A new $1\mathrm{/}(1-\rho )$-scaling bound for multiserver queues via a leave-one-out technique

Bounding the queue length in a multiserver queue is a central challenge in queueing theory. Even for the classical $G\mathrm{/}G\mathrm{/}n$ 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 $O(1\mathrm{/}(1-\rho ))$ that holds for any load $\rho <1$ 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 $O(1\mathrm{/}(1-\rho ))$ for the $G\mathrm{/}G\mathrm{/}n$ 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 $G\mathrm{/}G\mathrm{/}n$ queue, the stationarity of a quadratic test function, and a novel leave-one-out coupling technique. Finally, we also extend our method to $G\mathrm{/}G\mathrm{/}n$ queues with fully heterogeneous service-time distributions.