On Sojourn Times in the Finite Capacity $M\mathrm{/}M\mathrm{/}1$ Queue with Processor Sharing

We consider a processor shared $M\mathrm{/}M\mathrm{/}1$ 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 $K\to \mathrm{\infty }$, by locating the dominant singularity of the Laplace transform.