Introduction to Matrix Analytic Methods in Queues 1

3Discrete Phase Type Distributions

Computational aspects are not a luxury anymore. They are both an art and science. They have become an integral part of the analysis of any stochastic model and are needed to qualitatively study any stochastic model. We know how much Poisson processes played and still playing an important role in stochastic modeling. Poisson processes are very elegant mathematically and of course they have their own limitations in practice due to the memoryless property of the underlying exponential distribution. To get away from this restriction, Neuts developed the theory of phase type (PH) distributions and related point processes.

In this chapter, we give a brief overview of PH distributions including history behind the development. PH processes are natural extensions to geometric and negative binomial (in the case of discrete-time, the focus of this chapter) and exponential, Erlang and hyperexponential (in the case of continuous-time, the focus of the next chapter) distributions. The idea behind this development by Neuts is to enable one to get away from the restricted class of distributions yet preserve computationally tractable solutions in stochastic modeling. Traditionally, geometric and Poisson/exponential distributions have been used mainly for mathematical convenience more than for practical purposes. We also discuss several examples. This is the first of many building blocks for matrix-analytic methods (MAM). PH distributions lend themselves naturally ...