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Transient workload distribution in the $$M/G/1$$ M / G / 1 finite-buffer queue with single and multiple vacations

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  • Wojciech M. Kempa

    (Institute of Mathematics)

Abstract

A finite-buffer $$M/G/1$$ M / G / 1 -type queueing system with exhaustive service and server vacations is considered. The single and multiple vacation policies are investigated separately. In the former case a single vacation is being initialized every time when the system becomes empty. After the vacation the server waits on standby for packets to start the service process. In the multiple vacation policy the server begins successive single vacation periods until at least one packet is present in the system. For both vacation policies systems of integral equations for the transient virtual waiting time $$v(t)$$ v ( t ) distributions, conditioned by the numbers of packets present in the system at the start epoch, are found. The representations for the twofold Laplace transforms of the conditional distributions of $$v(t)$$ v ( t ) are obtained and written in compact forms, convenient in numerical treatment. From these formulae stationary distributions of $$v(t)$$ v ( t ) and their means can be computed via usual operations on transforms. Some numerical illustrations of theoretical results are attached as well.

Suggested Citation

  • Wojciech M. Kempa, 2016. "Transient workload distribution in the $$M/G/1$$ M / G / 1 finite-buffer queue with single and multiple vacations," Annals of Operations Research, Springer, vol. 239(2), pages 381-400, April.
    • Handle: RePEc:spr:annopr:v:239:y:2016:i:2:d:10.1007_s10479-015-1804-x
    DOI: 10.1007/s10479-015-1804-x
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    References listed on IDEAS

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    1. Jung Baek & Ho Lee & Se Lee & Soohan Ahn, 2008. "A factorization property for BMAP/G/1 vacation queues under variable service speed," Annals of Operations Research, Springer, vol. 160(1), pages 19-29, April.
    2. Hideaki Takagi, 1994. "M/G/1//N Queues with Server Vacations and Exhaustive Service," Operations Research, INFORMS, vol. 42(5), pages 926-939, October.
    3. Surendra Gupta & Ayse Kavusturucu, 2000. "Production systems with interruptions, arbitrary topology and finite buffers," Annals of Operations Research, Springer, vol. 93(1), pages 145-176, January.
    4. Jung Baek & Ho Lee & Se Lee & Soohan Ahn, 2013. "A MAP-modulated fluid flow model with multiple vacations," Annals of Operations Research, Springer, vol. 202(1), pages 19-34, January.
    5. Merav Shomrony & Uri Yechiali, 2001. "Burst arrival queues with server vacations and random timers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(1), pages 117-146, April.
    6. Wojciech Kempa, 2009. "GI/G/1/∞ batch arrival queueing system with a single exponential vacation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 81-97, March.
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    Cited by:

    1. Chakravarthy, Srinivas R. & Shruti, & Kulshrestha, Rakhee, 2020. "A queueing model with server breakdowns, repairs, vacations, and backup server," Operations Research Perspectives, Elsevier, vol. 7(C).
    2. G. K. Tamrakar & A. Banerjee, 2020. "On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation," OPSEARCH, Springer;Operational Research Society of India, vol. 57(4), pages 1337-1373, December.
    3. Madhu Jain & Sandeep Kaur & Parminder Singh, 2021. "Supplementary variable technique (SVT) for non-Markovian single server queue with service interruption (QSI)," Operational Research, Springer, vol. 21(4), pages 2203-2246, December.

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