > 0$ and $K \in \mathbb{N}$ such that

$$(26)\mathbb{E}(N_K\mathrm{\mid }{V}_0=v)\le \frac{1}{\gamma }\log (1+v),\mathrm{\forall }v\ge 0.(26)\; \backslash quad\; \backslash mathbb\{E\}(N\_K|V\_0\; =\; v)\; \backslash le\; \backslash frac\{1\}\{\backslash gamma\}\; \backslash log(1+v),\; \backslash quad\; \backslash forall\; v\; \backslash ge\; 0.$$

The Markov chain (V_n) is in particular positive recurrent.

Proof. For $V_0 = v \in \mathbb{N}$, Jensen's Inequality and Definition (25) give the relation

$$(27){\mathbb{E}}_v\log \left(\frac{1+V_1}{1+v}\right)\le {\mathbb{E}}_{(1,v)}\log \left[\frac{1+pZ(H_0)}{1+v}\right]\mathrm{.}(27)\; \backslash quad\; \backslash mathbb\{E\}\_v\; \backslash log\; \backslash left(\; \backslash frac\{1+V\_1\}\{1+v\}\; \backslash right)\; \backslash le\; \backslash mathbb\{E\}\_\{(1,v)\}\; \backslash log\; \backslash left[\; \backslash frac\{1+pZ(H\_0)\}\{1+v\}\; \backslash right].$$

From Proposition 3.5 and by using the same notations, one gets that, under $\mathbb{P}_{(1,v)}$,

$$Z(H_0)\le v+e^{{\mu }_w{H}_0}{M}_Y^{\ast },Z(H\_0)\; \backslash leq\; v\; +\; e^\{\backslash mu\_w\; H\_0\}\; M\_Y^*,$$

where $(Y(t))$ is a Yule process starting with one individual. By looking at the birth instants of $(Z(t))$, it is easily checked that the random variable $H_0$ under $\mathbb{P}{(1,v)}$ is stochastically bounded by $H_0$ under $\mathbb{P}{(1,0)}$. The integrability of $H_0$ under $\mathbb{P}_{(1,0)}$ (proved in Proposition 3.5) and of $M_Y^*$ give that the expression

$$\log \left(\frac{1+p(v+e^{{\mu }_w{H}_0}{M}_Y^{\ast })}{1+v}\right)\backslash log\; \backslash left(\; \backslash frac\{1\; +\; p(v\; +\; e^\{\backslash mu\_w\; H\_0\}\; M\_Y^*)\}\{1\; +\; v\}\; \backslash right)$$

bounding the right hand side of Relation (27) is also an integrable random variable under $\mathbb{P}_{(1,0)}$. Lebesgue's Theorem gives therefore that

Consequently, one concludes that $v \mapsto \log(1+v)$ is a Lyapunov function for the Markov chain $(V_n)$, i.e., if $\gamma = -(\log p)/2$, there exists $K$ such that for $v \ge K$,

$${\mathbb{E}}_v\log (1+V_1)-\log (1+v)\le -\gamma \mathrm{.}\backslash mathbb\{E\}\_v\; \backslash log\; (1\; +\; V\_1)\; -\; \backslash log\; (1\; +\; v)\; \backslash le\; -\backslash gamma.$$

Foster's criterion, see Theorem 8.6 of Robert [18], implies that $(V_n)$ is indeed ergodic and that Relation (26) holds. $\square$

4. ANALYSIS OF THE MULTI-CHUNK NETWORK

In this section it is assumed that a file of n chunks is distributed by the file-sharing network within the following framework, corresponding to Figure 1. Chunks are delivered in the sequential order, and, for k ≥ 1, requests with chunks 1, ..., k provide service for requests with one less chunk.

For $0 \le k < n$ and $t \ge 0$, the variable $X_k(t)$ denotes the number of requests downloading the $(k+1)$st chunk; for $k = n$, $X_n(t)$ is the number of requests having all the chunks. When taking into account the boundaries in the transition rates ---PAGE_BREAK---

described in Figure 1, one gets the following $Q$-matrix for the $(n+1)$-dimensional Markov process $(X_k(t), 0 \le k \le n)$:

where $x \in \mathbb{N}^{n+1}$, $f: \mathbb{N}^{n+1} \to \mathbb{R}_+$ is a function and for, $0 \le k \le n$, $e_k \in \mathbb{N}^{n+1}$ is the $k$th unit vector. Note that, as before, to avoid absorbing states, it is assumed that there is a server for the $k$th chunk when $x_k = 0$. The first section corresponds to the case $n = 2$ in a more general setting.

It is first shown in Proposition 4.1 that the network is stable for sufficiently small input rate $\lambda$. Proposition 4.2 studies the analog of the two-dimensional case with $\mu > \nu$, i.e., when $\mu_1 > \cdots > \mu_{n-1} > \mu_n - \nu > 0$, it is proved that the network is stable for any input rate $\lambda$. When this condition fails, it is shown that for $n = 2$ the network can only accommodate a finite input rate.

