  We consider Markovian multiclass multi-pool networks with heterogeneous
server pools, each consisting of many statistically identical parallel servers,
where the bipartite graph of customer classes and server pools forms a tree.
Customers form their own queue and are served in the first-come first-served
discipline, and can abandon while waiting in queue. Service rates are both
class and pool dependent. The objective is to study the limiting diffusion
control problems under the long run average (ergodic) cost criteria in the
Halfin--Whitt regime. Two formulations of ergodic diffusion control problems
are considered: (i) both queueing and idleness costs are minimized, and (ii)
only the queueing cost is minimized while a constraint is imposed upon the
idleness of all server pools. We develop a recursive leaf elimination algorithm
that enables us to obtain an explicit representation of the drift for the
controlled diffusions. Consequently, we show that for the limiting controlled
diffusions, there always exists a stationary Markov control under which the
diffusion process is geometrically ergodic. The framework developed in our
earlier work is extended to address a broad class of ergodic diffusion control
problems with constraints. We show that that the unconstrained and constrained
problems are well posed, and we characterize the optimal stationary Markov
controls via HJB equations.
