Stability and Heavy Traffic Limits for Queueing Networks

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. Two aspects of their evolution have been the object of considerable interest over the last two decades. The first is the question of when a network is 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. The other topic is the existence of heavy traffic limits for queueing networks. That is, when does a sequence of networks, under diffusive scaling, converge to a reflecting Brownian motion? We will be interested in both questions, while devoting more effort to the former. A unifying theme in our approach to both questions will be 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. Both the problems of stability and heavy traffic limits are 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 preliminary description of fluid models, and how they can be applied to provide 6 1 Introduction