Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

Abstract

Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda (N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (Math. Oper. Res. 43 (2018) 867–886) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin–Whitt regime, where $(N-\lambda (N))/\sqrt{N}\to \beta >0$ as $N\to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is nontrivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein’s method, Braverman (2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann. Appl. Probab. 29 (2018) 1262–1309) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion.

However, the analysis of the bulk behavior of the stationary distribution, namely, the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter $\beta$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady-state distribution, as $\beta $ tends to 0 and $\infty$. Of particular interest, we also establish a certain intermittency phenomena in the $\beta \to \infty$ regime and a surprising distributional convergence result in the $\beta \to 0$ regime.