  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 extrema of the process on large
time-interval. Our results imply that the asymptotic steady-state scaled number
of servers with queue length two or larger exhibits an Exponential tail,
whereas that for the number of idle servers turns out to be Gaussian. From the
methodological point of view, the diffusion process under consideration goes
beyond the state-of-the-art techniques in the study of the steady-state of
diffusion processes. Lack of any closed form expression for the steady state
and intricate interdependency of the process dynamics on its local times make
the analysis significantly challenging. We develop a technique involving the
theory of regenerative processes that provides a tractable form for the
stationary measure, and in conjunction with several sharp hitting time
estimates, acts as a key vehicle in establishing the results.
