Tail asymptotics for discriminatory processor-sharing queues with heavy-tailed service requirements

We derive the sojourn time asymptotics for a multi-class GI/GI/1 queue with regularly varying service requirements operating under the discriminatory processor-sharing (DPS) discipline. DPS provides a natural approach for modelling the flow-level performance of differentiated bandwidth-sharing mechanisms. Under certain assumptions, we prove that the service requirement and sojourn time of a given class have similar tail behaviour, independent of the specific values of the DPS weights. As a by-product, we obtain an extension of the tail equivalence for ordinary processor-sharing (PS) queues to non-Poisson arrivals. The results suggest that DPS offers a potential instrument for effectuating preferential treatment to high-priority classes, without inflicting excessive delays on low-priority classes. To obtain the asymptotics, we develop a novel method which only involves information of the workload process and does not require any knowledge of the steady-state queue length distribution. In particular, the proof method brings sufficient strength to extend the results to scenarios with a time-varying service capacity.