Queue-Length Asymptotics for Generalized Max-Weight Scheduling in the Presence of Heavy-Tailed Traffic

We investigate the asymptotic behavior of the steady-state queue-length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput-optimal max-weight-<formula formulatype="inline"><tex Notation="TeX">$\alpha $</tex> </formula> scheduling policies and derive an exact asymptotic characterization of the steady-state queue-length distributions. In particular, we show that the tail of the light queue distribution is at least as heavy as a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also shows that the celebrated max-weight scheduling policy leads to the worst possible tail coefficient of the light queue distribution, among all nonidling policies. Motivated by the above negative result regarding the max-weight-<formula formulatype="inline"><tex Notation="TeX">$\alpha $</tex></formula> policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail while still being throughput-optimal.

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