On Sojourn Times in the Finite Capacity $M/M/1$ Queue with Processor Sharing

TL;DR

This paper derives an exact Laplace transform expression for the sojourn time distribution in a finite capacity processor shared M/M/1 queue and analyzes its tail behavior as capacity grows large.

Contribution

It provides a novel exact formula for sojourn times in finite capacity queues and characterizes tail behavior in the limit of infinite capacity.

Findings

Exact Laplace transform for sojourn time distribution

Asymptotic tail behavior as capacity approaches infinity

Insights into queue performance limits

Abstract

We consider a processor shared $M/M/1$ queue that can accommodate at most a finite number $K$ of customers. We give an exact expression for the sojourn time distribution in the finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit $K\to\infty$, by locating the dominant singularity of the Laplace transform.