Many-server asymptotics for Join-the-Shortest Queue in the Super-Halfin-Whitt Scaling Window
Zhisheng Zhao, Sayan Banerjee, Debankur Mukherjee
TL;DR
This work analyzes JSQ in a many-server setting under the super-Halfin-Whitt scaling λ_N=1−β/N^{1/2+ε} with ε∈(0,1/2). It proves a diffusion-limit for the scaled total queue S^{(N)}(N^{2ε}t) that solves the Langevin-type SDE dX(t)=(1/X(t)−β)dt+√2 dW(t), and shows the stationary distribution converges to Gamma(2,β), with mean 2/β. A central achievement is the tightness and renewal-based interchange of limits for the pre-limit stationary distributions, established via detailed pathwise analysis of idleness and queue-length excursions, yielding universal limiting behavior independent of ε. The results reveal fundamental differences from Halfin-Whitt and Non-Degenerate Slowdown regimes, and provide a rigorous, quantitative bridge between pre-limit dynamics and limiting diffusion in a heavy-traffic, many-server JSQ system with universal characteristics. The methods combine martingale techniques, excursion analysis, and renewal theory to produce precise asymptotics and robust limit interchanges, with implications for the design and performance understanding of large-scale parallel-server pools.
Abstract
The Join-the-Shortest Queue (JSQ) policy is a classical benchmark for the performance of many-server queueing systems due to its strong optimality properties. While the exact analysis of the JSQ policy is an open question to date, even under Markovian assumption on the service requirements, recently, there has been a significant progress in understanding its many-server asymptotic behavior since the work of Eschenfeldt and Gamarnik (Math.~Oper.~Res.~43 (2018) 867--886). In this paper, we analyze the many-server limits of the JSQ policy in the \emph{super-Halfin-Whitt} scaling window when load per server $λ_N$ scales with the system size $N$ as $\lim_{N\rightarrow\infty}N^α(1-λ_N)=β$ for $α\in (1/2, 1)$ and $β>0$. We establish that the centered and scaled total queue length process converges to a certain Bessel process with negative drift and the associated centered and scaled steady-state total queue length, indexed by $N$, converges to a $\mathrm{Gamma}(2,β)$ distribution. Both the transient and steady-state limit laws are universal in the sense that they do not depend on the value of the scaling parameter $α$, and exhibit fundamentally different qualitative behavior from both the Halfin-Whitt regime ($α= 1/2$) and the Non-degenerate Slowdown (NDS) regime ($α=1$).