Parallel server networks are widely used to study many service systems (customer contact centers and hospital patient flows) and date networks. We focus on the optimal scheduling and routing problems for large-scale Markovian multiclass multi-pool networks under the long-run average (ergodic) cost criteria. The arrival processes are Poisson, service times are exponentially distributed with class and pool dependent rates, customer patience times are class dependent, and each server pool has a large number of statistically identical parallel servers. We consider two formulations: (i) both queueing and idleness costs are minimized, and (ii) the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. 
The optimal solution of each scheduling problem is approximated by that of the ergodic diffusion control in the limit via the HJB equations. We introduce a broad class of ergodic diffusion control problems for diffusions, which includes the limiting diffusions for a large class of multiclass multi-pool queueing networks. We prove the asymptotic convergence of the values for the multiclass queueing control problems to the value of the associated ergodic diffusion control problem. The mathematical challenge lies in the ergodicity properties of the limiting controlled diffusion and the diffusion-scaled state process of the stochastic networks. We have identified (sub)geometrically stable policies for the limit and diffusion-scaled state processes. The proof of asymptotic convergence relies on a spatial truncation technique and construction of concatenated scheduling policies by using the optimal solution to the diffusion control problem and the (sub)geometrically stable policies.