On transient queue-size distribution in the batch-arrivals system with a single vacation policy

Wojciech M. Kempa

Kybernetika (2014)

  • Volume: 50, Issue: 1, page 126-141
  • ISSN: 0023-5954

Abstract

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A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.

How to cite

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Kempa, Wojciech M.. "On transient queue-size distribution in the batch-arrivals system with a single vacation policy." Kybernetika 50.1 (2014): 126-141. <http://eudml.org/doc/261130>.

@article{Kempa2014,
abstract = {A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.},
author = {Kempa, Wojciech M.},
journal = {Kybernetika},
keywords = {batch Poisson arrivals; queue-size distribution; renewal theory; single vacation; transient state; batch Poisson arrivals; queue-size distribution; renewal theory; single vacation; transient state},
language = {eng},
number = {1},
pages = {126-141},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On transient queue-size distribution in the batch-arrivals system with a single vacation policy},
url = {http://eudml.org/doc/261130},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Kempa, Wojciech M.
TI - On transient queue-size distribution in the batch-arrivals system with a single vacation policy
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 1
SP - 126
EP - 141
AB - A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.
LA - eng
KW - batch Poisson arrivals; queue-size distribution; renewal theory; single vacation; transient state; batch Poisson arrivals; queue-size distribution; renewal theory; single vacation; transient state
UR - http://eudml.org/doc/261130
ER -

References

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  2. Bischof, W., 10.1023/A:1013992708103, Queueing Syst. 39 (2001), 4, 265-301. Zbl0994.60088MR1885740DOI10.1023/A:1013992708103
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  5. Bratiichuk, M. S., Kempa, W. M., 10.1081/STM-200033115, Stoch. Models 20 (2004), 4, 457-472. MR2094048DOI10.1081/STM-200033115
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  8. Kempa, W. M., 10.1007/s00186-008-0212-2, Math. Methods Oper. Res. 69 (2009), 1, 81-97. Zbl1170.60032MR2476049DOI10.1007/s00186-008-0212-2
  9. Kempa, W. M., 10.1080/07362990903417920, Stoch. Anal. Appl. 28 (2010), 1, 26-43. Zbl1189.60168MR2597978DOI10.1080/07362990903417920
  10. Kempa, W. M., On departure process in the batch arrival queue with single vacation and setup time., Ann. UMCS, AI 10 (2010), 1, 93-102. Zbl1284.60162MR3116951
  11. Kempa, W. M., Characteristics of vacation cycle in the batch arrival queueing system with single vacations and exhaustive service., Internat. J. Appl. Math. 23 (2010), 4, 747-758. Zbl1208.60096MR2731457
  12. Kempa, W. M., On main characteristics of the M / M / 1 / N queue with single and batch arrivals and the queue size controlled by AQM algorithms., Kybernetika 47 (2011), 6, 930-943. Zbl1241.90035MR2907852
  13. Kempa, W. M., 10.1007/978-3-642-30782-9_4, Lecture Notes Comp. Sci. 7314 (2012), 47-60. DOI10.1007/978-3-642-30782-9_4
  14. Prabhu, N. U., Stochastic Storage Processes., Springer 1998. Zbl0888.60073MR1492990
  15. Takagi, H., Queueing Analysis. A Foundation of Performance Evaluation. Volume 1: Vacation and Priority Systems. Part 1., North-Holland, Amsterdam 1991. Zbl0744.60114MR1149382
  16. Tang, Y., Tang, X., The queue-length distribution for M x / G / 1 queue with single server vacation., Acta Math. Sci. (Eng. Ed.) 20 (2000), 3, 397-408. Zbl0984.60097MR1793213
  17. Tian, N., Zhang, Z. G., Vacation Queueing Models. Theory and Applications., Springer, New York 2006. Zbl1104.60004MR2248264

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