A Stochastic Mean-Value Method for the Derivation of Delay Asymptotics in Heavy-Tailed Processor-Sharing Systems

We develop a stochastic mean-value method for the derivation of delay asymptotics in Processor-Sharing (PS) type systems with heavy-tailed service requirements. In order to demonstrate the strength of the approach, we apply the method to obtain the sojourn time asymptotics for a multi-class G/G/1 queue operating under the Discriminatory Processor-Sharing (DPS) discipline. Besides interesting from a queueing-theoretic perspective, DPS is also of practical relevance as it provides a useful paradigm for modelling the flow-level performance of differentiated resource-sharing mechanisms. Unlike for ordinary PS, however, the queue length for DPS does not have a simple distribution, and there are no manageable transform results available for the sojourn time. These circumstances seriously complicate the derivation of delay asymptotics using existing proof methods, and render DPS as a good ‘test case’ for judging the merits of alternative approaches. We use the stochastic mean-value method to show that under certain assumptions, the service requirement and sojourn time of a given class have similar tail behaviour, independent of the specific values of the DPS weights. The results suggest that DPS offers a useful instrument for effectuating preferential treatment to smaller service demands without inflicting excessive delays on larger requests. We also briefly discuss the potential applicability of the method for deriving the delay asymptotics under a broader class of resource-sharing strategies.