Stationary Birth-and-Death Processes Fit to Queues with Periodic Arrival Rate Functions

To better understand what queueing models are appropriate for complex service systems such as hospital emergency departments, we suggest fitting a general state-dependent birthand-death (BD) process to system data recording the number in system over a time interval To facilitate interpretation of the fitted BD rate functions, we investigate the consequences of fitting a BD process to a multi-server Mt/GI/s queue with a nonhomogeneous Poisson arrival process having a periodic time-varying rate function. The fitted death rates consistently have the same piecewise-linear structure previously found for the GI/GI/s model, independent of the service-time distribution, but the fitted birth rates have a very different structure, with a similar linear structure around the average occupancy, but constant limits at large and small arguments. Under minor regularity conditions, the fitted BD process has the same steady-state distribution as the original queue length process as the sample size increases. The steady-state distribution can be estimated efficiently by fitting a parametric function to the observed birth and death rates.