Rapid, efficient analysis of the λ(n)/Ck/r/N queue, with application to decomposition of closed queuing networks

Abstract

A queuing network model (QNM) of a logistics or manufacturing process could conceivably contain service stations with multiple homogeneous servers, each with a general service time distribution. The λ(n)/Ck/r/N queue, which forms at an N-capacity, r-server station where each server has a k-phase Coxian service distribution, plays an important role in the analysis of such models. In this paper, we propose a strategy for quickly and efficiently solving for the stationary probabilities of the related continuous time Markov chain (CTMC) of this queue. The proposed solution method makes more practical the analysis of queues with a much broader range of parameters than was previously possible, thereby making it easier to use a QNM to model industrial processes.In this paper, we exploit advances in computational linear algebra and computing power to solve for the stationary probability vector of the related continuous time Markov chain (CTMC) of a λ(n)/Ck/r/N queue. The insights gained from studying the transition rate matrix of this chain are used to develop a list of candidate solution methods, each of which is used to calculate stationary probabilities for a set of 46 representative large-dimension problems. Based on this computational experience, a preferred solution approach for this type of CTMC is proposed. The effectiveness of the solution approach is demonstrated by using Marie's method to decompose four queuing networks containing stations with multiple k-phase Coxian servers.