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as [MATH] -functions. It should be evident the enormous number of different PDFs that can be it depends from five parameters and three independent functions (essentially at least hundred free parameters). For the COG case, the equivalent of eq. depends from seven parameters and five independent functions, at large inci... |
2.4 The [MATH] Position Algorithms Two different position algorithms will be used in the following: the COG algorithm and the [MATH] algorithm. The results of the least squares are better with the [MATH] positioning algorithm than with the COG The [MATH] -algorithm is built to correct the COG systematic error. Assuming... |
[EQUATION] where [MATH] is the PDF of COG given by [MATH] as PDF of impact points. For uniform distribution of the impact point [MATH] |
and normalized to one on the strip length [MATH] [MATH] here), the differential equation can be integrated: [EQUATION] the positive sign of [MATH] is discussed in refs. |
With the exact initial constant [MATH] , the function [MATH] is a better estimation of the impact point than the simpler COG . In the [MATH] |
positioning algorithm, the function [MATH] is given by the experimental COG PDF inserted in eq. The extraction of the initial constant [MATH] from the data is discussed in refs. |
and tested in a dedicated test beam . The definition of eq. generalizes the [MATH] algorithm of ref. for any COG . Our [MATH] algorithm coincides with that of ref. |
with [MATH] This condition is true for an exact axial symmetry of the signal distribution and of the strip response function. It is evident that real detectors drastically deviate from this symmetry, and essential corrections must be implemented for [MATH] |
2.5 Track definition Given the novelty of this approach, our track definition must be very simple. The tracks are circles with a large radius to simulate the high momentum MIPs where the multiple scattering is negligible. The relation of the track parameters to the momenta is [MATH] (ref. |
) here [MATH] is in [MATH] [MATH] in Tesla and [MATH] , the track radius, in meters. The tracker model is formed by six parallel equidistant ( [MATH] detector layers as in the PAMELA tracker. The constant magnetic field is [MATH] . The simulated tracks are on a plane perpendicular to the the detector layers and to the ... |
[MATH] , the [MATH] axis is parallel to the layer planes and the [MATH] axis is perpendicular. The tracks are circles with center in [MATH] and [MATH] and the magnetic field is parallel to the analyzing strips. To simplify the geometry, the overall small rotation of the Lorentz angle ( [MATH] ) is neglected now, but it... |
) with a uniform random distribution on a strip, they are collected in groups of six to produce the tracks. The exact impact position [MATH] of each hit is subtracted from its reconstructed [MATH] position (as defined in refs. |
) and it is added the value of the fiducial track for the corresponding detector layer. In this way each group of six hits defines a track with our geometry and the error distribution of [MATH] -algorithm. Identically for the COG positioning algorithm. This hit collection simulates a set of tracks populating a large po... |
because it gives parameter distributions better than those obtained with the simplest COG positions, but even the results for the COG will be reported in the following. |
In the [MATH] -plane, the circular tracks are approximated, as usual, with parabolas linear in the track parameters: [EQUATION] The first line of eq. is the model track, the second line is the fit result. The circular track is the osculating circle of the parabola |
[MATH] , at our high momenta and tracker size the differences are negligible. The function [MATH] has [MATH] , and [MATH] is proportional to the track momentum. Due to the noise, the reconstructed track has equation [MATH] and the parameters [MATH] , given by the fit, are distributed around the model values. The non ga... |
[MATH] , the function [MATH] defined as the negative logarithm of the likelihood with the PDFs of eq. [EQUATION] The parameters [MATH] [MATH] (introduced in eq. ) are respectively: the COG [MATH] position, the sum of signal in the three strips for the hit [MATH] of the track [MATH] . The term |
[MATH] is the position of the detector plane [MATH] of the [MATH] th track. The [MATH] -dependence in the [MATH] is modified to [MATH] to place the impact points on the track. In real data, |
[MATH] is absent (and unknown) but the data are supposed to be on a track. We can easily use a non linear form for the function [MATH] , but in this case is of scarce meaning. In more complex cases, non linearities of various origin can be easily implemented, for example analytic expressions of the tracks that exactly ... |
We will reserve the definition of Maximum Likelihood Evaluation (MLE) to the results of eq. 10 The routine for the MLE is initialized as in ref. |
, the first track parameters are given by a weighted least squares with weights given by an effective variance ( [MATH] for each hit [MATH] and [MATH] as hit position. The |
[MATH] is obtained from eq. , but, for its form, the variance is an ill defined parameter even in [MATH] . Cuts in the integration limits suppress the tails and give finite results. |
[EQUATION] The parameter [MATH] is selected to reproduce the PDFs [MATH] with gaussian functions of variance [MATH] in the case of excellent hits. For a set of hits (excellent hits), eq. |
