  We study infinite-horizon asymptotic average optimality for parallel server
network 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 a
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.
