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This research refines theoretical bounds for queue peak behavior in stochastic networks, demonstrating a transition from square-root to logarithmic scaling once service slack exceeds specific geometric thresholds. By establishing self-normalization mechanisms that decouple fluctuation scales from capacity geometry, the study provides tighter performance guarantees for drift-minimizing scheduling policies like MaxWeight under dependent arrival processes. These findings are critical for optimizing high-dimensional resource allocation in complex AI-driven network systems where stability and latency constraints must be rigorously managed over finite horizons.
Finite-Time Queue Peak Laws in Stochastic Networks: Logarithmic Scaling After Geometric Thresholds
arXiv:2606.18218v1 Announce Type: cross Abstract: We study finite-horizon queue peaks in generalized switches, a standard stochastic-network model in which many queues share constrained service resources. Arrivals may be dependent, time-varying, and adapted to the past; the standing load condition is uniform interior slack, meaning the conditional mean arrival vector stays in a fixed contraction of the capacity region. We show that this slack reshapes the finite-time peak law for drift-minimizing scheduling policies such as MaxWeight. The square-root envelope that is sharp without slack persists only up to a geometry-dependent threshold; beyond that threshold, the running maximum grows only logarithmically with the horizon, both with high probability and in expectation. The mechanism is self-normalization: in the current queue direction, the projected fluctuation scale is normalized by the stabilizing drift scale. This removes capacity geometry from the logarithmic coefficient, while geometry remains in the threshold. Matching lower bounds show that both the logarithmic term and a geometric threshold are unavoidable. When finite-time state-space collapse is available, the threshold can be sharpened using local bottleneck geometry. For generalized input-queued switches, we obtain finite-time peak bounds with tight logarithmic coefficients. Simulations illustrate the two-phase envelope, local geometric refinements, and variance-sensitive improvements predicted by the theory.
Abstract:We study finite-horizon queue peaks in generalized switches, a standard stochastic-network model in which many queues share constrained service resources. Arrivals may be dependent, time-varying, and adapted to the past; the standing load condition is uniform interior slack, meaning the conditional mean arrival vector stays in a fixed contraction of the capacity region. We show that this slack reshapes the finite-time peak law for drift-minimizing scheduling policies such as MaxWeight. The square-root envelope that is sharp without slack persists only up to a geometry-dependent threshold; beyond that threshold, the running maximum grows only logarithmically with the horizon, both with high probability and in expectation. The mechanism is self-normalization: in the current queue direction, the projected fluctuation scale is normalized by the stabilizing drift scale. This removes capacity geometry from the logarithmic coefficient, while geometry remains in the threshold. Matching lower bounds show that both the logarithmic term and a geometric threshold are unavoidable. When finite-time state-space collapse is available, the threshold can be sharpened using local bottleneck geometry. For generalized input-queued switches, we obtain finite-time peak bounds with tight logarithmic coefficients. Simulations illustrate the two-phase envelope, local geometric refinements, and variance-sensitive improvements predicted by the theory.
Probability (math.PR); Machine Learning (cs.LG); Systems and Control (eess.SY); Optimization and Control (math.OC); Machine Learning (stat.ML)
arXiv:2606.18218 [math.PR]
(or arXiv:2606.18218v1 [math.PR] for this version)
https://doi.org/10.48550/arXiv.2606.18218 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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