Staffing many-server queues with autoregressive inputs

Recent studies reveal significant overdispersion and autocorrelation in arrival data at service systems such as call centers and hospital emergency departments. These findings stimulate the needs for more practical non-Poisson customer arrival models, and more importantly, new staffing formulas to account for the autocorrelative features in the arrival model. For this purpose, we study a multiserver queueing system where customer arrivals follow a doubly stochastic Poisson point process whose intensities are driven by a Cox-Ingersoll-Ross (CIR) process. The nonnegativity and autoregressive feature of the CIR process makes it a good candidate for modeling temporary dips and surges in arrivals. First, we devise an effective statistical procedure to calibrate our new arrival model to data which can be seen as a specification of the celebrated expectation-maximization algorithm. Second, we establish functional limit theorems for the CIR process, which in turn facilitate the derivation of functional limit theorems for our queueing model under suitable heavy-traffic regimes. Third, using the corresponding heavy traffic limits, we asymptotically solve an optimal staffing problem subject to delay-based constraints on the service levels. We find that, in order to achieve the designated service level, such an autoregressive feature in the arrival model translates into notable adjustment in the staffing formula, and such an adjustment can be fully characterized by the parameters of our new arrival model. In this respect, the staffing formulas acknowledge the presence of autoregressive structure in arrivals. Finally, we extend our analysis to queues having customer abandonment and conduct simulation experiments to provide engineering confirmations of our new staffing rules.