It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and first-buffer-first-served preemptive resume discipline in a re-entrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...

The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on long-run average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including re-entrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications. Keywords: Multiclass queueing networks, ergodicity, general state space Markov processes, polling models, generalized Jackson networks, stability, performance analysis. 1 Introduction The subject of this paper is open multiclass queueing networks, which are models of complex systems such as wafer fabri...

This paper studies the fluid approximation (also known as the functional strong law-of-large-numbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the first-order parameters (i.e., long-run average arrivals, services and routings) of the queueing network. It is proved that the fluid approximation for the queueing network exists if the corresponding linear fluid model is weakly stable, and that the queueing network is stable if the corresponding linear fluid model is (strongly) stable. Sufficient conditions are found for the stabilities of a linear fluid model. Keywords and phrases: Multiclass queueing networks, fluid models, fluid approximations, stability, positive Harris recurrent, and work-conserving service disciplines. Preliminary Versions: September 1993 Revisions: June 1994; September 1994; January 1995 To appear in Annals of Applied Probability AMS 1980 su...

Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for '-irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially bounded hitting times on one and then all suitably "small" sets; (b) the existence of "Foster-Lyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; (c) the existence of drift conditions on the extended generator e A of the process. We use the identity e AR fi = fi(R fi \Gamma I) connecting the extended generator and the resolvent kernels R fi , to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of (a)--(c). These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state ...

Re-entrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer re-entrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such netw...

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. Running head : Stability and Instability of Queueing Networks Keywords : Multiclass queueing networks, ergodicity, stability, performance analysis. 1 Introduction It has generally been taken for granted in queueing theory that stability of a network is guaranteed so long as the overall traffic intensity is less than unity and in recent years there has been much analysis which supports this belief for special classes of systems, such as single class queueing networks (see Borovkov [2], Sig...

In this paper we consider /-irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the solution is square integrable. These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to Poisson's equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes. AMS subject classifications: 68M20, 60J10 Running head: Poisson's Equation Keywords: Markov chain, Markov process, Poisson's equation, Lyapunov f...

In this paper we consider a continous time method of approximating a given distribution ß using the Langevin diffusion dL t = dW t + 1 2 r log ß(L t )dt: We find conditions under which this diffusion converges exponentially quickly to ß or does not: in one dimension, these are essentially that for distributions with exponential tails of the form ß(x) / exp(\Gammafljxj fi ), 0 ! fi ! 1, exponential convergence occurs if and only if fi 1. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretisations need not do so. We then consider a "Metropolis-adjusted" version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolised version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficultie...

In this paper we consider a #-irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedded chains with arbitrary sampling distributions. We show that the recurrence behavior of such generalized resolvents classifies the behavior of the continuous time process; from this we prove that hitting times on the small sets of a generalized resolvent chain provide criteria for, successively, (i) Harris recurrence of # (ii) the existence of an invariant probability measure # (or positive Harris recurrence of #) and (iii) the finiteness of #(f) for arbitrary f.

. This paper surveys recent work on the stability of open multiclass queueing networks via fluid models. We recapitulate the stability result of Dai [8]. To facilitate study of the converse of the stability result, we distinguish between the notion of fluid limit and that of fluid solution. We define the stability region of a service discipline and the global stability region of a network. Examples show that piecewise linear Lyapunov functions are powerful tools in determining stability regions. Key words. Stability, queueing networks, fluid models, scheduling, performance analysis, Harris recurrence, heavy traffic, Brownian models. 1. Introduction. There has been a recent surge in studying stability /instability of multiclass queueing networks. See, for example, Lu and Kumar [21], Rybko and Stolyar [24], Whitt [27], Bramson [2,3] and Seidman [25]. To show that the instability can occur even in a Kelly-type network, a network in which all customers visit a station have a common servi...