Proposition 4.1. Under the condition

the Markov process (X(t)) is ergodic for any $\nu > 0$.

Condition (28) is obviously not sharp as can be seen in the case $n=1$ analyzed in Section 2. But the proposition shows that there is always a positive threshold $\lambda^*$ such that the system is stable when $\lambda < \lambda^*$.

Proof. For $x \in \mathbb{N}^{n+1}$ and $(\alpha_k) \in \mathbb{R}^{n+1}$, define $f(x) = \alpha_0 x_0 + \dots + \alpha_n x_n$, then

$$Q(f)(x)=\lambda {\alpha }_0-\sum _{k=1}^n({\alpha }_{k-1}-{\alpha }_k){\mu }_k(x_k\vee 1)1_{\{x_{k-1}>0\}}-\nu x_n{\alpha }_n\mathrm{.}Q(f)(x)\; =\; \backslash lambda\backslash alpha\_0\; -\; \backslash sum\_\{k=1\}^\{n\}\; (\backslash alpha\_\{k-1\}\; -\; \backslash alpha\_k)\backslash mu\_k(x\_k\; \backslash vee\; 1)1\_\{\backslash \{x\_\{k-1\}>0\backslash \}\}\; -\; \backslash nu\; x\_n\; \backslash alpha\_n.$$

For $\varepsilon > 0$, one can choose $(\alpha_k)$ so that $\alpha_0 = 1$ and

$${\alpha }_{k-1}-{\alpha }_k=\frac{\lambda }{{\mu }_k}+\epsilon ,1\le k\le n,\backslash alpha\_\{k-1\}\; -\; \backslash alpha\_k\; =\; \backslash frac\{\backslash lambda\}\{\backslash mu\_k\}\; +\; \backslash varepsilon,\; \backslash quad\; 1\; \backslash le\; k\; \backslash le\; n,$$

hence

$${\alpha }_n=1-(n\epsilon +\sum _{i=1}^n\frac{\lambda }{{\mu }_k}),\backslash alpha\_n\; =\; 1\; -\; \backslash left(\; n\backslash varepsilon\; +\; \backslash sum\_\{i=1\}^\{n\}\; \backslash frac\{\backslash lambda\}\{\backslash mu\_k\}\; \backslash right),$$

so that, for $\varepsilon$ small enough, the $\alpha_k$'s, $0 \le k \le n$ are decreasing and positive under the condition of the proposition; in particular the set ${x : f(x) \le K}$ is finite for any $K \ge 0$.

Take $K = (1+\lambda)/\nu$, then if $x \in \mathbb{N}^{n+1}$ is such that $f(x) \ge K$, either $x_k > 0$ for some $0 \le k \le n-1$ and in this case

or $x_n \ge K$ so that

A Lyapunov function criteria for Markov processes shows that this implies that the Markov process $(X(t))$ is ergodic. See Proposition 8.14 of Robert [18] for example. □ ---PAGE_BREAK---

Decreasing Service Rates. The analog of the “good” case $\mu > \nu$ is proved in the next proposition.

Proposition 4.2. Under the condition $\mu_1 > \mu_2 > \cdots > \mu_{n-1} > \mu_n - \nu > 0$, the Markov process $(X(t)) = (X_k(t), 0 \le k \le n)$ describing the linear file-sharing network is ergodic for any $\lambda \ge 0$.

Proof. The proof proceeds in two steps: first coupling arguments with Yule processes allow to prove (30); then one can use the same technique as in the proof of Proposition 2.3, see Robert [18, Theorem 9.7].

Step 1 (coupling). Let $(W_n(t))$ be the process with $Q$-matrix defined by Relation (15) with $\mu_Z = \mu_n$ and starting at $W_n(0) = w_n \ge 1$. Since $\mu_n > \nu$, the process $(\exp(-(\mu_n-\nu)t)W_n(t))$ converges almost surely to a finite and positive random variable $M_{W_n}(\infty)$ by Corollary 3.2. Moreover, since $\mu_{n-1} > \mu_n - \nu > 0$, Corollary 3.1 entails that the birth instants $(\sigma_\ell^n)$ of this process are such that

Let $(Y_{n-1}(t))$ be an independent Yule process with parameter $\mu_{n-1}$ with initial condition $Y_{n-1}(0) = w_{n-1} \ge 1$ and $(W_{n-1}(t))$ the resulting process when its individuals are killed at the instants $(\sigma_\ell^n)$ of births of $(W_n(t))$: the previous equation and Proposition 3.4 show that $(W_{n-1}(t))$ can survive forever with a positive probability.

