In this chapter you will learn the mathematical analyses that underlie the control techniques used and the resulting network performance measures described in Chapter 8. Some sections of this chapter demand from the reader a sophisticated background in stochastic processes. A basic knowledge of multivariate random variables and Markov chains is essential for these sections. However, you can skip these sections and still find accessible the discussion on deterministic models. If you are able to follow the more mathematical material, you will be able to participate in the mathematical discussions on networking published in the research journals. But even if you are unable to follow the argument, Chapter 8 makes the conclusions of those discussions accessible.

We start by reviewing some key results on Markov chains in section 9.1. We apply these results to the study of circuit-switched networks in section 9.2 and of datagram networks in section 9.3. Section 9.4 explains the analysis of virtual circuit networks.

9.1.1 Overview

A Markov chain is a model of the random motion of an object in a discrete set of possible locations. Two versions of this model are of interest to us: discrete time and continuous time. In discrete time, the position of the object called the state of the Markov chain is recorded every unit of time, that is, at times 0, 1, 2, and so on. In continuous time, the state...