Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: An algorithmic approach |
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Authors: | Attahiru Sule Alfa |
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Institution: | (1) Department of Electrical & Computer Engineering, University of Manitoba, Winnipeg, Manitoba, R3T 5V6, Canada |
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Abstract: | In this paper, we show that the discrete GI/G/1 system with Bernoulli retrials can be analyzed as a level-dependent QBD process with infinite blocks; these blocks are finite when both the inter-arrival and service times have finite supports.
The resulting QBD has a special structure which makes it convenient to analyze by the Matrix-analytic method (MAM). By representing both the
inter-arrival and service times using a Markov chain based approach we are able to use the tools for phase type distributions
in our model. Secondly, the resulting phase type distributions have additional structures which we exploit in the development
of the algorithmic approach. The final working model approximates the level-dependent Markov chain with a level independent
Markov chain that has a large set of boundaries. This allows us to use the modified matrix-geometric method to analyze the
problem. A key task is selecting the level at which this level independence should begin. A procedure for this selection process
is presented and then the distribution of the number of jobs in the orbit is obtained. Numerical examples are presented to
demonstrate how this method works. |
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Keywords: | Matrix-analytic method Discrete-time GI/G/1 Retrial queue Bernoulli retrials QBD Infinite blocks Steady state distribution Discrete phase type distributions |
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