Let $(Y_{n-2}(t))$ be an independent Yule process starting from $w_{n-2} \ge 1$ with parameter $\mu_{n-2}$. Define $(W_{n-2}(t))$ the resulting process when the individuals of $(Y_{n-2}(t))$ are killed at the birth instants $(\tilde{\sigma}\ell^{n-1})$ of $(W{n-1}(t))$. Since $\mu_{n-2} > \mu_{n-1}$, the birth instants $(\tilde{\sigma}\ell^{n-1})$ of $(Y{n-1}(t))$ satisfy

almost surely by Equation (14) (which still holds for a Yule process starting with more than one particle). Since the birth instants $(\sigma_\ell^{n-1})$ of $(W_{n-1}(t))$ are a subsequence of $(\tilde{\sigma}\ell^{n-1})$, the same relationship holds for $(\sigma_\ell^{n-1})$, and therefore, with a positive probability, the three processes $(e^{-(\mu_n-\nu)t}W_n(t))$, $(e^{-\mu{n-1}t}W_{n-1}(t))$ and $(e^{-\mu_{n-2}t}W_{n-2}(t))$ converge simultaneously to positive and finite random variables $M_{W_n}(\infty)$, $M_{W_{n-1}}(\infty)$ and $M_{W_{n-2}}(\infty)$, respectively. This construction can be repeated inductively to give the existence of $n$ processes $(W_k(t), k = 1, \dots, n)$ such that $(\sigma_\ell^k)$ is the sequence of birth times of $W_k$, $W_n$ is the birth-and-death process with $Q$-matrix (15), $W_k$ for $1 \le k \le n-1$ is a Yule process with parameter $\mu_k$ killed at $(\sigma_\ell^{k+1})$, and the event $\mathcal{E} = {M_{W_1}(\infty) > 0, \dots, M_{W_n}(\infty) > 0}$ has a positive probability. On this event, $W_k(t) \ge 1$ for all $t \ge 0$ and $1 \le k \le n-1$, and

$$\underset{t\to +\mathrm{\infty }}{\lim }{W}_n(t)=+\mathrm{\infty }\mathrm{.}\backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; W\_n(t)\; =\; +\backslash infty.$$ ---PAGE_BREAK---

For $0 \le k \le n-1$, one defines $(X_k^S(t)) = (X_{k,n-k}^S(t), \dots, X_{k,n}^S(t))$, the $k$th saturated system, as the $(k+1)$-dimensional Markov process with generator

where $x \in \mathbb{N}^{k+1}$ and $f : \mathbb{N}^{k+1} \to \mathbb{R}+$ is an arbitrary function. Compared with the process $(X_\ell(t), 1 \le \ell \le n)$ with generator $Q$, it amounts to look at the $k+1$ last queues $(X{n-k}(t), \dots, X_n(t))$ under the assumption that the queue $n-k-1$ is saturated, i.e., $X_{n-k-1}(t) \equiv +\infty$ for all $t \ge 0$.

Note that for any $0 \le k \le n-1$, the transition rates of the Markov processes $(W_{n-\ell}(t), 0 \le \ell \le k)$ and $(X_{k,n-\ell}^S(t), 0 \le \ell \le k)$ are identical as long as no coordinate hits 0; one thus concludes that, with positive probability, the relation

$$\underset{t\to +\mathrm{\infty }}{\lim }{X}_{k,n}^S(t)=+\mathrm{\infty }\backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; X\_\{k,n\}^\{S\}(t)\; =\; +\backslash infty$$

holds when $X_{k,n-l}^S(0) \ge 1$, $l=0,\dots,k$. Consequently, since the set $(\mathbb{N}-{0})^{k+1}$ can be reached with positive probability from any initial state in $\mathbb{N}^{k+1}$ by $(X_k^S(t))$, then

$$(30)\underset{t\to +\mathrm{\infty }}{\lim }\mathbb{E}(X_{k,n}^S(t))=+\mathrm{\infty }\mathrm{.}(30)\; \backslash qquad\; \backslash lim\_\{t\; \backslash to\; +\backslash infty\}\; \backslash mathbb\{E\}(X\_\{k,n\}^S(t))\; =\; +\backslash infty.$$

Step 2 (Foster's criterion). We use Foster's criterion as stated in Theorem 9.7 of Robert [18]. First we inspect the case when $X_n(0)$ is large, then the case when $X_n(0)$ is bounded and $X_{n-1}(0)$ is large, etc... The key idea is that when $X_{n-k-1}(0)$ is large, then the process $(X_{n-k}(t), \dots, X_n(t))$ essentially behaves as the process $(X_k^S(t))$, for which Relation (30) ensures that the output rate is arbitrarily large.

Let $X(0) = x = (x_k) \in \mathbb{N}^{n+1}$, since the last queue serves at rate $\nu$ each request, for $t \ge 0$,

$$\mathbb{E}(\mathrm{\parallel }X(t)\mathrm{\parallel })\le \mathrm{\parallel }x\mathrm{\parallel }+\lambda t-x_n(1-e^{-\nu t}),\backslash mathbb\{E\}(\backslash |X(t)\backslash |)\; \backslash le\; \backslash |x\backslash |\; +\; \backslash lambda\; t\; -\; x\_n\; (1\; -\; e^\{-\backslash nu\; t\}),$$

where $|x| = x_0 + \dots + x_n$ for $x = (x_0, \dots, x_n) \in \mathbb{N}^{n+1}$. Define $t_n = 1$ and let $K_n$ be such that $\lambda t_n - K_1(1 - \exp(-\nu)) \le -1$, so that the relation

$${\mathbb{E}}_x(\mathrm{\parallel }X(t_n)\mathrm{\parallel })-\mathrm{\parallel }x\mathrm{\parallel }\le -1,\backslash mathbb\{E\}\_x(\backslash |X(t\_n)\backslash |)\; -\; \backslash |x\backslash |\; \backslash le\; -1,$$

holds when $x_n \ge K_n$.

