Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Botta RF, Harris CM, Marchal WG (1987) Characterization of generalized hyperexponential distribution functions. Stoch Model 3:115–148
Chaudhry ML, Templeton JGC (1983) A first course in bulk queues. Wiley, New York
Chaudhry ML (1991) Numerical issues in computing steady-state queueing-time distributions of single-server bulk-service qeueus: M/Gb/1 and M/G d/1. ORSA J Comput 4:300–311
Chaudhry ML, Gai J (2012) A simple and extended computational analysis of M/G\(_{j}^{(a, b)}\)/1 and M/G\(_{j}^{(a, b)}\)/1/(B+b) queues using roots. INFOR 50:72–79
Chaudhry ML, Samanta SK, Pacheco A (2012) Analytically explicit results for the GI/C-MSP/1 queueing system using roots. Probab Eng Inf Sci 26:221–244
Chaudhry ML, Singh G, Gupta UC (2013) A simple and complete computational analysis of MAP/R/1 queue using roots. Methodol Comput Appl Probab 15:563–582
Chaudhry ML, Kim JJ (2016) Analytically elegant and computationally efficient results in terms of roots for the GIX/M/c queueing system. Queueing Syst 82:237–257
Gross D, Harris CM (1985) Fundamentals of queueing theory, 2nd edn. Wiley, New York
Gupta UC, Singh G, Chaudhry ML (2016) An alternative method for computing system-length distributions of BMAP/R/1 and BMAP/D/1 queues using roots. Perform Eval 95:60–79
Le-Duc T, Koster RBM (2007) Travel time estimation and order batching in a 2-block warehouse. Eur J Oper Res 176:374–388
Pradhan S, Gupta UC, Samanta SK (2016) Analyzing an infinite buffer batch arrival and batch service queue under batch-size-dependent service policy. J Korean Stat Soc 45:137–148
Saaty TL (1961) Elements of queueing theory with appications. McGraw-Hill, New York
Samanta S, Chaudhry ML, Pacheco A, Gupta UC (2015) Analytic and computational analysis of the discrete-time GI/D-MSP/1 queue using roots. Comput Oper Res 56:33–40
Singh G, Chaudhry ML, Gupta UC (2012) Computing system-time and system-length distributions for MAP/D/1 queue using distributional Little’s law. Perform Eval 69:102–118
Singh G, Gupta UC, Chaudhry ML (2014) Analysis of queueing-time distributions for MAP/DN/1 queue. Int J Comput Math 91:1911–1930
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yu, M., Tang, Y. Analysis of the Sojourn Time Distribution for M/GL/1 Queue with Bulk-Service of Exactly Size L. Methodol Comput Appl Probab 20, 1503–1514 (2018). https://doi.org/10.1007/s11009-018-9635-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-018-9635-2