We consider the problem of classifying Markov chains on the quarter plane Z 2 + which possess a property of partial spatial homogeneity. Such chains arise frequently in the study of queueing and loss networks and have been previously studied, notably by Malyshev and Menshikov and by Fayolle. However existing results are either given under restrictive conditions, or lack complete proofs. We take a new approach to the construction of Lyapounov functions for such chains to give proofs which require weaker conditions, are complete in all cases and additionally are considerably simpler. We also describe the application of these results to the study of the dynamic and equilibrium behaviour of large loss networks. Keywords: Markov chains; random walks; ergodicity; transience; Lyapounov functions; loss networks; queueing networks. 1 Introduction In this paper we study discrete time irreducible Markov chains on the quarter plane Z 2 + (where Z+ is the set of nonnegative integers) whose tran...

: It is now well-known that multiclass networks may be unstable even under the "usual conditions" of stability (when the loads are less than one at all queues), but the proofs of transience (in the Markovian case) generally require a complex work based on the dynamics of an associated "fluid model". Here we develop a sample-path argument introduced in a previous paper, which provides new ergodicity conditions for networks ruled by priorities; when one of these conditions is violated, the network diverges at linear speed. Our approach is based on the identification of the essential faces, which are the sets of classes that can be simultaneously occupied in stationary regime. A graph is associated with the network, the existence of unessential faces being equivalent to the presence of cycles in this graph, in which case the usual conditions are not sufficient conditions of stability. As a by-product of our results, we recover the stability conditions and complete the analysis of two exem...

In this paper, we develop a systematic method for deriving bounds on the stationary distribution of stable Markov chains with a countably infinite state space. We assume that a Lyapunov function is available that satisfies a negative drift property (and, in particular, is a witness of stability), and that the jumps of the Lyapunov function corresponding to state transitions have uniformly bounded magnitude. We show how to derive closed form, exponential type, upper bounds on the steady-state distribution. Similarly, we show how to use suitably defined lower Lyapunov functions to derive closed form lower bounds on the steady-state distribution. We apply our results to homogeneous random walks on the positive orthant, and to multiclass single station Markovian queueing systems.

For a sequence of finite Markov chains L(N) we introduce the notion of "convergence time to equilibrium" T(N). For sequences that are obtained by truncating some countable Markov chain L we find the convergence time to equilibrium in terms of Lyapunov function of Markov chain L. We apply this result to queueing systems with a limited number of customers: a priority system with several customer types and the Jackson network.

Sufficient stability condition for the standard token passing ring is "known" since the seminal paper of Kuehn in 1979. However, this condition was derived without formal proof, and the proof seems to be of considerable interest to research community. In fact, Watson observed that in the performance evaluation of token passing rings "it is convenient to derive stability conditions ... (without proof)". Our intention is to fill this gap, and provide a formal proof of the sufficient and necessary stability condition for the token passing ring. In this paper we present the case when the arrival process to each queue is Poisson but service times and switchover times are generally distributed. We consider in depth gated `-limited (` 1) service discipline for each station. We also indicate that the basic steps of our technique can be used to study the stability of some other multiqueue systems. Keywords and Phrases Token passing rings, stability, substability, ergodicity, Markov chains, L...

A state-dependent 1-limited polling model with N queues is analyzed. The routing strategy generalizes the classical Markovian polling model, in the sense that two routing matrices are involved, the choice being made according to the state of the last visited queue. The stationary distribution of the position of the server is given. Ergodicity conditions are obtained by means of an associated dynamical system. Under rotational symmetry assumptions, average queue length and mean waiting times are computed. 1.

Introduction. Start with the following elegant interval packing (IP) problem. Let random subintervals arrive at a service facility in a rate- Poisson stream; a random subinterval is simply the interval between two independent uniform random draws from [0; 1]. Place an arrival at time t immediately into service if it is disjoint from each of the subintervals, if any, being served at time t; otherwise, place it into a queue. When a subinterval completes service, at time t say, scan the queue in arrival order, placing into service each subinterval encountered that is disjoint from all subintervals already accepted for service at time t. Determine the stability condition under the assumption that service times are independent unit-mean exponentials. A little reflection shows that, since the midpoint 1=2 is the most "congested" point of [0;

this paper, we apply our methodology to derive stability criteria for time-limited token passing rings introduced recently by Leung and Eisenberg [6], [7] (cf. Theorem 1). In such a system each station transmits messages for at most an amount of time ø . If the transmission time exceeds ø , the station completes the transmission of the message in progress and sends the token to the next station (the so called nonpreemptive time limited discipline). While we study this system as an important application, the technique can be applied almost without modification to a class of monotonic and contractive policies (cf Theorem 2). Our approach to the stability of token passing rings follows the idea discussed in our paper [3], and differs from the standard methodology of the Lyapunov test function (cf. [12], [18], [14]). (For other than test function approaches see also [2], [11] [14, 15].) It resembles the general idea of Malyshev's faces and induced Markov chains [10]. Our method is based on a simple idea of stochastic dominance technique, and application of Loynes [9] stability criteria for an isolated queue. We note that this approach is not restricted to token passing rings, and stability of several other distributed systems can be assessed by this methodology (cf. [14, 15]). We now summarize our main results. We shall analyze the token passing ring with Poisson arrivals with parameter i for the ith station, general distribution of service times fS

Abstract—We consider a wireless downlink shared by a dynamic population of flows. The flows of random size (bits) arrive at the base station at random times, and leave when they have been completely transmitted. The transmission rate supported by the wireless channel of each flow while the flow awaits transmission varies randomly over time and is independent of that of the other flows. The scheduling problem in this context is to select a flow for transmission based on the current system state (e.g., backlogs, wait times, and channel states of the contending flows). It has recently been shown that for such a system, the wellknown (backlog-driven) MaxWeight scheduler is not throughput optimal. That is to say, the MaxWeight scheduler will not stabilize a given system even though it is possible to construct a stabilizing scheduler using the various flow- and channel-related statistics. However, in this paper, we show that the delay-driven MaxWeight scheduler is, nevertheless, throughput optimal for such a system. The delay-driven MaxWeight, like its backlog-driven version, does not require any knowledge of the flow- or channel-related statistics. I.

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