The material on continuous-time Markov chains is divided between this chapter and the next. The theory takes some time to set up, but once up and running it follows a very similar pattern to the discrete-time case. To emphasise this we have put the setting-up in this chapter and the rest in the next. If you wish, you can begin with Chapter 3, provided you take certain basic properties on trust, which are reviewed in Section 3.1. The first three sections of Chapter 2 fill in some necessary background information and are independent of each other. Section 2.4 on the Poisson process and Section 2.5 on birth processes provide a gentle warm-up for general continuous-time Markov chains. These processes are simple and particularly important examples of continuous-time chains. Sections 2.6–2.8, especially 2.8, deal with the heart of the continuous-time theory. There is an irreducible level of difficulty at this point, so we advise that Sections 2.7 and 2.8 are read selectively at first. Some examples of more general processes are given in Section 2.9. As in Chapter 1 the exercises form an important part of the text.

Q-matrices and their exponentials

In this section we shall discuss some of the basic properties of Q-matrices and explain their connection with continuous-time Markov chains.