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...

Abstract--We consider the stability and performance of a model for networks supporting services that adapt their transmission to the available bandwidth. Not unlike real networks, in our model, connection arrivals are stochastic, each has a random amount of data to send, and the number of ongoing connections in the system changes over time. Consequently, the bandwidth allocated to, or throughput achieved by, a given connection may change during its lifetime as feedback control mechanisms react to network loads. Ideally, if there were a fixed number of ongoing connections, such feedback mechanisms would reach an equilibrium bandwidth al-location typically characterized in terms of its "fairness " to users, e.g., max-min or proportionally fair. In this paper we prove the sta-bility of such networks when the offered load on each link does not exceed its capacity. We use simulation to investigate performance, in terms of average connection delays, for various fairness criteria. Finally, we pose an architectural problem in TCP/IPs decoupling of the transport and network layer from the point of view of guaran-teeing connection-level stability, which we claim may explain con-gestion phenomena on the Internet. Index Terms--ABR service, bandwidth allocation, Lyapunov functions, performance analysis, proportional fairness, rate control, stability, TCP/IP, weighted max-min fairness. F I.

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...

The subject of this paper is stability properties of adversarial queueing networks. Such queueing systems are used to model packet switch communication networks, in which packets are generated and routed dynamically, and have become a subject of research focus recently. Adversarial queueing networks are defined to be stable, if the number of packets stays bounded over time. A central question is determining which adversarial queueing networks are stable, when an arbitrary greedy packet routing policy is implemented. In this paper we show how stability of a queueing network can be determined by considering an associated fluid models. Our main result is that the stability of the fluid model implies the stability of an underlying adversarial queueing network. This opens an opportunity for analyzing stability of adversarial networks, using established stability methods from continuous time processes, for example, the method of Lyapunov function or trajectory decomposition. We demonstrate t...

In this paper, we establish a sufficient condition for the stability of a multiclass fluid network and queueing network under priority service disciplines. The sufficient condition is based on the existence of a linear Lyapunov function, and it is stated in terms of the feasibility of a set of inequalities that are defined by network parameters. In all the networks we have tested, this sufficient condition actually gives a necessary and sufficient condition for their stability.

We develop an algorithm for simulating approximate random samples from the invariant measure of a Markov chain using backward coupling of embedded regeneration times. Related methods have been used effectively for finite chains and for stochastically monotone chains: here we propose a method of implementation which avoids these restrictions by using a "cycle-length" truncation. We show that the coupling times have good theoretical properties and describe benefits and difficulties of implementing the methods in practice. 1 Introduction There has been considerable recent work on the development and application of algorithms that will enable the simulation of the invariant measure ß of a Markov chain, either exactly (that is, by drawing a random sample known to be from ß) or approximately, but with computable order of accuracy. These were sparked by the seminal paper of Propp and Wilson [18], and several variations and extensions of this idea have appeared in the literature including rece...

It has been shown recently that fluid limit models provide the means to substantially simplify the stability analysis of multiclass queueing networks. This paper begins to address the most natural second question: do fluid models provide a tool for developing high performance scheduling policies for queueing networks? A linear program is constructed to give upper and lower bounds on achievable performance over all work conserving policies. The bounds are tested through simulations using Simulink. Stochastic translations of the control policies are tested through simulations using UltraSAN on a stochastic-discrete model of the network. Keywords: Queueing networks, stability, performance, linear programming. 1 Introduction The papers [6, 7, 14] show that stability of a multiclass queueing network is essentially equivalent to stability of an associated deterministic fluid model. Because stability is a property of the system which only depends on large values of the state, the s...

We introduce a new method to investigate stability of work-conserving policies in multiclass queueing networks. The method decomposes feasible trajectories and uses linear programming to test stability. We show that this linear program is a necessary and sufficient condition for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class (monotone piecewise linear convex) Lyapunov functions. Using linear programming duality, we show that for two-station systems the Lyapunov function approach is equivalent to ours and therefore characterizes stability exactly. 1 Introduction The problem of establishing conditions under which a multiclass queueing network is stable und...

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called “interchange-of-limits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

. 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...