Many-server queues with customer abandonment: A survey of diffusion and fluid approximations

The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on G / GI / n + GI parallel-server queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed (iid) service times with a general distribution (the first GI), and iid patience times with a general distribution (the +GI). Following the square-root safety staffing rule, this queue can be operated in the quality- and efficiency-driven (QED) regime, which is characterized by large customer volume, the waiting times being a fraction of the service times, only a small fraction of customers abandoning the system, and high server utilization. Operational efficiency is the central target in a system whose staffing costs dominate other expenses. If a moderate fraction of customer abandonment is allowed, such a system should be operated in an overloaded regime known as the efficiency-driven (ED) regime. We survey recent results on the many-server queues that are operated in the QED and ED regimes. These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.