Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

Abstract

We analyze a generalization of the Discriminatory Processor Sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume phase-type distributed service requirements and allow that customers have different weights in various phases of their service. We establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by [12] who considered a DPS queue with exponentially distributed service requirements. We finally discuss some implications for residual service requirements and monotonicity properties in the ordinary DPS model.