We wish now to utilize our knowledge of pmf's, pdf's, and probabilistic modeling to analyze a very important process in urban services: the Poisson process. As mentioned earlier in Section 2.9, when introducing the Poisson pmf, the Poisson process is most often applied to occurrences of events in time. In an urban services context, these events could be requests for service, breakdowns of equipment, arrivals of vehicles at an intersection, or any of numerous other entities. So as not to confuse a Poisson event with events having an algebra (e.g., union and intersection), hereafter we will refer to Poisson-type events as arrivals, such as customers arriving at a queue. As we will see later, the concepts of a Poisson process can be extended to spatial applications in order to model, for instance, the locations of demands for service throughout a city.

First, we list the postulates of a Poisson process so that we can see the underlying physical assumptions necessary to give rise to the process. As we will see in Chapter 3, these postulates carry over in a natural way to spatial applications.