Approximating a Point Process by a Renewal Process, I: Two Basic Methods

This paper initiates an investigation of simple approximations for stochastic point processes. The goal is to develop methods for approximately describing complex models such as networks of queues and multiechelon inventory systems. The proposed approach is to decouple or decompose the model by replacing all the component flows (point processes) by independent renewal processes. Here attention is focused on ways to approximate a single point process by a renewal process. This is done in two steps: First, properties of the point process are used to specify a few moments of the interval between renewals; then a convenient distribution is fit to these moments. Two different methods are suggested for specifying the moments of the renewal interval. The stationary-interval method equates the moments of the renewal interval with the moments of the stationary interval in the point process to be approximated. The asymptotic method, in an attempt to account for the dependence among successive intervals, determines the moments of the renewal interval by matching the asymptotic behavior of the moments of the sums of successive intervals. These two procedures are applied to approximate the superposition (merging) of point processes. The purpose here is to provide a better understanding of these procedures and a general framework for making new approximations. In particular, the two basic procedures can be used as building blocks to construct refined composite procedures. Composite procedures for the ∑ G i / G /1 queue (with a superposition arrival process) are discussed by Albin in Part II. Albin has developed a hybrid procedure for approximating the mean sequence length and other characteristics in the ∑ G i / G /1 queue for which the average error when compared with simulated values was 3% over a large number of test systems.