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

Abstract

This paper presents a novel methodology for approximating the queue length (thenumber of customers in the system) distributions of time-varying non-Markovian manyserverqueues (e.g., Gt/Gt/nt queues), where the number of servers (nt) is large. Ourmethodology consists of two steps. The first step uses phase-type distributions toapproximate the general inter-arrival and service times, thus generating an approximatingP ht/P ht/nt queue. The second step develops strong approximation theory toapproximate the P ht/P ht/nt queue with fluid and diffusion limits. However, by naivelyrepresenting the P ht/P ht/nt queue as a Markov process by expanding the state space,we encounter the lingering phenomenon even when the queue is overloaded. Lingeringtypically 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 notof zero measure. As a result, we develop an alternative representation for the queuelength process that avoids the lingering problem in the overloaded case, thus allowingfor the derivation of a Gaussian diffusion limit. Finally, we compare the effectivenessof our proposed method with discrete event simulation in a variety parameter settingsand show that our approximations are very accurate.