Universally optimal staffing of Erlang-A queues facing uncertain arrival rates

In many service systems, the staffing decisions must be made before the arrival rate is known with certainty. Thus, it is more appropriate to consider the arrival rate as a random variable at the time of the staffing decision. Motivated by this observation, we study the staffing problem in a service system modeled as an Erlang-A queue facing a random arrival rate. For linear staffing costs, linear waiting costs, and a cost per customer abandonment, we propose a policy that is based on modifying the well-known square-root safety staffing policy to explicitly account for the randomness in the arrival rate. Our primary contribution is to show that our proposed policy is “universally optimal”, i.e., irrespective of the magnitude of randomness in the arrival rate, the optimality gap between our proposed policy and the exact optimal policy remains bounded as the system size grows large. This is important because earlier performance guarantees for Erlang-A queues either (1) are not universal and offer performance guarantees that depend on the magnitude of uncertainty in the arrival rate or (2) are universal but assume a deterministic arrival rate. The practical relevance of this provable robustness is that our proposed policy is a “one-size-fits-all” as it is guaranteed to perform well for all levels of arrival rate uncertainty.