Max-Weight Scheduling in Queueing Networks with Heavy-Tailed Traffic

TL;DR

This paper analyzes how heavy-tailed traffic impacts delay stability in queueing networks under Max-Weight scheduling, revealing that heavy-tailed flows are delay unstable and that certain scheduling modifications can improve stability.

Contribution

It demonstrates that heavy-tailed traffic causes delay instability and proposes Max-Weight-a policies with adjustable parameters to improve queue length moments.

Findings

Heavy-tailed traffic is delay unstable under any policy.

Light-tailed flows conflicting with heavy-tailed flows are delay unstable.

Max-Weight-a policies can ensure finite moments of queue lengths.

Abstract

We consider the problem of packet scheduling in single-hop queueing networks, and analyze the impact of heavy-tailed traffic on the performance of Max-Weight scheduling. As a performance metric we use the delay stability of traffic flows: a traffic flow is delay stable if its expected steady-state delay is finite, and delay unstable otherwise. First, we show that a heavy-tailed traffic flow is delay unstable under any scheduling policy. Then, we focus on the celebrated Max-Weight scheduling policy, and show that a light-tailed flow that conflicts with a heavy-tailed flow is also delay unstable. This is true irrespective of the rate or the tail distribution of the light-tailed flow, or other scheduling constraints in the network. Surprisingly, we show that a light-tailed flow can be delay unstable, even when it does not conflict with heavy-tailed traffic. Furthermore, delay stability in this case may depend on the rate of the light-tailed flow. Finally, we turn our attention to the class of Max-Weight-a scheduling policies; we show that if the a-parameters are chosen suitably, then the sum of the a-moments of the steady-state queue lengths is finite. We provide an explicit upper bound for the latter quantity, from which we derive results related to the delay stability of traffic flows, and the scaling of moments of steady-state queue lengths with traffic intensity.