Steady-state analysis of the Join the Shortest Queue model in the Halfin-Whitt regime

Abstract

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.