Analysis of the Sojourn Time Distribution for M/G L /1 Queue with Bulk-Service of Exactly Size L

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Abstract

This paper presents a simple algorithm for computing the cumulative distribution function of the sojourn time of a random customer in an M/G L /1 queue with bulk-service of exactly size L. Both theoretical and numerical aspects related to this problem were not discussed by Chaudhry and Templeton in their monograph (1983). Our analysis is based on the roots of the so-called characteristic equation of the Laplace-Stieltjes transform (LST) of the sojourn time distribution. Using the method of partial fractions and residue theorem, we obtain a closed-form expression of sojourn time distribution, from which we can calculate the value of the distribution function for any given time t ∈ [0, + ). Finally, to ensure the reliability of the analytical procedure, employing the work done by Gross and Harris (1985), an effective way to validate the correctness of our results along with some numerical examples are also provided.

Keywords

Bulk-service Sojourn time Cumulative distribution function Roots Residue theorem 

Mathematics Subject Classification 2010

60K25 68M20 90B22 

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Notes

Acknowledgements

The authors acknowledge anonymous reviewers for their comments which were very helpful in improving the presentation of this paper. This research is supported by the National Natural Science Foundation of China (Nos. 71301111, 71571127), the Talent Introduction Foundation of Sichuan University of Science & Engineering (2017RCL55).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSichuan University of Science and EngineeringZigongChina
  2. 2.School of Mathematics, Software ScienceSichuan Normal UniversityChengduChina

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