Counting distributions for renewal processes

In works on classical renewal theory (for example, Cox, 1962), the basic ingredient is the distribution of life-times, or failure-times. This function, representing the stochastic duration of the article subject to renewal, can be conveniently measured in the specific applications which led to the development of the theory. From the abstract point of view, however, the importance of renewal processes goes far beyond the problems which contributed the name 'renewal'. In the family of point processes, a renewal process represents the first large step away from a Poisson process; being defined by adding together independent equi-distributed random variables, it is a process of natural simplicity. When we formulate probability models for special point processes, it is reasonable, there- fore, to consider various renewal processes, regardless of whether or not the renewal model has any relevance to the problem at hand. The present study is motivated by two such examples in which the distribution of life-times is theoretically or practically unknowable. The first example is accident distribution: accidents are given in statistical sources as so many per hour, or per day, or per year. If one wanted to measure the quantity analogous to life-time, one would have to record the exact instant of every accident, and most of the data from the past, or otherwise beyond the control of the statistician, would be useless. The other example is the instant in time, or position in space, of vehicles. The quantity corresponding to life-time is the time, or distance, between consecutive vehicles—a quantity difficult to measure without sophisticated instrumentation. As in the first example, the more convenient statistic (and the one which is usually given by traffic authorities) is the number of vehicles in a fixed time period or fixed length of roadway. We shall call the probability distributions formed by the number of events in a fixed interval counting distributions. The main purpose of this paper is to isolate those discrete distributions which can possibly serve as counting distributions for renewal processes. In developing the results, we will use terminology and notation which reflect a point of view somewhat more general than that of the original 'renewal-motivated' renewal theory.