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

To better understand how to interpret birth-and-death (BD) processes fit to service system data, we investigate the consequences of fitting a BD process to a multi-server queue with a periodic time-varying arrival rate function. We study how this fitted BD process is related to the original queue-length process. If a BD process is fit to a segment of the sample path of the queue-length process, with the birth (death) rates in each state estimated by the observed number of arrivals (departures) in that state divided by the total time spent in that state, then under minor regularity conditions that BD process has the steady-state distribution of the queue length process in the original Mt/GI/s queueing model 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.