Queues with Many Servers and Impatient Customers

The asymptotic many-server queue with abandonments, G/GI/N + GI, is considered in the quality- and efficiency-driven (QED) regime. Here the number of servers and the offered load are related via the square-root rule, as the number of servers increases indefinitely. QED performance entails short waiting times and scarce abandonments (high quality) jointly with high servers' utilization (high efficiency), which is feasible when many servers cater to a single queue. For the G/GI/N + GI queue, we derive diffusion approximations for both its queue-length and virtual-waiting-time processes. Special cases, for which closed-form analysis is provided, are the G/M/N + GI and G/D/N + GI queues, thus expanding and generalizing existing results.