Point and Renewal Processes

In the previous chapter, we saw one way to analyze a series of events in time, by conditioning on what happened before. Each event occurs in a given constant period of time. In fact, these periods were assumed small enough so that only one event, however defined, can occur in each period. Such data can also be studied in two further ways. If we group the data into longer periods, we can look at the rate of occurrence of the events. However, in such a case, we lose information since we also have available the actual times between successive events. Often, such varying intervals of time can be considered to be independently and identically distributed. In such a case, we have a renewal process. This term comes from industry where certain machines or parts must be replaced or renewed at varying intervals of time. But models for renewal processes have much wider application.