Scheduling strategies and long-range dependence

Abstract

Consider a single server queue with unit service rate fed by an arrival process of the following form: sessions arrive at the times of a Poisson process of rate \lambda, with each session lasting for an independent integer time \tau \geq 1, where P(\tau = k ) = p_k with p_k \sim \alpha k^{-(1 +\alpha)}L(k), where 1<\alpha<2 and L(\cdot) is a slowly varying function. Each session brings in work at unit rate while it is active. Thus the work brought in by each arrival is regularly varying, and, because 1 < \alpha < 2, the arrival process of work is long-range dependent. Assume that the stability condition \lambda E[\tau] < 1 holds. By simple arguments we show that for any stationary nonpreemptive service policy at the queue, the stationary sojourn time of a typical session must stochastically dominate a regularly varying random variable having infinite means this is true even if the duration of a session is known at the time it arrives. On the other hand, we show that there exist causal stationary preemptive policies, which do not need knowledge of the session durations at the time of arrival, for which the stationary sojourn time of a typical session is stochastically dominated by a regularly varying random variable having finite mean. These results indicate that scheduling policies can have a significant influence on the extent to which long-range dependence in the arrivals influences the performance of communication networks.