Bounding the queue length in a multiserver queue is a central challenge in queueing theory. Even for the classical $G/G/n$ queue with homogeneous servers, it is highly non-trivial to derive a simple and accurate bound for the steady-state queue length that holds for all problem parameters. A recent breakthrough by Li and Goldberg (2025) establishes a universal bound of order $O(1/(1-ρ))$ that holds for any load $ρ< 1$ and any number of servers $n$. This order is tight in many well-known scaling regimes, including classical 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 universal bound of order $O(1/(1-ρ))$ for the $G/G/n$ 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 $G/G/n$ queue, the stationarity of a quadratic test function, and a novel leave-one-out coupling technique. Finally, we also extend our method to $G/G/n$ queues with fully heterogeneous service-time distributions.

翻译：在多服务器队列中界定队列长度是排队论的核心挑战之一。即使对于经典的具有同质服务器的$G/G/n$队列，推导出对所有问题参数都成立的稳态队列长度的简单且精确的界也极具挑战性。Li和Goldberg（2025）近期取得的突破性进展建立了一个普适的$O(1/(1-\rho))$阶界，该界对任意负载$\rho<1$和任意服务器数量$n$均成立。该阶数在许多著名的标度区域（包括经典重流量、Halfin-Whitt以及非退化慢化区域）中是紧的。然而，他们的界含有巨大的常数因子且证明过程高度复杂，这表明还有进一步改进的空间。本文针对$G/G/n$队列提出了一个全新的普适$O(1/(1-\rho))$阶界。尽管我们的界仅限于轻尾情形和队列长度的一阶矩，但其领先常数更具可解释性且通常更紧。我们的证明相对简单，利用了修正的$G/G/n$队列、二次检验函数的平稳性以及一种新颖的留一法耦合技术。最后，我们还将该方法推广至服务时间分布完全异质的$G/G/n$队列。