Combined elapsed time and matrix-analytic method for the discrete time \(GI/G/1\) and \(GI^X/G/1\) systems. (English) Zbl 1046.90016

Summary: We show that the discrete \(GI/G/1\) system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The \(G\)-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the \(GI^X/G/1\) system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.


90B22 Queues and service in operations research
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