An ergodic control problem for many-server multi-class queueing systems with help

A $M/M/N+M$ queueing network is considered with $d$ independent customer classes and $d$ server pools in Halfin-Whitt regime. Class $i$ customers have priority for service in pool $i$ for $i=1, ..., d$, and may access other pool if the pool has an idle server. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is governed by a control set that depends on the state. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.