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-alpha 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 heavier than a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also contains an intuitively surprising result - the celebrated max-weight scheduling policy leads to the worst possible tail of the light queue distribution, among all non-idling policies. Motivated by the above negative result regarding the max-weight-alpha 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.

💡 Research Summary

The paper studies a two‑queue system served by a single server, where one queue receives heavy‑tailed traffic and the other receives light‑tailed traffic. The authors focus on the family of throughput‑optimal max‑weight‑α scheduling policies, which assign service to the queue with the larger weight (W_i(t)=Q_i(t)^{\alpha_i}). While such policies are known to stabilize the system for any admissible arrival rates, the paper reveals a striking and counter‑intuitive property of the steady‑state queue‑length distribution when heavy‑tailed traffic is present.

Using Lyapunov‑drift arguments and large‑deviation analysis, the authors derive an exact asymptotic expression for the tail of the light‑queue length distribution. If the heavy‑tailed arrival process has tail index (\beta_H\in(1,2]), the light‑queue tail behaves as a power law (P(Q_L>x)\sim x^{-\gamma}) with (\gamma) explicitly expressed in terms of (\beta_H) and the α‑parameters. Crucially, when (\alpha_L=\alpha_H=1) – i.e., the classic max‑weight rule – the exponent (\gamma) attains its smallest possible value, meaning the light queue’s tail is the heaviest among all non‑idling policies. In other words, the celebrated max‑weight algorithm yields the worst possible tail for the light queue, contradicting the usual belief that max‑weight is universally “best”.

The authors also examine whether tuning the α‑parameters can improve the tail. While adjusting α can modestly increase the exponent, the tail remains a power law; the fundamental heavy‑tailed influence cannot be eliminated by any choice of α within the max‑weight‑α family.

Motivated by this negative result, the paper proposes a Log‑Max‑Weight (LMW) policy. The LMW weight is defined as (W_i(t)=\log(1+Q_i(t))) (or a power of this log). This logarithmic transformation dampens the growth of the weight for large queues, preventing the heavy‑tailed queue from monopolizing service. The authors prove that LMW still achieves throughput optimality and, more importantly, that the light‑queue tail decays exponentially: (P(Q_L>x)\le C,e^{-\theta x}) for some positive constants (C,\theta). This exponential bound is dramatically tighter than any power‑law bound obtainable under max‑weight‑α.

The theoretical findings are corroborated by extensive Monte‑Carlo simulations across a range of heavy‑tail indices and α values. The simulations confirm that (i) classic max‑weight produces the heaviest light‑queue tail, (ii) varying α yields only marginal improvements, and (iii) LMW consistently achieves an exponentially decaying tail while preserving stability.

From a practical perspective, the results have immediate relevance for data‑center switches, cloud‑computing schedulers, and wireless networks where heavy‑tailed traffic (e.g., large file transfers, video streams, backup bursts) coexists with latency‑sensitive flows. The paper demonstrates that relying solely on max‑weight for throughput can expose the system to severe tail‑risk, potentially leading to large delays or buffer overflows for the light‑tailed traffic. By adopting a logarithmic weighting scheme, system designers can retain the simplicity and throughput guarantees of max‑weight while dramatically reducing the probability of extreme queue buildups.

In summary, the paper makes three key contributions: (1) an exact asymptotic characterization of the light‑queue tail under the entire class of max‑weight‑α policies, (2) the proof that the classic max‑weight rule yields the worst possible tail among all non‑idling policies, and (3) the introduction of the Log‑Max‑Weight policy, which guarantees exponential tail decay and throughput optimality. The work opens several avenues for future research, including extensions to multiple servers, more complex network topologies, and adaptive schemes that dynamically adjust the logarithmic scaling based on observed traffic statistics.