has the form of a narrow high peak and a gaussian, centered in [MATH] and variance [MATH] , can overlap well the [MATH] around the maximum. We selected few of them for the tuning of [MATH] . These gaussian approximations look good in linear plots, the logarithmic plots show marked differences even in these happy cases:... |
The [MATH] given by the weighted least squares are almost always near those given by the minimization of eq. 10 , thus accelerating the minimum search to the MATLAB |
[MATH] routine. The closeness of these approximations to the MLE supports the non criticality of the extraction of the functions [MATH] The approximate gaussian distributions are often very different from the hit PDFs but this partial information suffices to produce near optimal results. When the tails of the PDFs are ... |
can be reduced building a look-up table of [MATH] and calculating [MATH] with an interpolation. This use of eq. 11 to get weights to insert in the least squares is directly consistent with ref |
Having to plot the results of four type of reconstructions we will use the following color convention for the lines: red lines refer to our MLE (eq. 10 ), |
black lines are the weighted least squares with weight [MATH] and [MATH] position, blue lines for the least squares with the [MATH] position algorithm |
magenta lines for the least squares with the COG position algorithm. Low noise, high resolution, floating strip side The floating strip side is the best of the two sides of this type of strip detector. It is just this side that measures the track bending in the PAMELA magnetic spectrometer. In the test beam |
, the noise PDF of the "average" strips without signals is well reproduced by a gaussian with a standard deviation of 4 ADC counts. The PDF for the sum of three strip signals has its maximum at 142 ADC counts with the most probable signal-to-noise ratio ( [MATH] ) of [MATH] |
for a three strip cluster ( [MATH] ). The functions [MATH] are those of ref. for this strip type. In the simulations we will use a high momentum of [MATH] For this momentum and identical geometry, we have a report with some histograms of a CERN test beam of the PAMELA tracker before its installation in the satellite. T... |
, the curvature histogram turns out very similar to our, giving an excellent check of our simulations. In this case with the orthogonal incidence of the beam, the systematic error of [MATH] algorithm of ref. |
is constant (and small) and has no effect on the curvature reconstruction. In the left side of figure , the histogram values divided by the number of entries and the step size (frequency polygons called distributions in the following) of the differences of the fitted positions respect to the exact ones true residuals i... |
The PDFs of the residuals (as defined above) are not reported. Those for the COG and [MATH] are almost identical to the true residual PDFs of figure The residuals for the MLE and of the schematic model are very different from their true residuals of figure . Narrow and high peaks centered in zero are present in those t... |
distinguish good hits from bad hits and the fit optimization forces the track to pass near to good ones, giving to them an high frequency of small residuals. Often the residuals are used as a measure of the detector resolution, it is easy to show the inconsistency of this assumption almost in any case. For example in t... |
As illustrated in figure , the MLEs give the best results for the momentum reconstruction, and the weighted least squares, are very near to them. The fits of the standard least squares with [MATH] |
or COG positioning algorithms show a drastic decrease in resolution. The use of the simple COG algorithm is the worst one. Often the distributions of the left side of figure |
are reported as resolution of the momentum reconstruction, the k-value of ref. . For the [MATH] and COG least squares, the plots of the momentum distributions have appreciable shifts of the maxima (most probable value) respect to the fiducial value of 350 [MATH] , the shifts are negligible for the other two fits. These... |
3.1 Other track parameters The complete track reconstruction must consider even the other two parameters of a track, the [MATH] and [MATH] . Their fits give very similar distributions to those plotted in ref. |
The maxima of the distributions are now a little lower, in particular for the [MATH] parameter. This is not unexpected, in ref. we had 3 degree of freedom for two parameters, here we have 3 degree of freedom for three parameters, an effective reduction of the redundancy that has a slight effect on the results. |
High noise, low resolution, normal strip side The other side of the double sided silicon microstrip detector has very different properties respect to the floating strip side (the junction side). We will not recall the special treatments required to transform a ohmic side in strip detector and all the other particular s... |
[MATH] ). The absence of the floating strips gives to the histograms of the COG the normal aspect with a high central density and a drop around [MATH] No additional rises around [MATH] are present, they are typical of the charge spreads given by the floating strips. This absence reduces the efficiency of the positionin... |
[MATH] of ref. are similar to those of an interval function with a weak rounding to the borders, this rounding is mainly due to the convolution of the strip response with the charge spread produced by the drift in the collecting field. If a residual capacitive coupling is present, it is very small. In any case, it is j... |
