Approximating a Point Process by a Renewal Process, II: Superposition Arrival Processes to Queues

Abstract

We develop an approximation for a queue having an arrival process that is the superposition of independent renewal processes, i.e., ∑GI1/G/1. This model is useful, for example, in analyzing networks of queues where the arrival process to an individual queue is the superposition of departure processes from other queues. If component arrival processes are approximated by renewal processes, the ∑GI1/G/1 model applies. The approximation proposed is a hybrid that combines two basic methods described by Whitt. All these methods approximate the complex superposition process by a renewal process and yield a GI/G/1 queue that can be solved analytically or approximately. In the hybrid method, the moments of the intervals in the approximating renewal process are a convex combination of the moments determined by the basic methods. The weight in the convex combination is identified using the asymptotic properties of the basic methods together with simulation. When compared to simulation estimates, the error in hybrid approximations of the expected number in the queue is 3%; in contrast, the errors of the basic methods are 20–30%. The quality of the approximations suggests that the hybrid approach would be useful in approximating point processes in other contexts.