Universally optimal staffing for Erlang-A queues facing uncertain arrival rates: The case of constraint satisfaction

Motivated by service systems where staffing decisions must be made before the arrival rate becomes known, we study the constraint satisfaction problem in an Erlang-A queue facing a random arrival rate. The objective is to find the minimum staffing level subject to a service level constraint that is modeled either (1) via an average constraint formulation that ensures a given quality-of-service (QoS) target holds on average by bounding the average fraction of abandoning customers below the said QoS target or (2) via a chance constraint formulation that ensures the QoS target for the random fraction of abandoning customers is met with high probability. Our primary contribution, under each constraint formulation, is to propose a policy that is shown to be universally optimal, i.e., irrespective of the magnitude of randomness in the arrival rate, the staffing gap between the proposed policy and the exact optimal policy remains bounded as the system size grows large. To the best of our knowledge, this is the first universal performance guarantee for constraint satisfaction in Erlang-A queues with random arrival rates and complements a recent result on cost minimization. The practical importance of this universality is that our proposed policy is a “one-size-fits-all” that is guaranteed to perform well for all levels of arrival rate uncertainty.