Call centers with impatient customers:: Many-server asymptotics of the M/M/n plus G queue

The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate lambda, exponential service times at rate mu, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents). In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as lambda and n increase indefinitely and mu held fixed: Efficiency-Driven (ED): n approximate to (lambda/mu) . (1-gamma), gamma > 0, Quality-Driven (QD): n approximate to (lambda/mu) . (1+gamma), gamma > 0, and Quality and Efficiency Driven (QED): n approximate to lambda/mu + beta root lambda/mu, -infinity < beta < infinity. In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/root n). Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems. Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.