Sampling the Functional Kolmogorov Forward Equations for Nonstationary Queueing Networks

Nonstationary queueing networks are often difficult to approximate. Recent novel methods for approximating the moments of nonstationary queues use the functional version of the Kolmogorov forward equations in conjunction with orthogonal polynomial expansions. However, these methods require closed form expressions for the expectations that appear in the functional Kolmogorov forward equations. When closed form expressions cannot be easily derived, these methods cannot be used. In this paper, we present a new sampling algorithm to overcome this difficulty; our sampling algorithm accurately estimates the expectations using simulation. We apply our algorithm to priority queues, which are useful for modeling hospital triage systems. We show that our sampling algorithm accurately estimates the mean and variance of the priority queue without spending significantly more computational time than integrating ordinary differential equations. Last, we compare our sampling algorithm to the closed form analytical approximations for the Erlang-A queueing model and find that our method is comparable in time and accuracy.