Monotonicity Properties for Multi-Class Queueing Systems

We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.