Over the past four decades, the processor-sharing (PS) discipline has gained a prominent role within queueing theory. PS queues have been recently used as flow-level models for the Internet, and this has led to a renewed interest in the tail behavior of the sojourn time distribution. The sojourn time is the time a given (also called tagged) customer leaves the system after being served, assuming the customer arrives at time zero.

Two of the main measures of performance in a PS queueing system are a tagged customer's sojourn time distribution conditioned on that customer's service time, or conditioned on the number of other customers in the system when the tagged customer arrives. We consider M/ M/1-PS model as well as the M/G/1-PS model. We obtain the conditional sojourn time distribution in various asymptotic limits, which include large time and/or large service request, and/or large number of customers present, and heavy traffic.

In the M/M/1-PS queue, our results lead to a better understanding of what ranges of service time or number of customers present are important for removing the conditions. In the M/G/1-PS queue, we assume that the service density decays at least exponentially fast. 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).