  We study the infinite horizon optimal control problem for N-network queueing
systems, which consist of two customer classes and two server pools, under
average (ergodic) criteria in the Halfin-Whitt regime. We consider three
control objectives: 1) minimizing the queueing (and idleness) cost, 2)
minimizing the queueing cost while imposing a constraint on idleness at each
server pool, and 3) minimizing the queueing cost while requiring fairness on
idleness. The running costs can be any nonnegative convex functions having at
most polynomial growth.
  For all three problems we establish asymptotic optimality, namely, the
convergence of the value functions of the diffusion-scaled state process to the
corresponding values of the controlled diffusion limit. We also present a
simple state-dependent priority scheduling policy under which the
diffusion-scaled state process is geometrically ergodic in the Halfin-Whitt
regime, and some results on convergence of mean empirical measures which
facilitate the proofs.
