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...

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.

Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

. 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...

This paper proves a heavy traffic limit theorem for a multiclass service station with Markovian feedback. This result generalizes the one proved by Reiman (1988). Our approach also significantly simplifies Reiman's original proof. Numerical examples are presented to illustrate the effectiveness of the QNET method which is rooted in the theorem. Running Title: Multiclass Station in Heavy Traffic Revised on April 6, 1994 Mathematics of Operations Research, Vol. 20, pp 721--742 (1995) Keywords: Multiclass queueing network, heavy traffic, diffusion approximation, reflecting Brownian motion, performance analysis. AMS 1991 subject classifications: Primary 60K25, 60F17, 60G17; Secondary 60J15, 90B22, 68M20. 1 Research supported in part by two grants from Texas Instruments Corporation and by NSF grants DMS9209586 and DDM-9215233 2 Research supported in part by NSF grant DMS-8901464 1 Introduction We consider a multiclass single server station. There are c classes of customers, and ea...

The subject of this paper is the heavy traffic behavior of a general class of queueing networks with first-in-first-out (FIFO) service discipline. For special cases which require various assumptions on the network structure, several authors have proved heavy traffic limit theorems to justify the approximation of queueing networks by reflecting Brownian motions. Based on these theorems, some have conjectured that the Brownian approximation may in fact be valid for a more general class of queueing networks. In this paper, we prove that the Brownian approximation does not hold for such a general class of networks. Our findings suggest that it may be fruitful to consider a more general class of approximating processes. Contents 1. Introduction 2. Preliminaries 3. Conjecture and the Main Theorem 4. A Pseudo Heavy Traffic Limit Theorem 5. Proof of Theorem 3.1 6. Concluding Remarks and Open Problems Keywords: Multiclass queueing network, heavy traffic, diffusion approximation, reflecting Br...

This paper summarizes results of Dai and Vande Vate [15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of two-station open multiclass queueing networks with very general arrival and service processes. The conditions for pathwise global stability arise from two intuitively appealing phenomena: virtual stations and push starts. These phenomena shed light on the sources of bottlenecks in complicated queueing networks like those that arise in wafer fabrication facilities. We show that a two-station open multi-class queueing network is globally pathwise stable if and only if the corresponding fluid model is globally weakly stable. We further show that a two-station fluid model is globally (strongly) stable if and only if the average service times are in the interior of the global weak stability region. As a consequence, under stronger distributional assumptions on the arrival and service processes, the queue...

Consider a job-shop or batch-flow manufacturing system in which new jobs are introduced only as old ones depart, either because of physical constraints or as a matter of management policy. Assuming that there is never a shortage of new work to be done, the number of active jobs remains constant over time, and the system can be modeled as a kind of closed queueing network. With manufacturing applications in mind, we formulate a general closed network model and develop a mathematical method to estimate its steady-state performance characteristics. A restrictive feature of our network model is that all the job classes which are served at any given node or station share a common service time distribution. Our analytical method, which is based on an algorithm for computing the stationary distribution of an approximating Brownian model, is motivated by heavy traffic theory; it is precisely analogous to a method developed earlier for analysis of open queueing networks. The required inputs inc...

The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J \Theta J matrix G, known as the workload contents matrix, has a spectral radius less than unity, where J represents the number of service stations. The (j; `)th component of matrix G can be interpreted as the amount of future work for station j that is embodied in per unit of immediate work at station ` at time t. This class includes Rybko-Stolyar network with FIFO service discipline as a special case. The result extends existing diffusion limiting theorems to non-feedforward multiclass queueing networks. In establishing the diffusion limit theorem, a new approach is taken. The traditional approach is based on an oblique reflection mapping, but such a mapping is not well-defined for the network under consideration. Our approach takes two steps: f...