Sojourn time distribution in some processor-shared queues

We develop a technique for obtaining asymptotic properties of the sojourn time distribution in processor-sharing queues. We treat the standard M/M/1-PS queue and its finite capacity version, the M/M/1/K-PS queue. Using perturbation methods, we construct asymptotic expansions for the distribution of a tagged customer's sojourn time, conditioned on that customer's total required service. The asymptotic limit assumes that (i) the traffic intensity is close to one for the infinite capacity model, and (ii) that the system's capacity is large for the finite capacity queue.