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Stability of Queueing Networks Maury Bramson University of Minnesota Stability of Queueing Networks May 13, 2008 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo 2 This manuscript will appear as Springer Lecture Notes in Mathematics 1950, E´cole d’E´t´e de Probabilit´es de Saint-Flour XXXVI-2006. Copyright c 2008 Springer-Verlag Berlin Heidelberg ! Contents 1 Introduction . 5 1.1 The M/M/1 Queue . 6 1.2 Basic Concepts of Queueing Networks . 7 1.3 Queueing Network Equations and Fluid Models . 15 1.4 Outline of Lectures . 18 2 The Classical Networks . 21 2.1 Main Results . 22 2.2 Stationarity and Reversibility . 27 2.3 Homogeneous Nodes of Kelly Type . 31 2.4 Symmetric Nodes . 36 2.5 Quasi-Reversibility . 43 3 Instability of Subcritical Queueing Networks . 57 3.1 Basic Examples of Unstable Networks . 58 3.2 Examples of Unstable FIFO Networks . 64 3.3 Other Examples of Unstable Networks . 75 4 Stability of Queueing Networks . 81 4.1 Some Markov Process Background . 84 4.2 Results for Bounded Sets . 96 4.3 Fluid Models and Fluid Limits . 104 4.4 Demonstration of Stability . 120 4.5 Appendix . 131 5 Applications and Some Further Theory . 143 5.1 Single Class Networks . 144 5.2 FBFS and LBFS Reentrant Lines . 148 5.3 FIFO Networks of Kelly Type . 151 5.4 Global Stability . 159 5.5 Relationship Between QN and FM Stability . 167 4 Contents Acknowledgments . 179 References . 181 Index . 187 1 Introduction Queueing networks constitute a large family of models in a variety of settings, involving “jobs” or “customers” that wait in queues until being served. Once its service is completed, a job moves to the next prescribed queue, where it remains until being served. This procedure continues until the job leaves the network; jobs also enter the network according to some assigned rule. In these lectures, we will study the evolution of such networks and ad- dress the question: When is a network stable? That is, when is the underlying Markov process of the queueing network positive Harris recurrent? When the state space is countable and all states communicate, this is equivalent to the Markov process being positive recurrent. An important theme, in these lec- tures, is the application of fluid models, which may be thought of as being, in a general sense, dynamical systems that are associated with the networks. The goal of this chapter is to provide a quick introduction to queueing networks. We will provide basic vocabulary and attempt to explain some of the concepts that will motivate later chapters. The chapter is organized as follows. In Section 1.1, we discuss the M/M/1 queue, which is the “simplest” queueing network. It consists of a single queue, where jobs enter according to a Poisson process and have exponentially distributed service times. The problem of stability is not difficult to resolve in this setting. Using M/M/1 queues as motivation, we proceed to more general queueing networks in Section 1.2. We introduce many of the basic concepts of queueing networks, such as the discipline (or policy) of a network determining which jobs are served first, and the traffic intensity ρ of a network, which provides a natural condition for deciding its stability. In Section 1.3, we provide a pre- liminary description of fluid models, and how they can be applied to provide conditions for the stability of queueing networks. In Section 1.4, we summarize the topics we will cover in the remaining chapters. These include the product representation of the stationary distri- butions of certain classical queueing networks in Chapter 2, and examples of unstable queueing networks in Chapter 3. Chapters 4 and 5 introduce fluid 6 1 Introduction models and apply them to obtain criteria for the stability of queueing net- works. 1.1 The M/M/1 Queue The M/M/1 queue, or simple queue, is the most basic example of a queue- ing network. It is familiar to most probabilists and is simple to analyze. We therefore begin with a summary of some of its basic properties to motivate more general networks. The setup consists of a server at a workstation, and “jobs” (or “cus- tomers”) who line up at the server until they are served, one by one. After service of a job is completed, it leaves the system. The jobs are assumed to arrive at the station according to a Poisson process with intensity 1; equiva- lently, the interarrival times of succeeding jobs are given by independent rate -1 exponentially distributed random variables. The service times of jobs are given by independent rate-µ exponentially distributed random variables, with µ > 0; the mean service time of jobs is therefore m = 1/µ. We are inter- ested here in the behavior of Z(t), the number of jobs in the queue at time t, including the job currently being served (see Figure 1.1). Fig. 1.1. Jobs enter the system at rate 1 and depart at rate µ. There are currently 2 jobs in the queue. The process Z( ) can be interpreted in several ways. Because of the inde- pendent exponentia·lly distributed interarrival and service times, Z( ) defines a Markov process, with states 0, 1, 2, . .. (M/M/1 stands for Mark·ov input and Markov output, with one server.) It is also a birth and death process on 0, 1, 2, . , with birth rate 1 and death rate µ. Because of the latter interpre- t{ation, it is}easy to compute the stationary (or invariant) probability measure πm of Z( ) when it exists, since the process will be reversible. Such a measure satisfies · πm(n + 1) = mπm(n) for n = 0, 1, 2, . , since it is constant, over time, on the intervals [0, n] and [n + 1, ). It follows ∞ that when m < 1, πm is geometrically distributed, with π (n) = (1 m)mn, n = 0, 1, 2, . (1.1) m − 1.2 Basic Concepts of Queueing Networks 7 All states clearly communicate with one another, and the process Z( ) is 1 · positive recurrent. The mean of πm is m(1 m)− , which blows up as m 1. When m 1, no stationary probability m−easure exists for Z( ). Using sta↑ndard reasoni≥ng, one can show that Z( ) is null recurrent when m·= 1 and is transient when m > 1. · The behavior of Z( ) that was observed in the last paragraph provides the basic motivation for these· lectures, in the context of the more general queueing networks which will be introduced in the next section. We will investigate when the Markov process corresponding to a queueing network is stable, i.e., is positive Harris recurrent. As mentioned earlier, this is equivalent to positive recurrence when the state space is countable and all states communicate. For M/M/1 queues, we explicitly constructed a stationary probability measure to demonstrate positive recurrence of the Markov process. Typically, however, such a measure will not be explicitly computable, since it will not be reversible. This, in particular, necessitates a new, more qualitative, approach for showing positive recurrence. We will present such an approach in Chapter 4. 1.2 Basic Concepts of Queueing Networks The M/M/1 queue admits natural generalizations in a number of directions. It is unnecessary to assume that the interarrival and service distributions are exponential. For general distributions, one employs the notation G/G/1; or M/G/1 or G/M/1, if one of the distributions is exponential. (To emphasize the independence of the corresponding random variables, one often uses the notation GI instead of G.) The single queue can be extended to a finite system of queues, or a queueing network (for short, network), where jobs, upon leaving a queue, line up at another queue, or station, or leave the system. The queueing network in Figure 1.2 is also a reentrant line, since all jobs follow a fixed route. j=1 j=2 j=3 Fig. 1.2. A reentrant line with 3 stations. Depending on a job’s previous history, one may wish to prescribe different service distributions at its current station or different routing to the next 8 1 Introduction station. This is done by assigning one or more classes, or buffers, to each station. Except when stated otherwise, we label stations by j = 1, . , J and classes by k = 1, . , K; we use (j) to denote the set of classes belonging to station j and s(k) to denote theC station to which class k belongs. In Figure 1.3, there are 3 stations and 5 classes. Classes are labelled here in the order they occur along the route, with (1) = 1, 5 , (2) = 2, 4 , and (3) = 3 . C { } C { } C { } j=1 j=2 j=3 Fig. 1.3. A reentrant line with 3 stations and 5 classes. The stations are labelled by j = 1, 2, 3; the classes are labelled by k = 1, . 5, in the order they occur along the route. Other examples of queueing networks are given in Figures 1.4 and 1.5. Figure 1.4 depicts a network with 2 stations, each possessing 2 classes. The network is not a reentrant line but still exhibits deterministic routing, since each job entering the network at a given class follows a fixed route. When the individual routes are longer, it is sometimes more convenient to replace the above labelling of classes by (i, k), where i gives the route that is followed and k the order of the class along the route. j = 1 j = 2 Fig.
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