  We study a dynamic scheduling problem for a multi-class queueing network with
a large pool of statistically identical servers. The arrival processes are
Poisson, and service times and patience times are assumed to be exponentially
distributed and class dependent. The optimization criterion is the expected
long time average (ergodic) of a general (nonlinear) running cost function of
the queue lengths. We consider this control problem in the Halfin-Whitt (QED)
regime, that is, the number of servers $n$ and the total offered load
$\mathbf{r}$ scale like $n\approx\mathbf{r}+\hat{\rho}\sqrt{\mathbf{r}}$ for
some constant $\hat{\rho}$. This problem was proposed in [Ann. Appl. Probab. 14
(2004) 1084-1134, Section 5.2]. The optimal solution of this control problem
can be approximated by that of the corresponding ergodic diffusion control
problem in the limit. We introduce a broad class of ergodic control problems
for controlled diffusions, which includes a large class of queueing models in
the diffusion approximation, and establish a complete characterization of
optimality via the study of the associated HJB equation. We also prove the
asymptotic convergence of the values for the multi-class queueing control
problem to the value of the associated ergodic diffusion control problem. The
proof relies on an approximation method by spatial truncation for the ergodic
control of diffusion processes, where the Markov policies follow a fixed
priority policy outside a fixed compact set.
