Introduction to Matrix Analytic Methods in Queues 1

4Continuous Phase Type Distributions

In this chapter, we give a brief overview of the counterpart to discrete phase type (DPH) distributions. Continuous phase type (CPH) distributions include well-known distributions like exponential, Erlang and hyperexponential. CPH distributions have very nice closure properties and the fact that they need mostly matrix formalism makes them attractive for use in practice.

This chapter is organized as follows. In section 4.1, we discuss CPH distributions by providing the basics and key closure properties. The CPH renewal process, which plays a key role in this volume as well as in Volume 2, is presented in section 4.2. Both sections have several illustrative examples to further strengthen the understanding of the concepts as these are part of the building blocks for matrix-analytic methods (MAM).

4.1. Continuous phase type (CPH) distribution

At this stage, it is highly encouraged that readers review basic concepts and results from continuous-time Markov chain (CTMC).

Consider an irreducible CTMC, {Y_t : t ≥ 0}, with m transient states and one absorbing state, m + 1, with generator Q defined as:

[4.1]

where the column vector T⁰ is such that:

[4.2]

Suppose that α is the initial probability vector of the CTMC. That is, α_i = P(Y₀ = i), 1 ≤ i ≤ ...