Efficiency-Driven Heavy-Traffic Approximations for Many-Server Queues with Abandonments

To provide useful practical insight into the performance of service-oriented (non-revenue-generating) call centers, which often provide low-to-moderate quality of service, this paper investigates the efficiency-driven (ED), many-server heavy-traffic limiting regime for queues with abandonments. Attention is focused on the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces, exponential abandon times (the final +M), and the first-come–first-served service discipline. Both the number of servers and the arrival rate are allowed to increase, while the individual service and abandonment rates are held fixed. The key is how the two limits are related: In the now common quality-and-efficiency-driven (QED) or Halfin-Whitt limiting regime, the probability of initially being delayed approaches a limit strictly between 0 and 1, while the probability of eventually being served (not abandoning) approaches 1. In contrast, in the ED limiting regime, the probability of eventually being served approaches a limit strictly between 0 and 1, while the probability of initially being delayed approaches 1. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ > 1 (so that the arrival rate exceeds the maximum possible service rate). The ED regime can be realistic because with the abandonments, the delays need not be extraordinarily large. When the ED appropriations are appropriate, they are appealing because they are remarkably simple.