We have seen many examples of systems questions that can be answered by modeling the system as a Markov chain. For a system to be well modeled by a Markov chain, it is important that its workloads have the Markovian property. For example, if job sizes and interarrival times are independent and Exponentially distributed, and routing is probabilistic between the queues, then the system can typically be modeled easily using a CTMC. However, if job sizes or interarrival times are distributed according to a distribution that is not memoryless, for example Uniform(0, 100), then it is not at all clear how a Markov chain can be used to model the system.

In this chapter, we introduce a technique called “the method of stages” or “the method of phases.” The idea is that almost all distributions can be represented quite accurately by a mixture of Exponential distributions, known as a phase-type distribution (PH).We will see how to represent distributions by PH distributions in Section 21.1. Because PH distributions are made up of Exponential distributions, once all arrival and service processes have been represented by PH distributions, we will be able to model our systems problem as a CTMC, as shown in Section 21.2.

The Markov chains that result via the method of phases are often much more complex than Markov chains we have seen until now. They typically cannot be solved in closed form. Thus, in Section 21.3, we introduce the matrix-analytic method, a very powerful numerical method that allows us to solve many such chains that come up in practice.