Chapter 2: Poisson processes

Abstract

A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3.5. For the Bernoulli process, the arrivals can occur only at positive integer multiples of some given increment size (often taken to be 1). Section 1.3.5 characterized the process by a sequence of IID binary random variables (rv's), Y 1 , Y 2 ,. .. , where Y i = 1 indicates an arrival at increment i and Y i = 0 otherwise. We observed (without any careful proof) that the process could also be characterized by the sequence of interarrival times. These interarrival times are geometrically distributed IID rv's. For the Poisson process, arrivals may occur at arbitrary positive times, and the probability of an arrival at any particular instant is 0. This means that there is no very clean way of describing a Poisson process in terms of the probability of an arrival at any given instant. It is more convenient to define a Poisson process in terms of the sequence of interarrival times, X 1 , X 2 ,. .. , which are defined to be IID. Before doing this, we describe arrival processes in a little more detail. 2.1.1 Arrival processes An arrival process is a sequence of increasing rv's, 0 < S 1 < S 2 < , where 1 S i < S i+1 · · · means that S i+1 S i is a positive rv, i.e., a rv X such that F X (0) = 0. The rv's S 1 , S 2 ,. .. , are called arrival epochs (the word time is somewhat overused in this subject) and represent the times at which some repeating phenomenon occurs. Note that the process starts at time 1 These rv's Si can be viewed as sums of interarrival times. They should not be confused with the rv's Si used in Section 1.3.5 to denote the number of arrivals by time i for the Bernoulli process. We use Si throughout to denote the sum of i rv's. Understanding how such sums behave is a central issue of every chapter (and almost every section) of these notes. Unfortunately, for the Bernoulli case, the IID sums of primary interest are the sums of binary rv's at each time increment, whereas here the sums of primary interest are the sums of interarrival intervals. 69