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

TL;DR

The paper tackles finite-time scheduling in parallel queueing systems under nonstationary, uncertain workloads. It introduces a minimax framework that bounds the best possible finite-horizon backlog over a broad model class and proves a fundamental lower bound via Brownian coupling. Building on this framework, it designs LyapOpt, a policy that minimizes the full Lyapunov drift (including first- and second-order terms), achieving minimax-optimal finite-time performance in heavy traffic and retaining stability. A key finding is that the classic MaxWeight policy, which optimizes only the first-order drift, can incur significantly larger backlogs in finite time, especially with asymmetric scheduling sets. The work also provides extensive simulations, extensions to independent-queue settings, and practical guidance toward implementing second-order drift-aware scheduling in large-scale, time-varying environments.

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. To our knowledge, this provides the first firm foundation for evaluating and comparing scheduling policies in the finite-time regime, including nonstationary settings, and shows that the proposed policy can provably and empirically outperform classical MaxWeight 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 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 delineate the scope and limitations of classical drift-based scheduling and motivate new queueing-control methods with rigorous finite-time guarantees.

Figures (9)

• Figure 1: Scheduling sets induced by the service matrices in Table \ref{['tab:service-matrix']}.
• Figure 2: Construction of the instance $\mathcal{I}_{b,q}$
• Figure 3: Asymmetric-service shared queueing model
• Figure 4: Performance comparison of MaxWeight and LyapOpt policies versus $B$
• Figure 5: Performance comparison of MaxWeight and LyapOpt policies when $C_{\mathrm{a}}, C_{\mathrm{d}}>0$

Theorems & Definitions (40)

• Theorem 1: Minimax Lower Bound
• proof : Proof Sketch of Theorem \ref{['thm_lower_bound']}
• Theorem 2: Finite-Time Performance of the $\mathrm{LyapOpt}$ Policy
• proof
• Theorem 3: Upper Bound of MaxWeight Policy
• proof
• Theorem 4: Lower Bound of MaxWeight Policy
• proof : Proof Sketch
• proof : Proof Sketch
• Corollary 1: Performance Gap between $\mathrm{LyapOpt}$ and $\mathrm{MaxWeight}$