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...

We introduce a new approach to the study of dynamic (or continuous) packet routing, where packets are being continuously injected into a network. Our objective is to study what happens to packet routing under continuous injection

Usually, the stability of queueing networks is established by explicitly determining the invariant distribution. However, outside of the narrow class of queueing networks possessing a product form solution, such explicit solutions are rare, and consequently little is known concerning stability too. We develop here a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady--state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. For an example of an open re-entrant line, we show that all stationary non-idling policies are stable for all load factors less than one. This includes the well known First Com...

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...

We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM's) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d \Gamma 1)-dimensional faces that form the boundary of the polyhedron, the bounded variation part of the process increases in a given direction (constant for any particular face), so as to confine the process to the polyhedron. For historical reasons, this "pushing " at the boundary is called instantaneous reflection. For simple convex polyhedrons, we give a necessary and sufficient condition on the geometric data for the existence and uniqueness of an SRBM. For non-simple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the non-simple case. From the uniqueness, it follows that an SRBM defines a strong Markov process. Our results have application to the study of diffusions arising as heavy traffic limits of multiclass queueing networks and in particular, the non-simple case has application to multiclass fork and join networks. Our proof of weak existence uses a patchwork martingale problem introduced by T. G. Kurtz, whereas uniqueness hinges on an ergodic argument similar to that used by L. M. Taylor and R. J. Williams to prove uniqueness for SRBM's in an orthant.

This paper treats transience for queueing network models by considering an associated fluid model. If starting from any initial condition the fluid model explodes at a linear rate, then the associated queueing network with i.i.d. service times and a renewal arrival process explodes faster than any fractional power. Keywords: Queueing networks, stability. 1 Introduction There has been much recent interest in understanding the dynamics of queueing networks, and in particular their stability properties. Numerous techniques have been developed for verification of stability or ergodicity using a variety of methods. Of interest to us in the present paper is the recent approach based upon a fluid approximation. Rybko and Stolyar [15] have recently examined the stability properties of a particular example by studying the properties of the associated fluid approximation. Dupuis and Williams obtained results of this kind for reflected Brownian motion [8], and these ideas were subsequently gener...

: We consider flexible manufacturing systems using the `First Come, First Served' (FCFS or FIFO) scheduling policy at each machine. We describe and discuss in some detail simple deterministic examples which have adequate capacity but which, under FCFS, can exhibit instability: unboundedly growing WIP taking the form of a repeated pattern of behavior with the repetitions on an increasing scale. Key Words: scheduling policy, flexible manufacturing system, queueing network, stability, FIFO, `First Come First Served'. 1. Introduction We consider network models involving multiple flows with buffering/queuing at each node (processor). Specifying a queue discipline (i.e., a scheduling policy for the processing at nodes) then defines the dynamics for the system. A queue discipline is called stable if the queue lengths (WIP) remain uniformly bounded in time for any realization --- configuration, initial state --- with input rates subject to the obvious capacity limitations. We quote from [3...

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...

Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workload-relaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order | log(1 − ρ)|.