arXiv subjects

Pascal Moyal

Publications and source records attributed to Pascal Moyal.

35 records · Page 2Linked to original sources

arXiv2017

On the Instability of Matching Queues

A matching queue is described via a graph $G$ together with a matching policy. Specifically, to each node in the graph there is a corresponding arrival process of items which can either be queued, or matched with queued items in neighboring nodes. The matching policy specifies how items are matched whenever more than one matching is possible. Motivated by the increasing theoretical interest in such matching models, we investigate the question of (in)stability of matching queues which satisfy a natural necessary condition for stability, which can be thought of as an analogue of the usual traffic condition for traditional queueing networks (namely, $ρ_i < 1$ in each service station $i$). We employ fluid-stability arguments to show that matching queues can in general be unstable, even though the necessary stability condition is satisfied.

math.PR↗

arXiv2016

Stability of the stochastic matching model

We introduce and study a new model that we call the {\em matching model}. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be {\em matched}. There is a finite set of classes $\maV$ for the items, and the allowed matchings depend on the classes, according to a {\em matching graph} on $\maV$. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a {\em matching policy}. When the sequence of classes of the arriving items is i.i.d., the sequence of buffer-contents is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is non-bipartite.

math.PR↗

arXiv2015

A pathwise comparison result for parallel queues

We introduce the appropriate framework for pathwise comparison of multiple server queues under general stationary ergodic assumptions. We show in what sense it is better to have more servers for a system under FCFS ('First Come, First Served') or equivalently, more queues in a system of parallel queues under the JSW ('Join the Shortest Workload') allocation policy. This comparison result is based on the recursive representation of Kiefer and Wolfowitz, and on a non-mass conservative generalization of the Schur-Convex semi-ordering. We also show that the latter result does not hold true in general, for the larger class of systems applying the semi-cyclic allocation policy introduced by Scheller-Wolf in \cite{SW03}.

math.PR↗

arXiv2015

On the stability of a class of non-monotonic systems of parallel queues

We investigate, under general stationary ergodic assumptions, the stability of systems of $S$ parallel queues in which any incoming customer joins the queue of the server having the $p+1$-th shortest workload ($p < S$), or a free server if any. This change in the allocation policy makes the analysis much more challenging with respect to the classical FCFS model with $S$ servers, as it leads to the non-monotonicity of the underlying stochastic recursion. We provide sufficient conditions of existence of a stationary workload, which indicate a "splitting" of the system in heavy traffic, into a loss system of $p$ servers plus a FCFS system of $S-p$ servers. To prove this result, we show {\em en route} an original sufficient condition for existence and uniqueness of a stationary workload for a multiple-server loss system.

math.PR↗

arXiv2015

The Jamming Constant of Uniform Random Graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a "parking process" on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.

math.PR↗

arXiv2014

Estimating the Spatial Reuse with Configuration Models

We propose a new methodology to estimate the spatial reuse of CSMA-like scheduling. Instead of focusing on spatial configurations of users, we model the interferences between users as a random graph. Using configuration models for random graphs, we show how the properties of the medium access mechanism are captured by some deterministic differential equations, when the size of the graph gets large. Performance indicators such as the probability of connection of a given node can then be efficiently computed from these equations. We also perform simulations to illustrate the results on different types of random graphs. Even on spatial structures, these estimates get very accurate as soon as the variance of the interference is not negligible.

cs.NI↗

arXiv2012

Large graph limit for an SIR process in random network with heterogeneous connectivity

We consider an SIR epidemic model propagating on a configuration model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by Volz [Mathematical Biology 56 (2008) 293--310].

math.PR↗

arXiv2010

Measure-valued stochastic recurrences and the stability of queues

In this paper we present a stability criterion for finite measure-valued stochastic recursions, generalizing Loynes's Theorem to spaces of measures. This result provides conditions for the reach of a "total stationary state" for the queue with an infinity of servers and the single-server SRPT queue. Indeed, we give in both cases a condition of existence of a stationary measure-valued recursive sequence characterizing the queueing system exhaustively.

math.PR↗

arXiv2010

Construction of a stationary queue with impatient customers

In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying a non-monotonic stochastic recursion under the Palm settings, and by stochastic comparison of stochastic recursions.

math.PR↗

arXiv2010

Weak Solutions of stochastic recursions: an explicit construction

We propose an explicit construction of the solution of a stationary stochastic recursion of the form $X\circθ=ϕ(X)$ on a semi-ordered Polish space, when the monotonicity of $ϕ$ is not assumed. This solution exists on an enriched probability space (it is said \emph{weak}), provided the recursion is lattice-valued, and dominated by a proper monotonic stochastic recursion.

math.PR↗

arXiv2010

A generalized backwards scheme for solving non monotonic stochastic recursions

We propose an explicit construction of a stationary solution for a stochastic recursion of the form $X\circθ=ϕ(X)$ on a partially-ordered Polish space, when the monotonicity of $ϕ$ is not assumed. Under certain conditions, we show that an extension of the original probability space exists, on which a solution is well-defined, and construct explicitly this extension. We then provide conditions for the solution to be defined as well on the original space. We finally apply these results to the stability study of two non-monotonic queueing systems.

math.PR↗

arXiv2010

Stationarity of pure delay systems and queues with impatient customers via stochastic recursions

In this paper we solve a particular stochastic recursion in the stationary ergodic framework, and propose some applications of this result to the study of regenerativity (that is, finiteness of busy cycles) and stationarity of some queueing systems: pure delay systems, in which all customers are immediately served, and queues with impatient customers. In this latter case under the FIFO discipline, we prove as well the existence of a stationary workload on an enlarged probability space.

math.PR↗

arXiv2009

A functional central limit theorem for the M/GI/$\infty$ queue

In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/$\infty$. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\mathcal{S}^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\mathcal{S}^{\prime}$-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.

math.PR↗

arXiv2008

Stability of a processor sharing queue with varying throughput

In this paper, we present a stability criterion for Processor Sharing queues, in which the throughput may depend on the number of customers in the system (in such cases such as interferences between the users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

math.PR↗

arXiv2008

Construction of a stationary FIFO queue with impatient customers

In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying stochastic recursions under the Palm settings, and by stochastic comparison of stochastic recursions.

math.PR↗

arXiv2008

Convex comparison of service disciplines in real time queues

We present a comparison of the service disciplines in real-time queueing systems (the customers have a deadline before which they should enter the service booth). We state that giving priority to customers having an early deadline minimizes the average stationary lateness. We show this result by comparing adequate random vectors with the Schur-Convex majorization ordering.

math.PR↗

arXiv2006

Fluid limit of a heavily loaded EDF queue with impatient customers

In this paper we present the fluid limit of an heavily loaded Earliest Deadline First queue with impatient customers, represented by a measure-valued process keeping track of residual time-credits of lost and waiting customers. This fluid limit is the solution of an integrated transport equation. We then use this fluid limit to derive fluid approximations of the processes counting the number of waiting and already lost customers.

math.PR↗