Keywords: multiclass queueing network, di usion approximations, uid approximations, heavy tra c ..... En(t) = E0( nt), and the service process is Sn = fSn(t); t.
Di usion Approximations for Some Multiclass Queueing Networks with FIFO Service Disciplines Hong Chen1 Faculty of Commerce and Business Administration, UBC, Canada Hanqin Zhang2 Institute of Applied Mathematics, Academia Sinica, Beijing, China
Abstract
The di usion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy trac limit theorem, a key condition that characterizes this class is that a matrix , known as the workload contents matrix, has a spectral radius less than unity, where represents the number of service stations. The ( )th component of matrix can be interpreted as the amount of future work for station that is embodied in per unit of immediate work at station at time . This class includes Rybko-Stolyar network with FIFO service discipline as a special case. The result extends existing di usion limiting theorems to non-feedforward multiclass queueing networks. In establishing the di usion limit theorem, a new approach is taken. The traditional approach is based on an oblique re ection mapping, but such a mapping is not well-de ned for the network under consideration. Our approach takes two steps: rst establishing the -tightness of the scaled queueing processes, and then completing the proof for the convergence of the scaled queueing processes by invoking the weak uniqueness for the limiting processes, which are semimartingale re ecting Brownian motions. J
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Keywords: multiclass queueing network, di usion approximations, uid approximations,
heavy trac, semimartingale re ecting Brownian motion.
AMS 1991 Subject classi cations: Primary 60F17, 60K25, 60G17; Secondary 60J70, 90B10, 90B22. Current version: April 1996 Supported in part by a grant from NSERC (Canada). Supported in part by a grant from NSERC (Canada), and part of this work was done while the author was visiting the University of British Columbia. 1
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TABLE OF CONTENTS 1. Introduction 2. Queueing Network Model 2.1. Primitive Data 2.2. Performance Measures and Their Dynamics 3. Main Results 4. The Proof of the Main Theorem 4.1. Preliminaries 4.2. Fluid Approximations 4.3. Tightness for Queueing Processes 4.4. Convergence for Queueing Processes 5. Appendix 5.1. Pathwise Construction of Queueing Processes 5.2. Some Elementary Results
1 Introduction Ever since the pioneering work of Kingman (1965) and Iglehart and Whitt (1970a,b), there have been many publications in the area of di usion approximations [also known as functional central limit theorems] for queueing systems. A cornerstone in the development of a di usion approximation theory is the introduction of the oblique re ection mapping (ORM) by Harrison and Reiman (1981) and its use to prove the di usion approximation for a generalized open Jackson network by Reiman (1984). Equipped with the ORM, the di usion approximation has been extended to, among others, a multi-type network [under a special priority structure] by Johnson (1983), a feedforward multiclass (FIFO) network by Peterson (1991), a multiclass (FIFO) single station by Reiman (1988) and Dai and Kurtz (1995), a closed network by Chen and Mandelbaum (1991), and a time-dependent queue by Mandelbaum and Massey (1995). Recently, Pats (1994) has established the di usion approximation for a statedependent network. For surveys on the di usion approximation, readers are referred to Whitt (1974), Lemoine (1978), Glynn (1990), Harrison and Nguyen (1993), Chen and Mandelbaum (1994b), and Williams (1995). Over the past ten years or more, substantial e orts have been made by J. Michael Harrison and his associates (including many of his former students) to prove a di usion approximation 1
theorem for a general multiclass queueing network. These e orts were summarized in Harrison (1988) and Harrison and Nguyen (1990;1993), where a di usion approximation theorem was conjectured, and the limit was proposed as a Brownian model of a multiclass queueing network. The next signi cant development that motivates the current research on di usion approximations for multiclass queueing networks is the counterexample brought forward by Dai and Wang (1993). Their counterexample shows that the proposed Brownian network does not exist for some multiclass queueing networks. In fact, these queueing networks do not have a di usion limit under conventional heavy trac scaling (Dai and Nguyen (1994)). Whitt (1993) provided a similar example. More recently, Harrison and Williams (1995) provided an example of unconventional heavy trac limit theorem for a closed multiclass network. An important question is \for which classes of multiclass queueing networks does the diffusion approximation exist?" Except for those