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.

翻译：最短队列（JSQ）负载均衡方案因其能有效最小化同构服务器系统中任务的平均响应时间而被广泛认可。然而，当应用于服务器处理速度不同的异构系统时，JSQ方案表现出次优性能。最近，一种名为速度感知最短队列（SA-JSQ）的改进方案已被证明能在异构服务器系统中实现流体极限最优性。本文在Halfin-Whitt机制下研究异构服务器系统的SA-JSQ方案。我们首先证明系统状态的缩放与中心化版本弱收敛于一个由随机积分方程刻画的扩散过程。进一步，我们证明该扩散过程是正常返的，且缩放与中心化队长过程的稳态测度序列收敛于极限扩散过程的稳态测度。为实现这一结论，我们采用了基于生成元展开的Stein方法。