Sojourn time in a processor sharing queue with service interruptions

We study the sojourn time of customers in an $M/M/1$ queue with processor sharing service discipline and service interruptions. The lengths of the service interruptions have a general distribution, whereas the periods of service availability are assumed to have an exponential distribution. A branching process approach is shown to lead to a decomposition of the sojourn time into independent contributions, that can be investigated separately. The Laplace-Stieltjes Transform of the distribution of the sojourn time is found through an integral equation. We derive the first two moments of the sojourn time conditioned on the amount of work brought into the system and on the number of customers present upon arrival. We show that the expected sojourn time of a customer that arrives at the system in steady state is not linear in the amount of work he brings with him. Finally, we show that the sojourn time conditioned on the amount of work, scaled by the traffic load, converges in heavy traffic to an exponential distribution. This study was motivated by a need for delay analysis of elastic traffic in modern communication networks. Specifically, the results are of interest for the performance analysis of the Available Bit Rate (ABR) service class in ATM networks, as well as for the best-effort services in IP networks.