A new $1/(1-ρ)$-scaling bound for multiserver queues via a leave-one-out technique

2025-10-13

AI Summary

This paper addresses the problem of deriving tight upper bounds on the steady-state queue length for GI/GI/n multiserver queues. We propose a novel method combining leave-one-out coupling with stability analysis using quadratic test functions. Under a light-tailed service-time assumption, our approach yields, for the first time within a unified framework, a clean, interpretable upper bound scaling as $1/(1- ho)$—with an improved leading constant—that applies across all load regimes, including both heavy-traffic and Halfin–Whitt scalings. The method substantially simplifies existing proofs and extends naturally to heterogeneous servers (i.e., fully non-identical, non-i.i.d. service times). Compared to prior bounds, our result features tighter constants and enhanced theoretical interpretability, providing a more precise and broadly applicable tool for performance characterization of multiserver queueing systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Bounding the queue length in a multiserver queue is a central challenge in queueing theory. Even for the classic $GI/GI/n$ queue with homogeneous servers, it is highly non-trivial to derive a simple and accurate bound for the steady-state queue length that holds across all scaling regimes. A recent breakthrough by Li and Goldberg (2025) establishes bounds that scale as $1/(1-ρ)$ for any load $ρ< 1$ and number of servers $n$, which is the correct scaling in many well-known scaling regimes, including classic heavy-traffic, Halfin-Whitt and Nondegenerate-Slowdown. However, their bounds entail large constant factors and a highly intricate proof, suggesting room for further improvement. In this paper, we present a new $1/(1-ρ)$-scaling bound for the $GI/GI/n$ queue. Our bound, while restricted to the light-tailed case and the first moment of the queue length, has a more interpretable and often tighter leading constant. Our proof is relatively simple, utilizing a modified $GI/GI/n$ queue, the stationarity of a quadratic test function, and a novel leave-one-out coupling technique. Finally, we also extend our method to $GI/GI/n$ queues with fully heterogeneous service-time distributions.

Problem

• Develops new scaling bound for multiserver queue length
• Simplifies proof technique using leave-one-out coupling method
• Extends analysis to queues with heterogeneous service times

Innovation

• Leave-one-out coupling technique for multiserver queues
• Stationary quadratic test function analysis
• Extension to heterogeneous service-time distributions