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Asymptotic Expansions for the Sojourn Time Distribution in the $M/G/1$-PS Queue

We consider the $M/G/1$ queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, as well as the unconditional distribution, 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. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases $G=M$ and $G=D$ (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.