Approximating a Point Process by a Renewal Process: The View Through a Queue, an Indirect Approach

This paper investigates simple approximations for stochastic point processes. As in several previous studies, the approximating process is a renewal process characterized by the first two moments of the renewal interval. The approximating renewal-interval distribution itself is a convenient distribution with these two moments; it is constructed from exponential building blocks, e.g., the hyperexponential distribution. Here the moments of the renewal interval are chosen to produce the same level of congestion when the renewal process serves as an arrival process in a test queueing system as is produced when the general point process is the arrival process. The procedure can be applied to predict the behavior of a new service mechanism in a queueing system with a complicated arrival process; then we use the system with the old service mechanism as the test system. But the test system can also be an artificial device to approximate any point process. This indirect approximation procedure extends the equivalent random method and related techniques widely used in teletraffic engineering.