Finite-Time Queue Peak Laws in Stochastic Networks: Logarithmic Scaling After Geometric Thresholds
2026-06-16 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how queues that share limited service resources behave over a fixed time period when arrivals can vary and depend on past events. They focus on scheduling policies like MaxWeight and find that the usual way queues peak changes if there is some slack (extra capacity) available: initially, peaks grow like a square root function, but beyond a certain point related to system geometry, they grow only logarithmically. This happens because the system self-adjusts fluctuations by balancing with the stabilizing forces, leading to predictable peak growth regardless of capacity shape. They also provide precise bounds and examples where these patterns clearly show up.
generalized switchesfinite-horizon queue peaksMaxWeight schedulingcapacity regionconditional mean arrival vectorself-normalizationstate-space collapseinput-queued switcheslogarithmic boundsstochastic networks
Authors
Hao Liang, Cheng Tang, Yunzong Xu
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.