Figure illustrates the distributions of figure for this type of detector, that, as discussed above, are much lower than those of figure The true residuals of the four methods of track reconstruction in the left side, and the true residuals of the least squares compared with the position algorithms [MATH] and COG The hi... |
Discussion An easy comparison among the different fits is somewhat difficult. Often an effective variance (or a standard deviation) of the PDF is extracted interpolating a gaussian function on non gaussian distribution to avoid the effects of the tails. But even the variance itself is not free from arbitrariness as oft... |
Similarly the full width at half maximum, as suggested in ref. , does not characterize very well non-gaussian distributions. In any case the differences of the chosen parameters are not of easy interpretation from the physical point of view, and it is the physical point of view our essential interest. Here we will use ... |
and COG least squares reconstructions to overlap our best distributions (the red lines). This increase is the relative gain in resolution. |
5.1 Increasing the magnetic field With fixed momentum, the magnetic field is increased in the fit for the [MATH] least squares. The upper sectors of figure illustrate these results for the low noise side and the overlaps with our MLE. The red lines are identical to those of figure , the blue line are the |
[MATH] least squares for tracks with a magnetic field 1.5 times higher. The plots for the COG least squares are not reported to render easily legible the figures, in this case the magnetic field must be increased of factor 1.8 to overlap our red lines. The lowest sectors of figure report the noisy normal strip side, th... |
5.2 Increasing the signal-to-noise ratio Let us discuss now the effects of the increase of the signal-to-noise ratio. These modifications reproduce in part the results of the magnetic increase, but it is different in other parts. In fact, higher values of the magnetic field do not modify the form of the distributions o... |
and [MATH] have the magnetic field intensity as scaling factor and move the distributions to the forms of figure even for the COG case. But, as for the [MATH] parameters (with dimension [MATH] ), the distributions of the true residuals of figure and figure are not influenced by the increase of the magnetic field, a par... |
Keeping the magnetic field to [MATH] , the increase of the signal-to-noise ratio modifies the distributions of the [MATH] parameters and those of the true residuals and can bring them to overlap our red distributions. In our simulations, the increase of the signal-to-noise ratio is accomplished scaling the amplitude of... |
distributions, they remain essentially unchanged in any plot for a (any) reduction of the strip noise. The COG error distribution of figure |
(the green line) shows a slight modification becoming less rounded for the hard presence of the systematic error, unaffected by the random noise. |
For the higher noise side of the detector, a reduction to less of one half of the noise (from [MATH] to [MATH] -counts) is required to reproduce the low parts of figure With this doubling (2.2) of the signal-to-noise ratio, the blue line (the [MATH] best fit) overlaps the red line in figure increasing the most probable... |
. This fact renders of weak relevance the efforts to improve the signal-to-noise ratio in the detectors if the positioning algorithm remains the uncorrected COG. Unpleasant effects, due to the neglect of the systematic errors, were forecasted long time ago in the Gauss papers |
On the other side, this positioning method is stable respect to a decrease of the signal-to-noise ratio due to ageing or radiation damages, as far as the COG systematic error continues to dominate. |
A further point requires an explanation, i.e. the huge difference between the curvature PDF for the [MATH] least squares in the low part figure and the blue line of figure Now the two detector types have very similar signal-to-noise ratio (3.6 ADC counts in the first and 4 ADC counts in the second), but, to reach the o... |
of differences between the sizes of their strips. The main difference must be due to the beneficial charge sharing of the floating strip and the nearing of its detector architecture to the ideal detector defined in ref. |
Thus, detectors without this charge spreading mechanism are evidently disadvantaged for momentum measurements. 5.3 Different numbers of detecting layers |
Another important result of this new fitting method is the rapid grow of the momentum resolution with the number of detecting layers. The least squares demonstrations assume the identity of the variance of the data to be fitted. This assumption eliminates any statistical difference among the data and the resolution gro... |
The PDFs for the six layers set up are those of figure and of figure A very rapid increase, very near to a linear grow (as N), is observed for the maxima of the PDFs of our two methods. (For normalized PDFs the maxima are essentially proportional to the inverse of the full-width-at-half-maximum that is often defined as... |
[MATH] positioning algorithm. This coincidence is due to the absence of any redundancy (zero degree of freedom) and to the best quality of the [MATH] |
