On a single-server queueing system with refusal
Author:
I. K. Matsak
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 90 (2015), 153-160
MSC (2010):
Primary 60K25, 90B22
DOI:
https://doi.org/10.1090/tpms/956
Published electronically:
August 10, 2015
MathSciNet review:
3242027
Full-text PDF Free Access
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Abstract: A single-server queueing system is considered with refusal of a general type. Stationary probabilities are found and the central limit theorem is established for the sojourn time.
References
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- Thomas L. Saaty, Elements of queueing theory, with applications, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1961. MR 0133176
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- A. A. Borovkov, Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800
- Lajos Takács, Some probability questions in the theory of telephone traffic, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 8 (1958), 151–210 (Hungarian). MR 0096319
- Lajos Takács, Introduction to the theory of queues, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962. MR 0133880
- I. N. Kovalenko, Studies on the Analysis of Reliability of Compound Systems, “Naukova Dumka”, Kiev, 1975. (Russian)
- V. S. Korolyuk and A. F. Turbin, Polumarkovskie protsessy i ikh prilozheniya, Izdat. “Naukova Dumka”, Kiev, 1976 (Russian). MR 0420902
- V. V. Anisimov, Asymptotic Methods of Analysis of Stochastic Systems, Metsniereba, Tbilisi, 1984. (Russian)
- K. Yu. Zhernovyĭ, An investigation of an $M^{\theta }/G/1/m$ queueing system with service mode switching, Teor. Imovirnost. Matem. Statyst. 86 (2012), 56–68; English transl. in Theor. Probab. Math. Statist. 86 (2013), 65–78.
- K. Yu. Zhernovyĭ, Busy period and stationary distribution for the queueing system $\rm M^\theta /G/1/\infty $ with a threshold switching between service modes, Theory Probab. Math. Statist. 87 (2013), 51–63. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 46–57. MR 3241446, DOI https://doi.org/10.1090/S0094-9000-2014-00904-7
- D. R. Cox, Renewal theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962. MR 0153061
- Walter L. Smith, Renewal theory and its ramifications, J. Roy. Statist. Soc. Ser. B 20 (1958), 243–302. MR 99090
- B. V. Gnedenko, Yu. K. Belyaev, and A. D. Solov’yev, Mathematical Methods in Reliability Theory, “Nauka”, Moscow, 1965. (Russian)
- Alexandr A. Borovkov, Probability theory, Universitext, Springer, London, 2013. Translated from the 2009 Russian fifth edition by O. B. Borovkova and P. S. Ruzankin; Edited by K. A. Borovkov. MR 3086572
References
- B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, “Nauka”, Moscow, 1966; English transl., Birkhäser Boston, Inc., Boston, MA, 1989. (translated from the second Russian edition by Samuel Kotz) MR 0230395 (37:5957)
- T. L. Saaty, Elements of Queueing Theory, with Applications, McGraw–Hill, New York, 1961. MR 0133176 (24:A3010)
- J. Riordan, Stochastic Service Systems, Wiley, New York, 1962. MR 0133879 (24:A3703)
- A. A. Borovkov, Stochastic Processes in Queueing Theory, “Nauka”, Moscow, 1972; English transl., Springer-Verlag, New York–Berlin, 1976. MR 0315800 (47:4349)
- L. Takács, Some probability questions in the theory of telephone traffic, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 8 (1958), 151–210. (Hungarian) MR 0096319 (20:2803)
- L. Takács, Introduction to the Theory of Queues, Oxford University Press, New York, 1962. MR 0133880 (24:A3704)
- I. N. Kovalenko, Studies on the Analysis of Reliability of Compound Systems, “Naukova Dumka”, Kiev, 1975. (Russian)
- V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and their Applications, “Naukova Dumka”, Kiev, 1976. (Russian) MR 0420902 (54:8913)
- V. V. Anisimov, Asymptotic Methods of Analysis of Stochastic Systems, Metsniereba, Tbilisi, 1984. (Russian)
- K. Yu. Zhernovyĭ, An investigation of an $M^{\theta }/G/1/m$ queueing system with service mode switching, Teor. Imovirnost. Matem. Statyst. 86 (2012), 56–68; English transl. in Theor. Probab. Math. Statist. 86 (2013), 65–78.
- K. Yu. Zhernovyĭ, Busy period and stationary distribution for the queueing system $M^{\theta }/G/1/\infty$ with a threshold switching between service modes, Teor. Imovirnost. Matem. Statyst. 87 (2012), 46–57; English transl. in Theor. Probab. Math. Statist. 87 (2013), 51–63. MR 3241446
- D. R. Cox, Renewal Theory, Methuen, London, 1962. MR 0153061 (27:3030)
- W. L. Smith, Renewal theory and its ramifications, J. Roy. Statist. Soc. Ser. B 20 (1958), 243–302. MR 0099090 (20:5534)
- B. V. Gnedenko, Yu. K. Belyaev, and A. D. Solov’yev, Mathematical Methods in Reliability Theory, “Nauka”, Moscow, 1965. (Russian)
- A. A. Borovkov, Probability Theory, “Nauka”, Moscow, 1976; English transl. Springer, London, 2013. (translated from the 2009 Russian fifth edition) MR 3086572
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Additional Information
I. K. Matsak
Affiliation:
Department of Operations Research, Faculty for Cybernetics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
ivanmatsak@univ.kiev.ua
Keywords:
Queueing systems,
stationary probabilities,
central limit theorem
Received by editor(s):
June 7, 2013
Published electronically:
August 10, 2015
Article copyright:
© Copyright 2015
American Mathematical Society
