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

We consider the $M\mathrm{/}M\mathrm{/}1$-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.