From Equation (30) with $k=0$, one gets that there exists some $t_{n-1}$ such that for any $x_n \le K_n$,

$$\nu {\int }_0^{t_{n-1}}{\mathbb{E}}_{x_n}(X_{0,n}^S(u))du\ge \lambda t_{n-1}+2.\backslash nu\; \backslash int\_0^\{t\_\{n-1\}\}\; \backslash mathbb\{E\}\_\{x\_n\}\; (X\_\{0,n\}^S(u))\; du\; \backslash geq\; \backslash lambda\; t\_\{n-1\}\; +\; 2.$$

The two processes $(X_0^S(t))$ and $(X(t))$ can be built on the same probability space such that if they start from the same initial state, then the two processes $(X_{0,n}^S(t))$ and $(X_n(t))$ are identical as long as $X_{n-1}(t)$ stays positive. Since moreover the hitting time $\inf{t \ge 0 : X_{n-1}(t) = 0}$ goes to infinity as $x_{n-1}$ goes to infinity ---PAGE_BREAK---

for any $x_n \le K_n$, one gets that there exists $K_{n-1}$ such that if $x_{n-1} \ge K_{n-1}$ and $x_n < K_n$, then the relation

holds.

By induction, one gets in a similar way that there exist constants $t_n, \dots, t_0$ and $K_n, \dots, K_0$ such that for any $0 \le l \le n$, if $x_n \le K_n$, $x_{n-1} \le K_{n-1}$, $\dots$, $x_{n-l+1} \le K_{n-l+1}$ and $x_{n-l} > K_{n-l}$, then

$${\mathbb{E}}_x(\mathrm{\parallel }X(t_{n-l})\mathrm{\parallel })-\mathrm{\parallel }x\mathrm{\parallel }\le -1.\backslash mathbb\{E\}\_x\; (\backslash |X(t\_\{n-l\})\backslash |)\; -\; \backslash |x\backslash |\; \backslash le\; -1.$$

Theorem 8.13 of Robert [18] shows that (X(t)) is an ergodic Markov process. The proposition is proved. $\square$

Analysis of the Two-Chunk Network. In this subsection, one investigates the case when the monotonicity condition $\mu_1 > \cdots > \mu_{n-1} > \mu_n - \nu > 0$ fails. In general we conjecture the existence of bottlenecks which implies that the network can only accommodate a finite input rate. For instance, when $\mu_n - \nu < 0$, then it is easily seen that the network is unstable for $\lambda > \lambda^*$ where $\lambda^*$ is defined in Equation (32) below.

The first non-trivial case occurs for $n=2$, for which the monotonicity condition breaks in two situations, either when $\mu_2 - \nu > \mu_1$ or when $\mu_2 < \nu$. The latter case can be dealt in fact with the exact same arguments as before. See Proposition 4.4.

The actual difficulty is when $\mu_2 - \nu > \mu_1$: then the stationary behavior of $(X_2(t))$ is linked to the stationary behavior of the first saturated model $(X_1^S(t))$ defined through its Q-matrix (29). The difficulty in this case is that one needs to compare two processes which grow exponentially fast.

Proposition 4.3. Assume that $\mu_2 - \nu > \mu_1$, then the first saturated process $(X_1^S(t))$ with Q-matrix defined by Equation (29) is ergodic.

Corollary 4.1. If $\mu_2 - \nu > \mu_1$ and if

$${\lambda }_2^{\ast }\nu {\mathbb{E}}_{{\pi }^S}(X_{1,2}^S(0)),\backslash lambda\_2^*\; \backslash stackrel\{\backslash text\{def.\}\}\{=\}\; \backslash nu\; \backslash mathbb\{E\}\_\{\backslash pi^S\}\; (X\_\{1,2\}^S(0)),$$

where $\pi^S$ is the invariant distribution of the Markov process $(X_1^S(t))$, then the process $(X(t)) = (X_k(t), k = 0, 1, 2)$ describing the linear file-sharing network with parameters $\lambda, \mu_1, \mu_2$ and $\nu$ is ergodic for $\lambda < \lambda_2^*$ and transient for $\lambda > \lambda_2^*$.

