Renewal approximations of the doubly stochastic Poisson processes

Abstract

In this paper, we analyze two approaches to approximate a doubly stochastic Poisson (DSP) process by a renewal process. A DSP process consists of Poisson processes whose rates alternate between two levels. Each rate remains for renewal times forming an alternating renewal process. Such processes can be used to model situations in reliability, inventory, queueing and production systems. We develop two expressions for the Laplace-Stieltjes transforms of the interrenewal time distributions of the renewal processes that approximate a DSP process. We then use these approximations to develop expressions for the stationary loss probability in a G/M/1/0 system. We evaluate the quality of these approximations by comparing them against exact results. Our approximations should be of significant use in several practical applications.