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Chapter Stochastic 3 processes is to provide evaluation rigour. as outlined processes stochastic . We follow in Section discusses the necessary purposes. It is assumed in Appendix and classify process with A. them in Section , after which we with chains classes in more the study in which general detail. We start Markov chains special in Sections as arrival in Section chains. background that the reader in stochastic processes in this T about HE aim of this chapter for practical probability define a number processes performance theory, nor at mathematical stochastic of different in Section Then, . | Performance of Computer Communication Systems A Model-Based Approach. Boudewijn R. Haverkort Copyright 1998 John Wiley Sons Ltd ISBNs 0-471-97228-2 Hardback 0-470-84192-3 Electronic Chapter 3 Stochastic processes THE aim of this chapter is to provide the necessary background in stochastic processes for practical performance evaluation purposes. We do not aim at completeness in this chapter nor at mathematical rigour. It is assumed that the reader has basic knowledge about probability theory as outlined in Appendix A. We first define stochastic processes and classify them in Section after which we discuss a number of different stochastic process classes in more detail. We start with renewal processes in Section . We follow with the study of discrete-time Markov chains DTMCs in Section followed by Section in which general properties of Markov chains are presented. Then in Section continuous-time Markov chains CTMCs are discussed. Section then discusses semi-Markov processes. Two special cases of CTMCs the birth-death process and the Poisson process are discussed in Sections and respectively. In Section we discuss the use of renewal processes as arrival processes particular emphasis is given to phase-type renewal processes. Finally in Section we summarise the specification and evaluation of the various types of Markov chains. Overview of stochastic processes A stochastic process is a collection of random variables A t G T defined on a probability space and indexed by a parameter t usually assumed to be time which can take values in a set T. The values that X t assumes are called states. The set of all possible states is called the state space and is denoted Z. If the state space is discrete we deal with a discretestate stochastic process which is called a chain. For convenience it is often assumed that whenever we deal with a chain the state space Z 0 1 2 . The state space can also 32 3 Stochastic processes be continuous. We