An approximation for the sojourn-time distribution for the gi/g/1 processor-sharing queue

Abstract

In this paper, we propose an approximation for the sojourn-time distribution for the GI/G/1 processor-sharing queue and derive the heavy traffic theory for the special case of the M/G/1 queue. The main motivation for proposing this approximation is that existing known exact methods for the M/G/1 and GI/M/1 cases are far too complicated for use by practitioners who may want numerical information about the distribution. The approximation consists of expressing the sojourn-time as the product of two independent random variables. One of them is the service time and the other has the Gamma distribution. The approximation relies on the heavy traffic theory for the M/G/1 and GI/M/1 cases. We show that this distribution is exact in many limiting cases (such as heavy and light traffic) and has some other desirable properties. We compare the approximation against simulations and show that it performs very well.