Poisson and Related Processes

Most of the applications discussed are based on (generalizations of) the Poisson process. Most of our applications are based on (generalizations of) the Poisson process. There is a very good reason for this: the real world is a continuous-time one, and events in computer and communication systems are generally discrete valued: a message is sent, a task is completed etc. The simplest continuous-time, discrete-state-space stochastic process is the Poisson process. The chapter also presents some special tricks that apply to Poisson processes. While the statements of the theorems are precise, the derivation is intentionally heuristic. The purpose of this chapter is to provide sufficient information about these processes so that later chapters can be understood. A precise and more complete treatment is available. One of the most useful consequences of the introduction of generators is that they allow us to generate martingales that are related to the jump Markov processes of interest.