Diffusion Approximation for Efficiency-Driven Queues When Customers Are Patient

Motivated by call center practice, we propose a tractable approximate model for queues with general service and patience time assumptions in the efficiency-driven (ED) regime, when customers’ patience times are relatively long compared with their service times. We use a one-dimensional diffusion process to approximate the virtual waiting time process that is scaled in both magnitude and time, with the number of servers and the mean patience time as the respective scaling factors. Using this diffusion model, we obtain the steady-state distributions of virtual waiting time and queue length, which in turn, yield simple formulas for performance measures, such as service levels and effective abandonment fractions. These formulas can work well when the mean patience time is several times longer than the mean service time. To justify the diffusion model, we formulate an asymptotic framework by considering a sequence of queues, where both the number of servers and the mean patience time go to infinity. The centered and scaled virtual waiting time process converges in distribution to the one-dimensional diffusion process under this framework. To prove the diffusion limit, a functional central limit theorem is established for the superposition of renewal processes. From a macroscopic perspective, we demonstrate that the dynamics of a many-server queue in the ED regime could be as simple as the dynamics of a single-server queue when customers are patient.