position algorithm respect to COG . The resolution of the MLE and of the schematic model in the four layers set up results better than the resolution of the six layers set up for the [MATH] -least squares. Thus, if this resolution suffices, two detector layers could be eliminated with important saving of weight and ene... |
5.4 Momentum resolution for a track selection Up to now, we explored the momentum resolution for a generic track sample without any special attention to the hit quality. All the generated tracks are reconstructed identically. But our effective variance allow us to select subsets of tracks containing good or excellent h... |
The hit selections are defined in figure , the two horizontal lines isolate the (so called) good hits. The hits below the lowest line are defined excellent hits. It is evident that these are somewhat arbitrary definitions chosen to obtain easy track selections and, with a rough tuning, to have large increases of moment... |
The two last figures illustrate the momentum resolutions of a track selections. For the floating strip side, we select tracks with two excellent and three good hits. The [MATH] of all the tracks has this hit combination, and a very well defined momentum and curvature, drastically better than that without hit selection ... |
Identically we can proceed with the high noise side. Now we select tracks with two excellent hits and four other hits of random type, [MATH] of tracks has this hit combination. Even in this case the momentum resolution has a great improvement respect to the case without the hit selection. Figure 10 shows this compariso... |
5.5 A gift of heteroscedasticity The large variations of the effective variances observed in figure have effects even in the standard least squares. The heteroscedasticity couples the track variance to the parameter distributions. Tracks, richer of good or excellent hits (as we defined above), have an higher probabilit... |
[MATH] -PDF) has the distribution of the reconstructed momenta overlapping our best PDFs of figure and of figure . The price is an efficiency below [MATH] but these improvements can be obtained without any modifications of the fitting methods. For the COG hit positions, the track residual variance has PDF near to a [MA... |
Conclusions The hit heteroscedasticity of a tracker system is carefully explored for the momentum reconstruction of minimum ionizing particles in a simulated uniform magnetic field of [MATH] Other important effects, as [MATH] -rays, multiple scattering (negligible at high momenta), energy loss, etc., are explicitly exc... |
To reach the overlap of the two standard fits with the best momentum distributions, the magnetic field must grow by a factor 1.5 for the [MATH] |
positioning, and 1.8 for the two strip Center of Gravity in the low noise side and 1.8 and 2 for the higher noise side. The increase of signal-to-noise ratio is effective only for the [MATH] position algorithm, the overlaps are obtained with factors 1.6 and 2.2 for the two detector sides. Any increase of the signal-to-... |
Assuming an optimistic view, this increase in resolution can be spent in different ways either for better results on running experiments or in reducing the complexity of future experiments if the baseline fits (almost always based on the COG positioning) are estimated sufficient. Further details about this method will ... |
# Source: arxiv 1806.08016 # Title: Equilibrium and Learning in Queues with Advance Reservations # Sections: all # Downloaded: 2026-03-02T09:23:15.541852+00:00 |
Equilibrium and Learning in Queues with Advance Reservations Abstract Consider a multi-class preemptive-resume [MATH] queueing system that supports advance reservations (AR). In this system, strategic customers must decide whether to reserve a server in advance (thereby gaining higher priority) or avoid AR. Reserving a... |
Introduction Many services, such as health care, cloud computing and banking, combine both a first-come-first-served policy and advance reservations (AR). Advance reservations benefit a service provider since knowledge about future demand can improve resource management and quality-of-service (e.g., Charbonneau and Vok... |
Since the decision of a customer, about reserving a server in advance or not, affects the waiting time of other customers, game theory is the solution of choice for studying such systems. Although there exists a rich literature on advance reservations, works that study advance reservation systems as a game are rare. Th... |
We assume that the time axis is divided into two time-periods: a reservation period and a service period . This restriction simplifies the analysis and is common in the literature of advance reservations (e.g., Virtamo ( 1992 Yessad et al. ( 2007 and Syed et al. ( 2008 ). It can also be found in real life applications.... |
During the reservation period, each customer realizes that he/she will need service at a specific future time point. Upon such a realization, the customer decides whether or not to make a reservation. Customers are assumed to be strategic and rational. Thus, a customer will make a reservation only if it reduces his/her... |
We start the analysis by finding the equilibrium structure of the game. We show that there are two possible types of equilibria. In the first type, none of the customers makes AR, while in the second type customers that realize early enough that they will need future service make AR. We refer to those two types of equi... |
