Stochastic grey-box modeling of queueing systems: fitting birth-and-death processes to data

摘要

This paper explores grey-box modeling of queueing systems. A stationary birth-and-death (BD) process model is fitted to a segment of the sample path of the number in the system in the usual way. The birth (death) rates in each state are estimated by the observed number of arrivals (departures) in that state divided by the total time spent in that state. Under minor regularity conditions, if the queue length (number in the system) has a proper limiting steady-state distribution, then the fitted BD process has that same steady-state distribution asymptotically as the sample size increases, even if the actual queue-length process is not nearly a BD process. However, the transient behavior may be very different. We investigate what we can learn about the actual queueing system from the fitted BD process. Here we consider the standard (GI/GI/s) queueing model with (s) servers, unlimited waiting room and general independent, non-exponential, interarrival-time and service-time distributions. For heavily loaded (s)-server models, we find that the long-term transient behavior of the original process, as partially characterized by mean first passage times, can be approximated by a deterministic time transformation of the fitted BD process, exploiting the heavy-traffic characterization of the variability.