Many-server asymptotics for Join-the-Shortest Queue in the Super-Halfin-Whitt Scaling Window

TL;DR

This paper investigates the asymptotic behavior of the Join-the-Shortest Queue policy in many-server systems within the super-Halfin-Whitt scaling window, revealing universal limit laws for queue length distributions that differ from classical regimes.

Contribution

It provides the first analysis of JSQ in the super-Halfin-Whitt regime, establishing convergence to Bessel and Gamma distributions, and highlights the regime's unique qualitative behavior.

Findings

Total queue length converges to a Gamma distribution in steady state.

The scaled queue length process converges to a Bessel process with negative drift.

Results are universal across the super-Halfin-Whitt scaling parameter.

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 $\lambda_N$ scales with the system size $N$ as $\lim_{N\rightarrow\infty}N^{\alpha}(1-\lambda_N)=\beta$ for $\alpha\in (1/2, 1)$ and $\beta>0$. 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 $\mathrm{Gamma}(2,\beta)$ 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 $\alpha$, and exhibit fundamentally different qualitative behavior from both the Halfin-Whitt regime ($\alpha = 1/2$) and the Non-degenerate Slowdown (NDS) regime ($\alpha=1$).