Capacity Sizing Under Parameter Uncertainty: Safety Staffing Principles Revisited

We study a capacity sizing problem in a service system that is modeled as a single-class queue with multiple servers and where customers may renege while waiting for service. A salient feature of the model is that the mean arrival rate of work is random (in practice this is a typical consequence of forecasting errors). The paper elucidates the impact of uncertainty on the nature of capacity prescriptions, and relates these to well established rules-of-thumb such as the square root safety staffing principle. We establish a simple and intuitive relationship between the incoming load (measured in Erlangs) and the extent of uncertainty in arrival rates (measured via the coefficient of variation) which characterizes the extent to which uncertainty dominates stochastic variability or vice versa. In the former case it is shown that traditional square root safety staffing logic is no longer valid, yet simple capacity prescriptions derived via a suitable newsvendor problem, are surprisingly accurate.