Ergodic diffusion control of multiclass multi-pool networks in the Halfin-Whitt regime

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 firstserved 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 [Ann. Appl. Probab. 25 (2015) 3511-3570] is extended to address a broad class of ergodic diffusion control problems with constraints. We show that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.