On a redundant system with renewals
Authors:
B. V. Dovgaĭ and I. K. Matsak
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 94 (2017), 63-76
MSC (2010):
Primary 60K25, 90B22
DOI:
https://doi.org/10.1090/tpms/1009
Published electronically:
August 25, 2017
MathSciNet review:
3553454
Full-text PDF
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Additional Information
Abstract: A two units redundant system with renewals of general type is considered. Stationary probabilities are found and the average duration of busy periods are established for the stationary regime.
References
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- T. P. Mar′janovič, Some problems of reliability of systems with reserves, Ukrain. Mat. Ž. 15 (1963), 145–157 (Russian, with English summary). MR 0158758
- Ī. K. Matsak, On a single-server queueing system with losses, Teor. Ĭmovīr. Mat. Stat. 90 (2014), 135–141 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 90 (2015), 153–160. MR 3242027, DOI https://doi.org/10.1090/S0094-9000-2015-00956-X
- 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
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References
- B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev, Mathematical Methods of Reliability Theory, “Nauka”, Moscow, 1965; English transl. Scripta Technica, Inc. Probability and Mathematical Statistics, Translation edited by Richard E. Barlow, vol. 6, Academic Press, New York–London, 1969. MR 0345234
- B. Epstein and T. Hosford, Reliability of some two units redundant systems, Proc. 6-th Nat. Symp. on Reliability and Quality Control, 1960, pp. 466–476.
- B. V. Gnedenko, On a two-unit redundant system, Izv. Akad. Nauk SSSR Tehn. Kibernet. 4 (1964), 3–12. (Russian) MR 0191630
- B. V. Gnedenko, Doubling with renewal, Izv. Akad. Nauk SSSR Tehn. Kibernet. 5 (1964), 111–118. (Russian) MR 0173541
- I. N. Kovalenko, Studies in the Reliability Analysis of Complex Systems, “Naukova Dumka”, Kiev, 1975. (Russian) MR 0451435
- I. N. Kovalenko and N. Yu. Kuznetsov, Modelling highly reliable systems, Reliability of Technical Systems (I. A. Ushakov, ed.), “Radio i svyaz”, Moscow, 1985. (Russian)
- V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and their Applications, “Naukova Dumka”, Kiev, 1976. (Russian) MR 0420902
- V. S. Korolyuk and A. F. Turbin, Markov Renewal Processes in Problems of the Reliability of Systems, “Naukova Dumka”, Kiev, 1982. (Russian) MR 695006
- V. S. Korolyuk, Stochastic Models of Systems, “Naukova Dumka”, Kiev, 1989. (Russian) MR 1001967
- A. D. Solovyev, Reliability of systems with renewal, Cybernetics for Service to Communism, vol. 2, “Energiya”, Moscow, 1964. (Russian)
- A. D. Solovyev, Methods for evaluation of reliability of systems with renewal, Reliability of Technical Systems (I. A. Ushakov, ed.), “Radio i svyaz”, Moscow, 1985. (Russian)
- T. P. Maryanovich, Some problems of reliability of systems with reserves, Ukrain. Mat. Zh. 15 (1963), no. 2, 145–157. (Russian) MR 0158758
- I. K. Matsak, On a single-server queueing system with refusal, Teor. Imovirnost. Matem. Statyst. 90 (2014), 136–142; English transl. in Theor. Probability and Math. Statist. 90 (2015), 153–160. MR 3242027
- 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 0099090
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Theory of Probability and Mathematical Statistics, “Vyshcha shkola”, Kiev, 1988. (Russian)
- A. N. Kolmogorov, Probability Theory and Mathematical Statistics. Selected Works, “Nauka”, Moscow, 1986. (Russian) MR 861120
- V. V. Anisimov, O. K. Zakusylo, and V. S. Donchenko, Elements of the Queueing Theory and Asymptotic Analysis of Systems, “Vyshcha shkola”, Kiev, 1987. (Russian)
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Additional Information
B. V. Dovgaĭ
Affiliation:
Faculty for Computer Science and Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 4d, Kyiv 03680, Ukraine
Email:
bogdov@gmail.com
I. K. Matsak
Affiliation:
Faculty for Computer Science and Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 4d, Kyiv 03680, Ukraine
Email:
ivanmatsak@univ.kiev.ua
Keywords:
Redundant system,
limit theorems,
stationary probabilities
Received by editor(s):
April 11, 2016
Published electronically:
August 25, 2017
Article copyright:
© Copyright 2017
American Mathematical Society