networks with special structure such as feedforward networks and single stations as cited above, little is known. In this paper, we establish a sucient condition for the existence of a di usion approximation for a multiclass queueing network under a FIFO service discipline (Theorem 3.1). In addition to the usual assumptions for a heavy trac limit theorem, the sucient condition is stated as the spectral radius of a J J matrix G being less than unity, where J represents the number of service stations, and matrix G is known as the workload contents matrix whose (j; `)th component 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 sucient condition covers networks that may be neither feedforward nor a single station. In particular, it covers the Rybko-Stolyar network with FIFO service discipline as a special case. (This is a two-station network with four classes. In addition to establishing the instability of this network under a priority service discipline, Rybko and Stolyar (1992) also established the stability of this network under a FIFO service discipline.) The analysis for establishing a di usion approximation of a general multiclass queueing network is much harder. One of the reasons is that the oblique re ection mapping, which has been used for almost all of the previous di usion approximation results, is not uniquely de ned in this case (see, for example, Mandelbaum (1989) and Bernard and El Kharroubi (1991)). Our approach takes two steps: rst establishing the C -tightness [which is de ned in Subsection 4.1] of the scaled queueing processes, and then completing the proof for the convergence of the scaled queueing processes by invoking the weak uniqueness for the limiting processes. The foundation for this second step is the existence and the uniqueness of a class of semimartingale re ecting Brownian motions (SRBMs) [which is described in Section 3]. Reiman and Williams (1988) established a necessary condition for the existence of an SRBM, and Taylor and Williams (1993) proved that condition is also sucient for the existence of an SRBM and that the SRBM is unique in law. In order to use the above uniqueness result and to establish that the limiting process is a well-de ned SRBM on a ltered probability space, a substantial e ort is made in showing that a limiting input process has a martingale property (Proposition 4.13); this result has a potential use in establishing more general sucient conditions for the existence of the di usion approximation for a multiclass queueing network. We note that at the same time of the preparation for this paper, Dai and Dai (1995) used the same approach to establish a di usion approximation for single class nite bu er queueing networks. 2
Closely related to this research is the notion of a uid approximation and the problem of stability of a multiclass queueing network. Essentially, all work on a di usion approximation implicitly implies a uid approximation, though more assumptions are made. Establishing a
uid approximation is usually a convenient rst step towards establishing a di usion approximation, as rst demonstrated by Johnson (1983). Readers are referred to Chen and Mandelbaum (1994a) for a survey on uid approximations. The stability of a multiclass queueing network has attracted a lot of attention in the applied probability community in recent years. This is due to several counterexamples where a queueing network that has a trac intensity less than unity at each station is not stable (Kumar and Seidman (1990), Lu and Kumar (1991), Rybko and Stolyar (1992), Bramson (1994) and Seidman (1994)). Our simulation suggest that those counterexamples in Dai and Wang (1993) whose di usion approximations do not exist also be counterexamples for the stability. In addition, the sucient condition established in this paper for the di usion approximation matches one of the sucient conditions established in Chen and Zhang (1994) for the stability of multiclass FIFO queueing networks. On the other hand, it was established that the stability of all uid approximations is sucient for the stability of the associated multiclass queueing network (with some mild distributional assumptions on the interarrival and service times) (Rybko and Stolyar (1992) and Dai (1995a)). (Readers are referred to Dai (1995b) for references on the stability of multiclass queueing networks.) Therefore, we believe that close relations may exist among uid approximations, di usion approximations and stability of queueing networks, and that understanding any one of these areas would help our understanding of the other areas. The paper is organized as follows. The queueing network model and its dynamics are described in Section 2. The main theorem is stated in Section 3, and is proved in Section 4. To conclude this section, we introduce some notation and make some de nitions that are used throughout the paper. We denote by
