Many-Server Asymptotics for Join-the-Shortest-Queue in the Super-Halfin-Whitt Scaling Window

Join-the-shortest queue (JSQ) is a classical benchmark for the performance of parallel-server queueing systems because of its strong optimality properties. Recently, there has been significant progress in understanding its large-system asymptotic behavior. In this paper, we analyze the JSQ policy in the super-Halfin-Whitt scaling window when load per server ${\lambda }_N$ scales with the system size N as ${\lim }_{N\to \infty }{N}^{\alpha }(1-{\lambda }_N)=\beta$ 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 $\text{gamma}(2,\beta )$ distribution. The limit laws are universal in the sense that they do not depend on the value of $\alpha$ and exhibit fundamentally different behavior from both the Halfin–Whitt regime ($\alpha =1/2$) and the nondegenerate slowdown (NDS) regime ($\alpha =1$).

Funding: This work was supported by the National Science Foundation to S. Banerjee [Grants CAREER DMS-2141621 and RTG DMS-2134107] and D. Mukherjee and Z. Zhao [Grants CIF-2113027 and CPS-2240982].