This paper presents a simple algorithm for computing the cumulative distribution function of the sojourn time of a random customer in an M/GL/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.
Bulk-service Sojourn time Cumulative distribution function Roots Residue theorem
Mathematics Subject Classification 2010
60K25 68M20 90B22
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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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