Analysis of a two-node closed queueing cyclic network with one non-exponential node

Abstract

Consider a two-node closed cyclic queueing network with a total of M customers. The first node consists of N⩽M parallel identical servers having exponential service times. Node 2 consists of a single server having a general service time distribution. Waiting space at each node is sufficient to accommodate all M customers.We derive an exact solution for the steady state system size probabilities. Our approach is based on developing and solving an imbedded Markov chain at Node 2 service completion epochs coupled with a birth process over Node 1 service times. In that sense, our methodology is intuitively appealing and easy to apply. We focus on analytically and numerically investigating the effect of Node 2 service time variability on system performance. We find that closed queueing networks, like the one considered here, are less vulnerable to variability than open networks. However, the effect of variability is still significant to merit considerable attention. Numerical results are presented to illustrate the viability of our method for many service time distributions. As by-products, our work yield new results on birth processes and combinatorial identities, which can be useful in other contexts.A primary application of this model is the well-known machine repair model where a set of identical machines are attended by a single repairman. Other applications include performance evaluation of manufacturing and computer networks, as well as reliability studies where our model can be easily used to compute system availability.