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Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations

Author

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  • Qingqing Ye

    (Nanjing University of Science and Technology)

  • Liwei Liu

    (Nanjing University of Science and Technology)

Abstract

In this paper, we consider a new class of the GI/M/1 queue with single working vacation and vacations. When the system become empty at the end of each regular service period, the server first enters a working vacation during which the server continues to serve the possible arriving customers with a slower rate, after that, the server may resume to the regular service rate if there are customers left in the system, or enter a vacation during which the server stops the service completely if the system is empty. Using matrix geometric solution method, we derive the stationary distribution of the system size at arrival epochs. The stochastic decompositions of system size and conditional system size given that the server is in the regular service period are also obtained. Moreover, using the method of semi-Markov process (SMP), we gain the stationary distribution of system size at arbitrary epochs. We acquire the waiting time and sojourn time of an arbitrary customer by the first-passage time analysis. Furthermore, we analyze the busy period by the theory of limiting theorem of alternative renewal process. Finally, some numerical results are presented.

Suggested Citation

  • Qingqing Ye & Liwei Liu, 2017. "Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 685-714, September.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:3:d:10.1007_s11009-016-9496-5
    DOI: 10.1007/s11009-016-9496-5
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    References listed on IDEAS

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    1. Naishuo Tian & Zhe George Zhang, 2006. "Vacation Queueing Models Theory and Applications," International Series in Operations Research and Management Science, Springer, number 978-0-387-33723-4, April.
    2. Teghem, J., 1986. "Control of the service process in a queueing system," European Journal of Operational Research, Elsevier, vol. 23(2), pages 141-158, February.
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