Sketch of Proof. The proof of the transience when $\lambda > \lambda_2^*$ follows similarly as in Section 2: when $X_0(0)$ is large, the process $(X_1(t), X_2(t))$ can be coupled for some time with the second saturated system $(X_1^S(t))$. Since the output rate $\lambda_2^*$ of this system is smaller than the input rate $\lambda$, this implies that $(X_0(t))$ builds up, and it can indeed be shown that $X_0(t)/t$ converges almost surely to $\lambda - \lambda_2^*$.

The ergodicity when $\lambda < \lambda_2^*$ is slightly more complicated, but it involves the same arguments as the ones employed in the proof of Proposition 4.2. The details are omitted. $\square$ ---PAGE_BREAK---

Proof of Proposition 4.3. Denote $(X_1^S(t)) = (X_{1,1}^S(t), X_{1,2}^S(t))$, then as long as the first coordinate $X_{1,1}^S$ is positive, the process $(X_1^S(t))$ has the same distribution as $(W(t), Z(t))$ introduced in Section 3: $(Z(t))$ is a Bellman-Harris process with Malthusian parameter $\mu_2 - \nu$ and $(W(t))$ is a Yule process with parameter $\mu_1$ killed at times of births of $(Z(t))$.

By Proposition 3.5 and since $\mu_2 - \nu > \mu_1$, one has that $(X_{1,1}^S(t))$ returns infinitely often to 0. When $(X_{1,1}^S(t))$ is at 0 it jumps to 1 after an exponential time with parameter $\mu_1$, one denotes by $(E_{\mu_1,n})$ the corresponding i.i.d. sequence of successive residence times at 0. One defines the sequence $(S_n)$ by induction, $S_0 = 0$ and then

$$S_{n+1}=\inf \{t>S_n:X_{1,1}^S(t)=0\}+E_{{\mu }_1,n+1},n\ge 0.S\_\{n+1\}\; =\; \backslash inf\backslash \{t\; >\; S\_n\; :\; X\_\{1,1\}^S(t)\; =\; 0\backslash \}\; +\; E\_\{\backslash mu\_1,\; n+1\},\; \backslash quad\; n\; \backslash ge\; 0.$$

For $n \ge 1$, $X_{1,1}^S(S_n) = 1$ and for $n \ge 0$, define $M_n \stackrel{\text{def.}}{=} X_{1,2}^S(S_n)$. With the notations of Proposition 3.5, $(X_{1,1}^S(t))$ hits 0 after a duration of $H_{0,n}$ and at that time $(X_{1,2}^S(t))$ is at $Z(H_{0,n})$ with the initial condition $Z(0) = M_n$; while $X_{1,1}^S$ is still at 0, the dynamics of $X_{1,2}^S$ is simple, since it just empties. Finally, at time $S_{n+1} = S_n + H_{0,n} + E_{\mu_1,n+1}$, $(X_{1,1}^S(t))$ returns to 1 and at this instant the location of $(X_{1,2}^S(t))$ is given by

$$X_{1,2}^S(S_{n+1})=M_{n+1}=\sum _{i=1}^{Z(H_{0,n})}{1}_{\{E_{\nu ,i}>E_{{\mu }_1,n+1}\}},X\_\{1,2\}^\{S\}(S\_\{n+1\})\; =\; M\_\{n+1\}\; =\; \backslash sum\_\{i=1\}^\{Z(H\_\{0,n\})\}\; 1\_\{\backslash \{E\_\{\backslash nu,i\}>E\_\{\backslash mu\_1,n+1\}\backslash \}\},$$

where $(E_{\nu,i})$ are i.i.d. exponential random variables with parameter $\nu$, the ith variable being the residence time of the ith request in node 2. Consequently, $(M_n, n \ge 1)$ is a Markov chain whose transitions are defined by Relation (25) with $p = \nu / (\nu + \mu_1)$; note that $(M_n, n \ge 0)$ has the same dynamics only when $X_{1,1}^S(0) = 1$.

Define for any $K > 0$ the stopping time $T_K$

$$T_K=\inf \{t\ge 0:X_{1,2}^S(t)\le K,X_{1,1}^S(t)=1\}\mathrm{.}T\_K\; =\; \backslash inf\backslash \{t\; \backslash ge\; 0\; :\; X\_\{1,2\}^S(t)\; \backslash le\; K,\; X\_\{1,1\}^S(t)\; =\; 1\backslash \}.$$

The ergodicity of $(X_1^S(t))$ will follow from the finiteness of $\mathbb{E}_{(x_1,x_2)}(T_K)$ for some $K$ large enough and for arbitrary $x = (x_1, x_2) \in \mathbb{N}^2$. The strong Markov property of $(X_1^S(t))$ applied at time $S_1$ gives

$${\mathbb{E}}_{(x_1,x_2)}(T_K)\le 2{\mathbb{E}}_{(x_1,x_2)}(S_1)+{\mathbb{E}}_{(x_1,x_2)}[{\mathbb{E}}_{(1,X_{1,2}^S(S_1))}(T_K)],\backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(T\_K)\; \backslash le\; 2\backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(S\_1)\; +\; \backslash mathbb\{E\}\_\{(x\_1,x\_2)\}\backslash left[\backslash mathbb\{E\}\_\{(1,X\_\{1,2\}^S(S\_1))\}(T\_K)\backslash right],$$

and so one only needs to study $T_K$ conditioned on ${X_{1,1}^S(0) = 1}$ since $\mathbb{E}_{(x_1,x_2)}(S_1)$ is finite in view of Proposition 3.5.

