Queue Tests for Renewal Processes

In many operations research applications we want to know if a stochastic point process (series of events) can be modeled as a renewal process or a Poisson process. In other words, we want to know if the intervals between points are approximately lid (independent and identically distributed) and, if so, whether they are approximately exponentially distributed. For example, we may be considering arrivals to a queue, demands for inventory, or the occurrence of failures. Whether we start with data from a physical system or a mathematical model we want to know if the point process can be modeled as a renewal process or a Poisson process. Then we can do further work with a more elementary analytic model or a more elementary simulation. Of course, there are standard ways to analyze these problems. With data, we can apply standard statistical tests for renewal processes and Poisson processes; see Cox and Lewis [6, Chapter 6]. With mathematical models, we can do further analysis to see whether the point process of interest is actually renewal or Poisson; e.g., Disney, Farrel and DeMorais [7] and Melamed [I i] have characterized when flows in a network of queues are renewal or Poisson. Alternatively, with mathematical models we can simulate, collect data and then perform the standard statistical tests. The purpose of this paper is to suggest a different approach. We suggest using a queuing model to test whether a point process can be adequately modeled as a renewal process or a Poisson process. (This may be in addition to other statistical tests.) The idea is to analyze the queuing model using the process of interest to generate the arrivals (or the service times). If we have data, then we use that data to generate arrivals to the queue in a simulation; otherwise, we either solve the queuing model analytically or simulate it. Then we compare vari. ous congestion measures with theoretical results in the case of a renewal or Poisson arrival process. In this approach, the queuing model is an artificial device introduced to perform the test. The actual application may have nothing to do with queues. Even though we consider only queuing models, other models (e.g., inventory and reliability) could also be used to perform the tests. Obviously, however, it Should be desirable to perform the test with a model closely related to the intended application. To make the motivation clear, consider the case of a stationary departure process from an M / G / I queue in equilibrium. The theory may demonstrate that the process is actually not renewal [7]. However, in many applications we will be satisfied if the process is only approximately renewal. Of course, we can apply the standard statistical tests, but how should we interpret the results? With enough data, we would reject the renewal hypothesis. Obviously what matters is how the de-