Staffing large-scale service systems with distributional uncertainty

This paper analyzes a staffing level problem for large-scale single-station queueing systems. The system manager operates an Erlang-C queueing system with a quality-of-service constraint on the probability that a customer is queued. However, in this model, the arrival rate is uncertain in the sense that even the arrival-rate distribution is not completely known to the manager. Rather, the manager has an estimate of the support of the arrival-rate distribution and the mean. The goal is to determine the number of servers needed to satisfy the quality-of-service constraint. Two cases are explored. First, the constraint is enforced on an overall delay probability, given the probability that different feasible arrival-rate distributions are selected. In the second case, the constraint has to be satisfied by every possible distribution. For both problems, asymptotically optimal solutions are developed based on Halfin---Whitt type scalings.