Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Tail Asymptotics and Scaling of Extrema. (arXiv:1803.03306v1 [math.PR])

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 (2015) established that in the Halfin-Whitt regime where $(N-\lambda(N))/\sqrt{N}\to\beta>0$ as $N\to\infty$, appropriately scaled occupancy measure of the system under the JSQ policy converges weakly on any finite time interval to a certain diffusion process as $N\to\infty$. Recently, it was further established by Braverman (2018) that the stationary occupancy measure of the system converges weakly to the steady state of the diffusion process as $N\to\infty$. In this paper we perform a detailed analysis of the steady state of the above diffusion process. Specifically, we establish precise tail-asymptotics of the stationary distribution and scaling of 查看全文>>