  This paper studies the steady-state properties of the Join the Shortest Queue
model in the Halfin-Whitt regime. We focus on the process tracking the number
of idle servers, and the number of servers with non-empty buffers. Recently,
Eschenfeldt & Gamarnik (2015) proved that a scaled version of this process
converges, over finite time intervals, to a two-dimensional diffusion limit as
the number of servers goes to infinity. In this paper we prove that the
diffusion limit is exponentially ergodic, and that the diffusion scaled
sequence of the steady-state number of idle servers and non-empty buffers is
tight. Our results mean that the process-level convergence proved in
Eschenfeldt & Gamarnik (2015) implies convergence of steady-state
distributions. The methodology used is the generator expansion framework based
on Stein's method, also referred to as the drift-based fluid limit Lyapunov
function approach in Stolyar (2015). One technical contribution to the
framework is to show that it can be used as a general tool to establish
exponential ergodicity.

Probability