Optimal service rate perturbations of many server queues in heavy traffic

An optimal control problem for a single customer class, many server queueing system of the type \( G/M/n+GI \) is considered, where the control corresponds to the service rate. An infinite horizon discounted cost functional which consists of a convex control cost, linear delay and idle server costs, and a linear abandonment cost is formulated. We study this problem in the heavy traffic regime originally proposed by Halfin and Whitt, where the arrival rates and the number of servers grow to infinity in concert. First we address the diffusion control problem (DCP) associated with the heavy traffic limit. By constructing a smooth solution to the associated Hamilton–Jacobi–Bellman equation, we obtain a feedback type optimal control for the DCP. We show that the value function of the DCP is an asymptotic lower bound for the value functions of the corresponding queueing control problems. We use the optimal control of the DCP to obtain an asymptotically optimal control policy for the queueing control problem.