On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users

TL;DR

This paper derives a new formula for the conditional sojourn time distribution in an $M/M/1$ processor sharing queue, analyzing its behavior under various asymptotic conditions and connecting it to classic results from the 1940s.

Contribution

It introduces a novel formula for the conditional sojourn time distribution using a discrete Green's function, linking it to historical results and exploring asymptotic behaviors.

Findings

New formula for conditional sojourn time distribution

Asymptotic analysis under large time and heavy traffic

Connection to classic Pollaczeck and Vaulot results

Abstract

We consider the $M/M/1$-PS queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green's function. This is shown to be equivalent to some classic results of Pollaczeck and Vaulot from 1946. Then various asymptotic limits are studied, including large time and/or large number of customers present, and heavy traffic, where the arrival rate is only slightly less than the service rate.