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 ≤ M servers, each having an exponential service time with rate λ . The second node consists of a single server with a general service time distribution function B . . 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.
Highlights
Computing the steady-state probability distribution of a nonMarkovian two-node closed queueing cyclic network is known to be computationally challenging
The first node consists of N ≤ M parallel identical servers having independent and identically distributed (i.i.d.) exponential service times with rate λ
The second node consists of a single server with i.i.d. service requirements Si, i = 1, 2, ⋯, having a general distribution function Bð:Þ
Summary
Computing the steady-state probability distribution of a nonMarkovian two-node closed queueing cyclic network is known to be computationally challenging. A key contribution of this paper is to develop an easy to understand efficient polynomial algorithm to compute the stationary distribution of the number of customers at node 2 and to determine the measures of performance of this two-node network model For this purpose, we develop and solve an imbedded Markov Chain (MC) with states given by the number of customers in the second node immediately after departure epochs. We develop and solve an imbedded Markov Chain (MC) with states given by the number of customers in the second node immediately after departure epochs Such an approach is standard in analyzing the general single server model with Markovian arrivals, M/G/1. Maddah and El-Taha [1] use the imbedded Markov chain approach representing node 2 system size probabilities at departure epochs They compute the transition probabilities using an exponential algorithm. In Appendix A we give a second direct proof of our main result in Theorem 3, and in Appendix B, we describe in detail the algorithm to compute the stationary probabilities
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