An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

Abstract

Consider a closed cyclic queueing model that consists of two nodes and a total of $M$ customers. Each node buffer can accommodate all $M$ customers. Node 1 has $N\le M$ servers, each having an exponential service time with rate $\lambda$. The second node consists of a single server with a general service time distribution function $B\left(.\right)$. The well-known machine repair model with spares, where a set of identical machines, $N$, is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.