DIFFUSION APPROXIMATION FOR EFFICIENCY-DRIVEN QUEUES UNDER REFINED PATIENCE TIME SCALING

Abstract

We propose a tractable model for GI/GI/n+GI queues that model call centers in the efficiency-driven (ED) regime. Delay announcements made upon customer arrival providing expected delay may cause a rapid change in the patience time hazard rate after expected delay. Unlike diffusion approximations in the literature that depend on the patience time hazard rate only at the mean virtual waiting time, our diffusion model includes the patience time hazard rate in a neighborhood of the mean virtual waiting time.

A one-dimensional diffusion process is used to approximate the virtual waiting time process that is scaled in both space 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 generates simple formulas for performance measures. We further use the diffusion model to solve optimization problem for staffing. A key approach is the new patience time scaling which combines the space-time scaling approach and the hazard rate scaling approach. This joint scaling approach simplifies the many-server analysis when service times follow a general distribution, and captures the rapid change in patience time hazard rate. In consequence, the approximate formulas derived from the diffusion model depend on the patience time hazard rate in a neighborhood of the mean virtual waiting time, which are more general and precise than those depending on assumptions of exponential service times or a smooth patience time distribution.