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On Stochastic Decomposition in M / G /1 Type Queues with Generalized Server Vacations

Author

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  • J. George Shanthikumar

    (University of California, Berkeley, California)

Abstract

Recently, S. W. Fuhrmann and R. B. Cooper showed that the stationary distribution of the number of customers in an M / G /1 queueing system with generalized server vacation is a convolution of the distribution functions of two independent positive random variables. One of these is the stationary distribution of the number of customers in an ordinary M / G /1 queueing system without server vacations. They use an elegant, intuitive approach to establish this result. In this paper, a mechanistic (analytic) proof of this result is given for systems more general than that discussed in Fuhrmann and Cooper. Such systems allow customer arrivals in bulk and some variations with reneging, balking, and in arrival rate that is dependent on system state.

Suggested Citation

  • J. George Shanthikumar, 1988. "On Stochastic Decomposition in M / G /1 Type Queues with Generalized Server Vacations," Operations Research, INFORMS, vol. 36(4), pages 566-569, August.
    • Handle: RePEc:inm:oropre:v:36:y:1988:i:4:p:566-569
    DOI: 10.1287/opre.36.4.566
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    Cited by:

    1. Atencia, I., 2017. "A Geo/G/1 retrial queueing system with priority services," European Journal of Operational Research, Elsevier, vol. 256(1), pages 178-186.
    2. B. Krishna Kumar & R. Rukmani & A. Thanikachalam & V. Kanakasabapathi, 2018. "Performance analysis of retrial queue with server subject to two types of breakdowns and repairs," Operational Research, Springer, vol. 18(2), pages 521-559, July.
    3. Renbin Liu & Attahiru Sule Alfa & Miaomiao Yu, 2019. "Analysis of an ND-policy Geo/G/1 queue and its application to wireless sensor networks," Operational Research, Springer, vol. 19(2), pages 449-477, June.
    4. Kailash C. Madan, 1999. "An M/G/1 queue with optional deterministic server vacations," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3-4), pages 83-95.
    5. Ioannis Dimitriou, 2013. "A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 542-571, October.
    6. Jianjun Li & Liwei Liu & Tao Jiang, 2018. "Analysis of an M/G/1 queue with vacations and multiple phases of operation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 51-72, February.
    7. I. Atencia & G. Bouza & P. Moreno, 2008. "An M [X] /G/1 retrial queue with server breakdowns and constant rate of repeated attempts," Annals of Operations Research, Springer, vol. 157(1), pages 225-243, January.

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    Keywords

    queues: stochastic decomposition;

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