Diffusion Approximations for G/M/n + GI Queues with State-Dependent Service Rates

We consider a sequence of many-server queueing systems with impatient customers of the type G/M/n+GI in heavy traffic. This sequence is indexed by n, where the parameter n represents the number of servers in the nth system. The state process is considered to be the diffusion-scaled total customer count in the system and the service rate is a state-dependent perturbation of a given basic service rate Œ0 > 0. When the system is critically loaded in the Halfin-Whitt heavy traffic regime, we obtain the limiting diffusion for the state processes. We also establish the asymptotic relationships among the diffusion-scaled processes representing the total customer count, virtual waiting time, and the number of customer abandonments. Motivated by the cost structures of telephone call centers, we formulate a cost functional and show that the expected value of this cost functional in the nth system converges to that of the limiting diffusion under mild assumptions.