Customer Abandonment in Many-Server Queues

We study G/G/n + GI queues in which customer patience times are independent, identically distributed following a general distribution. When a customer's waiting time in queue exceeds his patience time, the customer abandons the system without service. For the performance of such a system, we focus on the abandonment process and the queue length process. We prove that under some conditions, a deterministic relationship between the two stochastic processes holds asymptotically under the diffusion scaling when the number of servers n goes to infinity. These conditions include a minor assumption on the arrival processes that can be time-nonhomogeneous and a key assumption that the sequence of diffusion-scaled queue length processes, indexed by n, is stochastically bounded. We also establish a comparison result that allows one to verify the stochastic boundedness by studying a corresponding sequence of systems without customer abandonment.