Approximations for the Queue Length Distributions of Time-Varying Many-Server Queues

This paper presents a novel and computationally efficient methodology for approximating the queue length (the number of customers in the system) distributions of time-varying non-Markovian many-server queues (e.g., G t / G t / n t queues), where the number of servers ( n t ) is large. Our methodology consists of two steps. The first step uses phase-type distributions to approximate the general interarrival and service times, thus generating an approximating Ph t / Ph t / n t queue. The second step develops strong approximation theory to approximate the Ph t / Ph t / n t queue with fluid and diffusion limits whose mean and variance can be computed using ordinary differential equations. However, by naively representing the Ph t / Ph t / n t queue as a Markov process by expanding the state space, we encounter the lingering phenomenon even when the queue is overloaded . Lingering typically occurs when the mean queue length is equal or near the number of servers, however, in this case it also happens when the queue is overloaded and this time is not of zero measure. As a result, we develop an alternative representation for the queue length process that avoids the lingering problem in the overloaded case, thus allowing for the derivation of a Gaussian diffusion limit. Finally, we compare the effectiveness of our proposed method with discrete event simulation in a variety parameter settings and show that our approximations are very accurate.