Chapter 2POISSON PROCESSES2.1 IntroductionA Poisson process is a simple and widely used sto chastic process for modeling the times atwhich arrivals enter a system. It is in many ways the continuous-time version of the Bernoulliprocess that was described in Section 1.3.5. For the Bernoulli process, the arrivals can occuronly at positive integer multiples of some given increment size (often taken to be 1). Section1.3.5 characterized the Bernoulli process by a sequence of IID binary random variables(rv’s), Y1, Y2, . . . , where Yi= 1 indicates an arrival at increment i and Yi= 0 otherwise.We observed (without any careful proof) that the process could also be characterized by asequence of geometrically distributed interarrival times.For the Poisson process, arrivals may occur at arbitrary positive times, and the probabilityof an arrival at any particular instant is 0. This means that there is no very clean way ofdescribing a Poisson process in terms of the probability of an arrival at any given instant. Itis more convenient to define a Poisson process in terms of the sequence of interarrival times,X1, X2, . . . , which are defined to be IID. Before doing this, we describe arrival processes ina little more detail.2.1.1 Arrival processesAn arrival process is a sequence of increasing rv’s, 0 < S1< S2< ···, where Si< Si+1means that Si+1Siis a positive rv, i.e., a rv X such that FX(0) = 0. The rv’s S1, S2, . . . ,are called arrival epochs1and represent the successive times at which some random rep eatingphenomenon occurs. Note that the process starts at time 0 and that multiple arrivalscan’t occur simultaneously (the phenomenon of bulk arrivals can be handled by the simpleextension of associating a positive integer rv to each arrival). We will sometimes permitsimultaneous arrivals or arrivals at time 0 as events of zero probability, but these can beignored. In order to fully specify the process by the sequence S1, S2, . . . of rv’s, it is necessaryto specify the joint distribution of the subsequences S1, . . . , Snfor all n > 1.1Epoch is a synonym for time. We use it here to avoid overusing the word time in overlapping ways.7374 CHAPTER 2. POISSON PROCESSESAlthough we refer to these processes as arrival processes, they could equally well modeldepartures from a system, or any other sequence of incidents. Although it is quite common,esp ecially in the simulation field, to refer to incidents or arrivals as events, we shall avoidthat here. The nth arrival epoch Snis a rv and {Sn t}, for example, is an event. Thiswould make it confusing to refer to the nth arrival itself as an event.rrrN(t) = n for Sn t < Sn+1X1-X2-X3-6tN(t)0 S1S2S3Figure 2.1: A sample function of an arrival process and its arrival epochs {S1, S2, . . . },its interarrival timess {X1, X2, . . . }, and its counting process {N(t); t > 0}As illustrated in Figure 2.1, any arrival process {Sn; n  1} can also be specified by eitherof two alternative stochastic processes. The first alternative is the sequence of interarrivaltimes, {Xi; i  1}. The Xihere are positive rv’s defined in terms of the arrival epochs byX1= S1and Xi= Si Si1for i > 1. Similarly, each arrival epoch Snis specified by{Xi; i  1} asSn=Xni=1Xi. (2.1)Thus the joint distribution of X1, . . . , Xnfor all n > 1 is sufficient (in principle) to specifythe arrival process. Since the interarrival times are IID in most cases of interest, it is usuallysimpler to specify the joint distribution of the Xithan of the Si.The second alternative is the counting process {N(t); t > 0}, where for each t > 0, the rvN(t) is the aggregate number of arrivals2up to and including time t.The counting pro cess {N(t); t > 0}, illustrated in Figure 2.1, is an uncountably infinitefamily of rv’s where the counting rv N(t), for each t > 0, is the aggregate number ofarrivals in the interval (0, t], i.e., the number of arrivals in the set {⌧ : 0 < ⌧  t}. Byconvention, a parenthesis is used to indicate that the corresponding end point of an intervalis not contained in the interval and a bracket is used if it is. Thus, as another example,(a, b) = {⌧ : a < ⌧ < b}. The rv N(0) is defined to be 0 with probability 1, which meansthat we are considering only arrivals at strictly positive times.The counting process {N(t); t > 0} for any arrival process has the property that N(⌧) N(t) for all ⌧  t > 0 (i.e., N(⌧)  N (t) is a nonnegative random variable for ⌧  t > 0).For any given integer n  1 and time t > 0, the nth arrival epoch, Sn, and the countingrandom variable, N(t), are related by{Sn t} = {N(t)  n}. (2.2)2Thus, for the Bernoulli process with an increment size of 1, N (n) is the rv denoted as Snin Section1.3.52.2. DEFINITION AND PROPERTIES OF A POISSON PROCESS 75To see this, note that {Sn t} is the event that the nth arrival occurs at some epoch ⌧  t.This event implies that N(⌧) = n, and thus that {N(t)  n}. Similarly, {N(t) = m} forsome m  n implies {Sm t}, and thus that {Sn t}. This equation is essentially obviousfrom Figure 2.1, but is one of those peculiar obvious things that is often difficult to see. Analternate form, which is occasionally more transparent, comes from taking the complementof both sides of (2.2), getting{Sn> t} = {N(t) < n}. (2.3)For example, the event {S1> t} means that the first arrival occurs after t, which means{N(t) < 1} (i.e., {N(t) = 0}). These relations will be used constantly in going back andforth between arrival epochs and counting rv’s. In principle, (2.2) or (2.3) can be used tospecify joint distribution functions of arrival epochs in terms of joint distribution functionsof counting variables and vice versa, so either characterization can be used to specify anarrival process.In summary, then, an arrival process can be specified by the joint distributions of the arrivalepochs, or of the interarrival times, or of the counting rv’s. In principle, specifying any oneof these specifies the others also.32.2 Definition and properties of a Poisson processThe nicest, and in many ways most important, arrival processes are those for which theinterarrival times are IID. These arrival processes are called renewal processes and form thesubject of Chapter 5.Poisson processes are the nicest, and in many ways most important, class of renewal pro-cesses. They are characterized by having exponentially distributed interarrival times. Wewill soon see why these exponentially