Then, on this event and with $N_K$ defined in Proposition 3.6, the identity

$$(31)T_K=\sum _{i=0}^{N_K}(H_{0,i}+E_{{\mu }_1,i})(31)\; \backslash qquad\; T\_K\; =\; \backslash sum\_\{i=0\}^\{N\_K\}\; (H\_\{0,i\}\; +\; E\_\{\backslash mu\_1,i\})$$

holds. For $i \ge 0$, the Markov property of $(M_n, n \ge 0)$ gives

$${\mathbb{E}}_{(x_1,x_2)}(H_{0,i}{1}_{\{i\le N_K\}})={\mathbb{E}}_{(x_1,x_2)}({\mathbb{E}}_{(1,M_i)}(H_0)1_{\{i\le N_K\}})\backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(H\_\{0,i\}\; 1\_\{\backslash \{i\; \backslash le\; N\_K\backslash \}\})\; =\; \backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(\backslash mathbb\{E\}\_\{(1,M\_i)\}(H\_0)\; 1\_\{\backslash \{i\; \backslash le\; N\_K\backslash \}\})$$

With the same argument as in the proof of Proposition 3.6, one has

---PAGE_BREAK---

with Equations (31) and (26) of Proposition (3.6), one gets that for some $\gamma > 0$ and some $K > 0$,

$${\mathbb{E}}_{(x_1,x_2)}(T_K)\le 2{\mathbb{E}}_{(x_1,x_2)}(S_1)+C(1+{\mathbb{E}}_{(x_1,x_2)}[\log (1+X_{1,2}^S(S_1))])\backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(T\_K)\; \backslash leq\; 2\backslash mathbb\{E\}\_\{(x\_1,x\_2)\}(S\_1)\; +\; C\; \backslash left(1\; +\; \backslash mathbb\{E\}\_\{(x\_1,x\_2)\}\backslash left[\backslash log\backslash left(1\; +\; X\_\{1,2\}^S(S\_1)\backslash right)\backslash right]\backslash right)$$

with the constant $C = (\mathbb{E}_{(1,0)}(H_0) + 1/\mu_2)/\gamma$. This last term is finite for any $(x_1, x_2)$ in view of Proposition 3.5, which proves the proposition. $\square$

Proposition 4.4. If $\nu > \mu_2$ and

$$(32){\lambda }^{\ast }\frac{{\mu }_2}{(1-{\mu }_2\mathrm{/}\nu )(1-\log (1-{\mu }_2\mathrm{/}\nu ))},(32)\; \backslash qquad\; \backslash lambda^*\; \backslash stackrel\{\backslash text\{def.\}\}\{=\}\; \backslash frac\{\backslash mu\_2\}\{(1\; -\; \backslash mu\_2/\backslash nu)(1\; -\; \backslash log(1\; -\; \backslash mu\_2/\backslash nu))\},$$

then the Markov process $(X(t)) = (X_k(t), k = 0, 1, 2)$ is transient if $\lambda > \lambda^*$ and ergodic if $\lambda < \lambda^*$.

Sketch of Proof. The result for transience comes directly from the fact that the last coordinate is stochastically dominated by the birth-and-death process $(Y_1^1(t))$ of Section 2.

As before, the arguments employed in the proof of Proposition 4.2 to prove ergodicity can also be used, for this reason they are only sketched. One has in fact to consider the following situations.

— If there are many customers in the last queue, then the total number of customers instantaneously decreases.

— If there are many customers in the second queue, then the last queue has time to get close to stationarity, the input rate is $\lambda$ and the output rate is $\lambda^*$.

— Finally, if there are many customers in the first queue, then it is easily seen that the second queue builds up, since it grows like a Yule process killed at times $(\sigma_n)$ where the sequence $(\sigma_n)$ essentially grows linearly since the last queue is stable. Hence the second queue reaches high values and the last queue offers an output rate of $\lambda^*$.

Hence when $\lambda < \lambda^*$, the Markov process $(X(t))$ is ergodic. $\square$

APPENDIX A. PROOF OF PROPOSITION 3.3

In this appendix the notations of Section 3 are used. Since the random variable $(B_\sigma(t) | Z(0) = 0)$ is stochastically smaller than $(B_\sigma(t) | Z(0) = z)$ for any $z \in \mathbb{N}$, it is enough to show that for $\eta < \eta^*(\nu/\mu_Z)$

where $\alpha = \mu_Z - \nu > 0$.

