Analysis of GI/M/1 queues using the trial solution approach

Abstract

The subject of M/G/1 queue has been studied extensively in the literature by a number of researchers. Specially, M/G/1 queue with generalized vacations model has a very useful property named stochastic decomposition. On the contrary, the subject of GI/M/1 queue has been studied by quite fewer researchers. So the main objective of this paper is to present a simple method for analyzing GI/M/1 queue. For GI/M/1 queue, if we pick imbedded point by the instance prior to an arrival time, we can define Markov chain. Then from the transition probability matrix P associated with the imbedded Markov chain we make the balance equations. When ρ<1, the balance equations have one unique solution. So once we have a proper trial solution for the balance equations, we can easily the trial solution is right by substituting the trial solution into the balance equations. And we suggest the trial solutions using the regenerative process. Using this trial solution approach, we obtain the distribution of number of customers for N-policy GI/M/1 queue with EMV. Also From these results, we obtain the distribution of number of customers for N-policy GI/M/1 queue and GI/M/1 queue with EMV. And the decomposition does not hold for the N-policy GI/M/1 queue with EMV. We hope that we can use this trials solution approach in order to analyze other vacation types of the GI/M/1 queue and evaluate this tools for analyzing Multi-server GI/M/c queues.