Bayesian Analysis of Stochastic Process Models

5

Poisson processes and extensions

5.1 Introduction

Poisson processes are one of the simplest and most applied types of stochastic processes. They can be used to model the occurrences (and counts) of rare events in time and/or space, when they are not affected by past history. In particular, they have been applied to describe and forecast incoming telephone calls at a switchboard, arrival of customers for service at a counter, occurrence of accidents at a given place, visits to a web site, earthquake occurrences, and machine failures, to name but a few applications. Poisson processes are a special case of continuous time Markov chains, described in Chapter 4, in which jumps are possible only to the next higher state. They are also a particular case of birth–death processes, introduced in Example 4.1, that is, pure birth processes, as well as being the model for the arrival process in M/G/c queueing systems presented in Chapter 7. The simple mathematical formulation of the Poisson process, along with its relatively straightforward statistical analysis, makes it a very practical, if approximate, model for describing and forecasting many random events.

After introducing the basic concepts and results in Section 5.2, homogeneous and nonhomogeneous Poisson processes are analyzed in Sections 5.3 and 5.4, respectively. Compound Poisson processes are presented in Section 5.5 and other related processes are discussed in Section 5.6. A case study based on the analysis of earthquake data ...