Finite-Time Queue Peak Laws in Stochastic Networks: Logarithmic Scaling After Geometric Thresholds

Problem
This work addresses the gap in understanding finite-time queue peaks in generalized switches, a common model in stochastic networks where multiple queues share limited service resources. The authors investigate scenarios with dependent, time-varying arrivals that are adapted to past states, which complicates traditional analyses. The study is particularly relevant as it is a preprint and has not undergone peer review, indicating that the findings should be interpreted with caution.

Method
The authors develop a theoretical framework to analyze the finite-time peak behavior of drift-minimizing scheduling policies, specifically the MaxWeight policy. They introduce a standing load condition characterized by uniform interior slack, which constrains the conditional mean arrival vector within a fixed contraction of the capacity region. The core contribution is the identification of a geometric threshold beyond which the running maximum of queue lengths grows logarithmically with the time horizon, rather than following the square-root envelope observed without slack. The mechanism of self-normalization is central to their analysis, where the projected fluctuation scale in the current queue direction is normalized by the stabilizing drift scale. This approach effectively decouples the logarithmic coefficient from the capacity geometry while retaining the geometric threshold’s influence. The authors also provide matching lower bounds to demonstrate the necessity of both the logarithmic term and the geometric threshold. Additionally, they refine their results for generalized input-queued switches, achieving tighter logarithmic coefficients under finite-time state-space collapse conditions.

Results
The paper presents significant findings, showing that beyond the geometry-dependent threshold, the peak queue length grows logarithmically with the time horizon, both in expectation and with high probability. The authors illustrate this two-phase envelope through simulations, which confirm the theoretical predictions. The results indicate that the logarithmic growth is a robust feature of the system under the specified conditions, with the authors providing empirical evidence of variance-sensitive improvements and local geometric refinements.

Limitations
The authors acknowledge that their analysis is contingent on the assumptions of uniform interior slack and the specific structure of the scheduling policy. They do not address potential extensions to more complex arrival processes or the implications of non-uniform slack conditions. Additionally, the reliance on theoretical constructs may limit the applicability of the findings to real-world scenarios where such idealized conditions may not hold.

Why it matters
This research has significant implications for the design and analysis of scheduling policies in stochastic networks, particularly in environments with constrained resources and complex arrival patterns. The logarithmic scaling behavior identified could inform the development of more efficient scheduling algorithms that account for the nuanced dynamics of queue interactions. The findings contribute to the broader literature on queueing theory and stochastic processes, as discussed in related works, and are available on arXiv.