Asymptotic expansions for the conditional sojourn time distribution in the M/M/1-PS queue
We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152-167, [1985]) and Flatto (Ann. Appl. Probab. 7:382-409, [1997]).
asymptotics sojourn time m/m/1 queue processor sharing