Note that the process $(B_\sigma(t+\sigma_1), t \ge 0)$ under $\mathbb{P}_0$ has the same distribution as $(B_\sigma(t)+1, t \ge 0)$ under $\mathbb{P}_1$, and by independence of $\sigma_1$, an exponentially random variable with parameter $\mu_Z$, and $(B_\sigma(t+\sigma_1), t \ge 0)$, one gets

$${\mathbb{E}}_0[\underset{t\ge {\sigma }_1}{\sup }(e^{\eta \alpha t}{B}_{\sigma }(t{)}^{-\eta })]={\mathbb{E}}_0(e^{\eta \alpha {\sigma }_1}){\mathbb{E}}_1[\underset{t\ge 0}{\sup }(e^{\eta \alpha t}(B_{\sigma }(t)+1{)}^{-\eta })]\mathrm{.}\backslash mathbb\{E\}\_0\; \backslash left[\; \backslash sup\_\{t\; \backslash ge\; \backslash sigma\_1\}\; (e^\{\backslash eta\; \backslash alpha\; t\}\; B\_\backslash sigma(t)^\{-\backslash eta\})\; \backslash right]\; =\; \backslash mathbb\{E\}\_0\; (e^\{\backslash eta\; \backslash alpha\; \backslash sigma\_1\})\; \backslash mathbb\{E\}\_1\; \backslash left[\; \backslash sup\_\{t\; \backslash ge\; 0\}\; (e^\{\backslash eta\; \backslash alpha\; t\}\; (B\_\backslash sigma(t)\; +\; 1)^\{-\backslash eta\})\; \backslash right].$$

Since $\alpha < \mu_Z$ and $\eta^*(\nu/\mu_Z) < 1$, then $\mathbb{E}_0(\exp(\eta\alpha\sigma_1))$ is finite, and all one needs to prove is that the second term is finite as well.

Define $\tau$ as the last time $Z(t) = 0$:

$$\tau =\sup \{t\ge 0:Z(t)=0\},\backslash tau\; =\; \backslash sup\backslash \{t\; \backslash ge\; 0\; :\; Z(t)\; =\; 0\backslash \},$$ ---PAGE_BREAK---

with the convention that $\tau = +\infty$ if $(Z(t))$ never returns to 0. Recall that, because of the assumption $\mu_Z > \nu$, with probability 1, the process $(Z(t))$ returns to 0 a finite number of times.

Conditioned on the event ${\tau = +\infty}$, the process $(Z(t))$ is a $(p, \lambda)$-branching process conditioned on survival, with $\lambda = \mu_Z + \nu$ and $p = \mu_Z/\lambda$. Such a branching process conditioned on survival can be decomposed as $Z = Z_{(1)} + Y$, where $(Y(t))$ is a Yule process $(Y(t))$ with parameter $\alpha$. See Athreya and Ney [3]. Consequently, for any $0 < \eta < 1$,

$${\mathbb{E}}_1[\underset{t\ge 0}{\sup }(e^{\eta \alpha t}(B_{\sigma }(t)+1{)}^{-\eta })\mathrm{\mid }\tau =+\mathrm{\infty }]\le {\mathbb{E}}_1\left[\underset{t\ge 0}{\sup }(e^{\eta \alpha t}Y(t{)}^{-\eta })\right]\mathrm{.}\backslash mathbb\{E\}\_1\; \backslash left[\; \backslash sup\_\{t\; \backslash ge\; 0\}\; \backslash left(\; e^\{\backslash eta\; \backslash alpha\; t\}\; (B\_\backslash sigma(t)\; +\; 1)^\{-\backslash eta\}\; \backslash right)\; |\; \backslash tau\; =\; +\backslash infty\; \backslash right]\; \backslash le\; \backslash mathbb\{E\}\_1\; \backslash left[\; \backslash sup\_\{t\; \backslash ge\; 0\}\; \backslash left(\; e^\{\backslash eta\; \backslash alpha\; t\}\; Y(t)^\{-\backslash eta\}\; \backslash right)\; \backslash right].$$

Since the nth split time $t_n$ of $(Y(t))$ is distributed like the maximum of n i.i.d. exponential random variables, $Y(t)$ for $t \ge 0$ is geometrically distributed with parameter $1 - e^{-\alpha t}$, hence,

For $0 < u < 1$, the relation

holds, hence

The process $(e^{-\alpha t}Y(t))$ being a martingale, by convexity the process $(e^{\eta\alpha t}Y(t)^{-\eta})$ is a non-negative sub-martingale. For any $\eta \in (0, 1)$ Doob's $L_p$ inequality gives the existence of a finite $q(\eta) > 0$ such that

The following result has therefore been proved.