Next, we assume that the AR cost is a fee charged by the service provider. We analyze the game from the prospective of a provider aiming to maximize its revenue from AR fees. We show that if the utilization is greater than [MATH] , then the revenue maximizing fee leads to multiple equilibria. Thus, charging that fee ma... |
Finally, we study a dynamic version of the game. We use best response dynamics (as in Fudenberg ( 1998 ) and distinguish between strategy-learning and action-learning . In strategy-learning , customers obtain information about strategies adopted at previous steps, while in action-learning , customers estimate the previ... |
The rest of the paper is structured as follows. In Section , we review related work. In Section , we formally define the game. In Section , we find the equilibrium structure of the game. In Section , we derive the revenue maximizing fee and resulting equilibria. In Section , we define and analyze dynamic versions of th... |
Related Work Strategic behavior in queues (also known as queueing games) was pioneered by Naor ( 1969 and has been studied extensively since. In that seminal paper, the author studies an [MATH] queue where customers decide whether to join or balk after observing the queue length. Hassin and Haviv ( 2003 and Hassin ( 20... |
Advance reservations have been researched from various other perspectives in the literature, including scheduling and routing algorithms for communication networks, methods for revenue maximization, and performance analysis of queueing systems. The work in Wang et al. ( 2013 describes a distributed architecture for adv... |
Simhon and Starobinski ( 2014 introduces AR games. In that paper, the authors consider a loss system (i.e., customers that finds all servers busy leave). The authors show that the game may have multiple equilibria, where in one equilibrium the number of reservations is a random variable, while in the other equilibrium,... |
The concept of learning an equilibrium is rooted in Cournot’s duopoly model Cournot ( 1897 and has been extensively researched since. Traditionally, learning models are used for fixed-player games (i.e., the same players participate at each iteration), see Lakshmivarahan ( 1981 ), Fudenberg ( 1998 and Milgrom and Rober... |
Different learning models differ by their learning rules. A learning rule defines what kind of information players gain and how they use it. In this paper, we focus on best response dynamics . According to this rule, which is rooted in Cournot’s work, players observe the most recent actions adopted by other players and... |
Other relevant work includes Niu et al. ( 2012 , which presents a theoretical model for pricing cloud bandwidth reservations, in order to maximize social welfare. The reservation fee of each customer is a function of his/her guaranteed portion instead of the actual amount of resources reserved, as considered in our mod... |
Game Description We consider a preemptive-resume [MATH] queue that supports advance reservations. In our model, there is a reservation period which covers [MATH] . Each customer [MATH] is associated with a request time [MATH] and a desired service starting time (shortly noted as arrival time) [MATH] . That is, if [MATH... |
The request times are derived from a general continuous distribution with cumulative distribution function [MATH] . The arrivals follow a Poisson process with rate [MATH] . The service time is [MATH] and we assume that [MATH] |
Each customer, at his/her request time, decides whether to make a reservation or not. We denote those two actions by [MATH] and [MATH] , respectively. If a customer makes a reservation but his/her desired service time is already reserved, the nearest future available time will be reserved for that customer. A customer ... |
The total cost of each customer consists of the reservation cost [MATH] (if making AR) and the cost of waiting which is a linear function of the waiting time. Without loss of generality, we assume that the cost of waiting is equal to the waiting time. Note that the waiting time when making AR is smaller than when not m... |
In a preemptive-resume queue, if a job is interrupted, then it later resumes and is not restarted. Due to this property, if the server is idle and a customer is waiting for service, the customer will be served even if service cannot be completed due to an existing reservation (in this case the service will be preempted... |
Note that customers do not know a-priori what will be their waiting time if making or if not making AR. The decision is based on statistical information only, namely the values of [MATH] [MATH] and [MATH] . However, once a customer decides to make a reservation, the system can provide him/her with the start and end tim... |
Equilibrium Analysis We can analyze this system as a priority queue where a priority between [MATH] (lowest priority) and [MATH] (highest priority) is assigned to each customer. A customer with request time [MATH] has priority [MATH] if not making AR and priority [MATH] if making AR. Customers that share the same prior... |
Since customers are statistically identical, we consider only symmetrical behavior. Thus, a decision of a tagged customer is a mapping of his/her potential priority [MATH] to the probability of making AR. We denote this strategy function by [MATH] . Consider a tagged customer with potential priority [MATH] . We define ... |
[EQUATION] Next, we define a threshold strategy and show that this is the only strategy that can lead to equilibria. Definition 1 |
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