Finite-Time Minimax Bounds and an Optimal Lyapunov Policy in Queueing Control

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arXiv:2506.18278v4 Announce Type: replace-cross Abstract: We introduce an original minimax framework for finite-time performance analysis in queueing control and propose a surprisingly simple Lyapunov-based scheduling policy with superior finite-time performance. The framework quantitatively characterizes how the expected total queue length scales with key system parameters, including the capacity of the scheduling set and the variability of arrivals and departures across queues. This characterization provides a systematic quantitative basis for evaluating and comparing scheduling policies in the finite-time regime, including nonstationary settings under certain assumptions on the model, and shows that the proposed policy provably and empirically outperforms the classical MaxWeight strategy in finite time. Within this framework, we establish three main sets of results. First, we derive minimax lower bounds on the expected total queue length for parallel-queue scheduling via a novel Brownian coupling argument. Second, we propose a new policy, LyapOpt, which minimizes the full quadratic Lyapunov drift-capturing both first- and second-order terms-and achieves optimal finite-time performance under the dominated region condition in heavy traffic while retaining classical stability guarantees. Third, we identify a key limitation of the classical MaxWeight policy, which optimizes only the first-order drift: its finite-time performance depends suboptimally on system parameters, leading to substantially larger backlogs in explicitly characterized settings. Together, these results …

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