  The Join-the-Shortest-Queue (JSQ) load balancing scheme is widely
acknowledged for its effectiveness in minimizing the average response time for
jobs in systems with identical servers. However, when applied to a
heterogeneous server system with servers of different processing speeds, the
JSQ scheme exhibits suboptimal performance. Recently, a variation of JSQ called
the Speed-Aware-Join-the-Shortest-Queue (SA-JSQ) scheme has been shown to
attain fluid limit optimality for systems with heterogeneous servers. In this
paper, we examine the SA-JSQ scheme for heterogeneous server systems under the
Halfin-Whitt regime. Our analysis begins by establishing that the scaled and
centered version of the system state weakly converges to a diffusion process
characterized by stochastic integral equations. Furthermore, we prove that the
diffusion process is positive recurrent and the sequence of stationary measures
for the scaled and centered queue length processes converge to the stationary
measure for the limiting diffusion process. To achieve this result, we employ
Stein's method with a generator expansion approach.
