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Consider a system of $N$ parallel single-server queues with unit-exponentialservice time distribution and a single dispatcher where tasks arrive as aPoisson process of rate $\lambda(N)$. When a task arrives, the dispatcherassigns it to one of the servers according to the Join-the-Shortest Queue (JSQ)policy. Eschenfeldt and Gamarnik (Math. Oper. Res., 43(3):867-886, 2018)identified a novel limiting diffusion process that arises as the weak-limit ofthe appropriately scaled occupancy measure of the system under the JSQ policyin 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 theart techniques, and even proving its ergodicity is non-trivial, and was left asan open question. Recently, exploiting a generator expansion framework via theStein's method, Braverman (arXiv:1801.05121, 2018) established its exponentialergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann.Appl. Probab., 29(2):1262-1309, 2018) analyzed the tail properties of thestationary distribution and path fluctuations of the diffusion.  However, the analysis of the bulk behavior of the stationary distribution,viz., the moments, remained intractable until this work. In this paper, weperform a thorough analysis of the bulk behavior of the stationary distributionof the diffusion process, and discover that it exhibits different qualitativebehavior, depending on the value of the heavy-traffic parameter $\beta$.Moreover, we obtain precise asymptotic laws of the centered and scaled steadystate 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.

Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Sensitivity on the Heavy-traffic Parameter