Poisson Processes

Abstract

Poisson processes are used extensively in applied probability models. Their importance is due to their versatility for representing a variety of physical processes, and because a Poisson process is a natural model for a sum of many sparse point processes. The most basic Poisson process, introduced in the preceding chapter, is a renewal process on the time axis with exponential inter-renewal times. This type of process is useful for representing times at which an event occurs, such as the times at which items arrive to a network, machine components fail, emergencies occur, a stock price takes a large jump, etc. The first part of the present chapter continues the discussion of this basic Poisson process by presenting several characterizations of it, including the result that its point locations (i.e., occurrence times) on a finite time interval are equal in distribution to order-statistics from a uniform distribution on the interval.