Abstract-We consider a multi-class queueing system operating under the Discriminatory Processor-Sharing (DPS) discipline. The DPS discipline provides a natural approach for modeling the flow-level performance of differentiated bandwidth-sharing mechanisms. Motivated by the extreme diversity in flow sizes observed in the Internet, we examine the system performance in an asymptotic regime where the flow dynamics of the various classes occur on separate time scales. Specifically, from the perspective of a given class, the arrival and service completions of some of the competing classes (called mice) evolve on an extremely fast time scale. In contrast, the flow dynamics of the remaining classes (referred to as elephants) occur on a comparatively slow time scale. Assuming a strict separation of time scales, we obtain simple explicit expressions for various performance measures of interest, such as the distribution of the numbers of flows, mean delays, and flow throughputs. In particular, the latter performance measures are insensitive, in the sense that they only depend on the service requirement distributions through their first moments. Numerical experiments show that the limiting results provide remarkably accurate approximations in certain cases 1 .

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We study the joint queue length distribution of the Discriminatory Processor Sharing model, assuming all classes have phase-type service requirement distributions. We show that the moments of the joint queue length distribution can be obtained by solving linear equations. We use this to study the system in two asymptotic regimes. In the first regime, the different user classes operate on strictly separated time scales. Then we study the system in heavy traffic.

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