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

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 heavytailed 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-α scheduling policies; we show that if the α-parameters are chosen suitably, then the sum of the α-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.

Delay Stability of Back-Pressure Policies in the Presence of Heavy-Tailed Traffic

“…Since the network has single-hop traffic, the Back-Pressure policy reduces to Max-Weight scheduling. The findings of [9] imply that the light-tailed flow is delay unstable.…”

“…It is not hard to see that this model is equivalent to a single-server system of two parallel queues, where the Back-Pressure policy reduces to MaxWeight scheduling. Then, Theorem 2 of [9] implies that the light-tailed flow 2 is delay unstable. The main idea behind this result is the following: queue (1, 1) is, occasionally, very long due to the heavy-tailed arrivals that it receives exogenously.…”

“…Moreover, an arrival rate vector λ belongs to the stability region Λ if there exists a policy under which the network is stable. An explicit characterization of the stability region of the above multi-hop network can be found in [8].…”

“…Closer to the present paper comes our earlier work [9], which studies the delay stability of Max-Weight policies (the single-hop equivalent of Back-Pressure) in networks with a mix of heavy-tailed and light-tailed traffic. There, the decision problem is "one-dimensional," boiling down to a linkscheduling problem: which subset of servers/communication links to activate at any given time slot.…”

“…This can be interpreted as a joint scheduling and routing decision. Since the link-scheduling part of the decision problem has been analyzed extensively in [9], and in the follow-up work [10], here we consider a wireline multi-hop network, where only the flow-scheduling part remains relevant. Thus, in the present paper we focus only on phenomena and insights that arise due to the multi-hop nature of the network and the possibility of multiple source-destination routes.…”