Infinite horizon asymptotic average optimality for large-scale parallel server networks

Abstract

We study infinite-horizon asymptotic average optimality for parallel server networks with multiple classes of jobs and multiple server pools in the Halfin–Whitt regime. Three control formulations are considered: (1) minimizing the queueing and idleness cost, (2) minimizing the queueing cost under constraints on idleness at each server pool, and (3) fairly allocating the idle servers among different server pools. For the third problem, we consider a class of bounded-queue, bounded-state (BQBS) stable networks, in which any moment of the state is bounded by that of the queue only (for both the limiting diffusion and diffusion-scaled state processes). We show that the optimal values for the diffusion-scaled state processes converge to the corresponding values of the ergodic control problems for the limiting diffusion. We present a family of state-dependent Markov balanced saturation policies (BSPs) that stabilize the controlled diffusion-scaled state processes. It is shown that under these policies, the diffusion-scaled state process is exponentially ergodic, provided that at least one class of jobs has a positive abandonment rate. We also establish useful moment bounds, and study the ergodic properties of the diffusion-scaled state processes, which play a crucial role in proving the asymptotic optimality.