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 address 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 lectures, 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.