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

2026-06-17 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

A new study on arXiv:2606.18218, "Finite-Time Queue Peak Laws in Stochastic Networks," reveals a two-phase behavior for finite-horizon queue peaks in generalized switches. For drift-minimizing scheduling policies like MaxWeight, the traditional square-root envelope for queue growth, observed without slack, only holds up to a specific geometry-dependent threshold. Beyond this point, the running maximum of the queue length transitions to a significantly slower logarithmic growth with the time horizon, a phenomenon observed both with high probability and in expectation. This shift is attributed to a self-normalization mechanism, where the projected fluctuation scale is normalized by the stabilizing drift scale. This mechanism removes capacity geometry from the logarithmic coefficient while retaining it in the threshold. The research provides matching lower bounds, confirming the inevitability of both the logarithmic term and the geometric threshold, and demonstrates these theoretical predictions through simulations, including local geometric refinements and variance-sensitive improvements.

Key takeaway

For network architects and systems engineers designing or optimizing stochastic networks, understanding finite-time queue peak behavior is crucial. You should account for the two-phase queue growth, where initial square-root scaling transitions to logarithmic growth after a geometry-dependent threshold. This insight allows for more accurate capacity planning and resource allocation, particularly for systems utilizing drift-minimizing policies like MaxWeight, preventing over-provisioning or unexpected performance bottlenecks in long-horizon operations.

Key insights

Queue peaks in stochastic networks exhibit logarithmic growth after a geometric threshold, driven by self-normalization.

Principles

• Uniform interior slack reshapes finite-time peak laws.
• Self-normalization drives logarithmic queue growth.
• Geometric thresholds are unavoidable in queue peak scaling.

Method

The study employs theoretical analysis of generalized switches with dependent, time-varying arrivals, validated by simulations illustrating two-phase envelopes and geometric refinements.

In practice

• Model queue behavior in generalized input-queued switches.
• Predict long-term queue growth using logarithmic scaling.
• Account for geometry-dependent thresholds in network design.

Topics

• Stochastic Networks
• Queueing Theory
• MaxWeight Scheduling
• Finite-Time Analysis
• Logarithmic Scaling
• Network Optimization

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.