Lemma A.1. For any $0 < \eta < 1$,

---PAGE_BREAK---

On the event ${\tau < +\infty}$, $(Z(t))$ hits a geometric number of times 0 and then couples with a $(p, \lambda)$-branching process conditioned on survival. On this event,

where $B'_\sigma(t)$ for $t \ge \tau$ is the number of births in $(\tau, t]$ of a $(p, \lambda)$-branching process conditioned on survival and independent of the variable $\tau$, consequently

In view of Lemma A.1, the proof of Proposition 3.3 will be finished if one can prove that

which actually comes from the following decomposition: under $\mathbb{P}_1(\cdot | \tau < +\infty)$, the random variable $\tau$ can be written as

$$\tau =\sum _{k=1}^{1+G}(T_k+E_{{\mu }_Z,k})\backslash tau\; =\; \backslash sum\_\{k=1\}^\{1+G\}\; (T\_k\; +\; E\_\{\backslash mu\_Z,k\})$$

where G is a geometric random variable with parameter q = ν/μ_Z, (T_k) is an i.i.d. sequence with the same distribution as the extinction time of a (p, λ)-branching process starting with one particle and conditioned on extinction and (E_μZ,k) are i.i.d. exponential random variables with parameter μ_Z.

Since q is the probability of extinction of a (p, λ)-branching process started with one particle, G + 1 represents the number of times (Z(t)) hits 0 before going to infinity. This representation entails

A (p, λ)-branching process conditioned on extinction is actually a (1 − p, λ)-branching process. See again Athreya and Ney [3]. Thus T₁ satisfies the following recursive distributional equation:

$$T_1^{\text{dist.}}:=E_{\lambda }+1_{\{\xi =2\}}(T_1\vee T_2),T\_1^\{\backslash text\{dist.\}\}\; :=\; E\_\{\backslash lambda\}\; +\; 1\_\{\backslash \{\backslash xi=2\backslash \}\}(T\_1\; \backslash lor\; T\_2),$$

where $\mathbb{P}(\xi = 2) = 1-p$ and $E_\lambda$ is an exponential random variable with parameter $\lambda$. This equation yields

$$\mathrm{\P }(T_1\ge t)\le e^{-\lambda t}+2\lambda (1-p){\int }_0^t\mathrm{\P }(T_1\ge t-u)e^{-\lambda u}du,\backslash P(T\_1\; \backslash ge\; t)\; \backslash le\; e^\{-\backslash lambda\; t\}\; +\; 2\backslash lambda(1-p)\; \backslash int\_0^t\; \backslash P(T\_1\; \backslash ge\; t-u)\; e^\{-\backslash lambda\; u\}\; du,$$

and Gronwall's Lemma applied to the function $t \mapsto \exp(\lambda t)\mathbb{P}(T_1 \ge t)$ gives that

$$\mathrm{\P }(T_1\ge t)\le e^{(\lambda -2{\lambda }_p)t}=e^{(\nu -{\mu }_Z)t}\backslash P(T\_1\; \backslash ge\; t)\; \backslash le\; e^\{(\backslash lambda\; -\; 2\backslash lambda\_p)t\}\; =\; e^\{(\backslash nu\; -\; \backslash mu\_Z)t\}$$ ---PAGE_BREAK---

hence for any $0 < \eta < 1$,

$${\mathbb{E}}_1(e^{\eta \alpha T_1})\le \frac{1}{1-\eta }\mathrm{.}\backslash mathbb\{E\}\_1(e^\{\backslash eta\backslash alpha\; T\_1\})\; \backslash le\; \backslash frac\{1\}\{1-\backslash eta\}.$$

Since G is a geometric random variable with parameter q, $\mathbb{E}(\gamma(\eta)^G)$ is finite if and only if $\gamma(\eta) < q$. Since finally

$$\gamma (\eta )=\frac{{\mu }_Z}{{\mu }_Z-\eta \alpha }\mathbb{E}(e^{\eta \alpha T_1})\le \frac{{\mu }_Z}{(1-\eta )({\mu }_Z-\eta \alpha )}\backslash gamma(\backslash eta)\; =\; \backslash frac\{\backslash mu\_Z\}\{\backslash mu\_Z\; -\; \backslash eta\backslash alpha\}\; \backslash mathbb\{E\}\; (e^\{\backslash eta\backslash alpha\; T\_1\})\; \backslash le\; \backslash frac\{\backslash mu\_Z\}\{(1-\backslash eta)(\backslash mu\_Z\; -\; \backslash eta\backslash alpha)\}$$

one can easily check that $\gamma(\eta) < q$ for $\eta < \eta^*(\nu/\mu_Z)$ as defined by Equation (19), which concludes the proof of Proposition 3.3.

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(L. Leskelä) HELSINKI UNIVERSITY OF TECHNOLOGY, DEPARTMENT OF MATHEMATICS AND SYSTEMS ANALYSIS, PO BOX 1100, 02015 TKK, FINLAND

E-mail address: lasse.leskela@iki.fi

URL: http://www.iki.fi/lsl

(Ph. Robert, F. Simatos) INRIA PARIS — ROCQUENCOURT, DOMAINE DE VOLUCEAU, BP 105, 78153 LE CHESNAY, FRANCE.

E-mail address: Philippe.Robert@inria.fr

E-mail address: Florian.Simatos@inria.fr

URL: http://www-rocq.inria.fr/~robert