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On Steady State Reliability and Sensitivity Analysis of a k-out-of-n System Under Full Repair Scenario

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Distributed Computer and Communication Networks: Control, Computation, Communications (DCCN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13766 ))

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Abstract

The paper is devoted to the investigation of a k-out-of-n system under full repair scenario after its failure. This regime implies the restoration of all failed components for some random time. As a result, the system runs like a new one. A repairable system is one that is repaired not only after a component failure (partial repair), but also after the failure of the entire system (full repair). It is supposed that these repair times have both arbitrary and different distributions, while the components’ lifetime is exponentially distributed.

In some previous works, time-dependent reliability characteristics have been obtained with the theory of decomposable semi-regenerative processes [1] and method of characteristics [2] in the same assumptions about life and repair time distributions. Markovization method in particular method of supplementary variables [3] has been applied for some special cases of parameters k and n for calculation of the stationary characteristics. In the current paper, the closed-form representations for the steady state system reliability characteristics for arbitrary k and n under similar assumptions about life and repair time distributions are presented. The obtained expressions are demonstrated in terms of Laplace transform of components’ partial repair time. The results are validated on an example of a 3-out-of-6 system by substituting the exponential distribution of repair time. In addition, the probabilistic characteristics of this system in the case of rare failures, as well as some numerical example to show their insensitivity are considered.

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References

  1. Rykov, V., Ivanova, N., Kozyrev, D.: Application of decomposable semi-regenerative processes to the study of \(k\)-out-of-\(n\) systems. Mathematics 9(16), 1933 (2021). https://doi.org/10.3390/math9161933

    Article  Google Scholar 

  2. Dimitrov, B., Rykov, V.: On k-out-of-n system under full repair and arbitrary distributed repair time. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2021. LNCS, vol. 13144, pp. 323–335. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92507-9_26

    Chapter  MATH  Google Scholar 

  3. Rykov, V.V., Ivanova, N.M., Kozyrev, D.V.: Sensitivity analysis of a k-out-of-n:F system characteristics to shapes of input distribution. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2020. LNCS, vol. 12563, pp. 485–496. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-66471-8_37

    Chapter  Google Scholar 

  4. Trivedi, K.S., Bobbio, A.: Reliability and Availability Engineering - Modeling, Analysis, and Applications. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781316163047

    Book  Google Scholar 

  5. Zuo, M.J., Tian, Z.: \(k\)-out-of-\(n\) Systems. Wiley Encyclopedia of Operations Research and Management Science. Ed. by James J. Cochran (2010)

    Google Scholar 

  6. Wang, Y., Hu, L., Yang, L., Li, J.: Reliability modeling and analysis for linear consecutive-\(k\)-out-of-\(n\): F retrial systems with two maintenance activities. Reliab. Eng. Syst. Saf. 226, 108665 (2022). https://doi.org/10.1016/j.ress.2022.108665

    Article  Google Scholar 

  7. Zhang, T., Zhang, Y., Du, X.: Reliability analysis for k-out-of-n systems with shared load and dependent components. Struct. Multidiscip. Optim. 57(3), 913–923 (2018). https://doi.org/10.1007/s00158-017-1893-z

    Article  MathSciNet  Google Scholar 

  8. Eruguz, A.S., Tan, T., van Houtum, G.-J.: Optimizing usage and maintenance decisions for \(k\)-out-of-\(n\) systems of moving assets. Nav. Res. Logist. 64, 418–434 (2017). https://doi.org/10.1002/nav.21764

    Article  MathSciNet  MATH  Google Scholar 

  9. Rykov, V.V., Ivanova, N.M.: Reliability and sensitivity analysis of a repairable \(k\)-out-of-\(n\): F system with general life- and repair times distributions. In: Baraldi, P., Di Maio, F., Zio, E. (eds.) Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference. Research Publishing Services, Singapore (2020). https://doi.org/10.3850/978-981-14-8593-05750-cd

  10. Hu, L., Liu, S., Peng, R., Liu, Z.: Reliability and sensitivity analysis of a repairable \(k\)-out-of-\(n\): G system with two failure modes and retrial feature. Commun. Stat. Theory Methods (2020). https://doi.org/10.1080/03610926.2020.1788083

    Article  Google Scholar 

  11. Amari, S., Bergman, R.: Reliability analysis of \(k\)-out-of-\(n\) load-sharing systems. Ann. Reliab. Maintainab. Sympos. 440–445 (2008). https://doi.org/10.1109/RAMS.2008.4925836

  12. Krishnamoorthy, A., Ushakumari, P., Lakshmi, B.: \(K\)-out-of-\(n\) system with repair: the \(N\)-policy. Asia Pacific J. Oper. Res. 19, 47–61 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Ushakumari, P., Krishnamoorthy, A.: \(k\)-out-of-\(n\) system with repair: the \(max(N, T)\) policy. Perform. Eval. 57(2), 221–234 (2004). https://doi.org/10.1016/j.peva.2003.10.006

    Article  Google Scholar 

  14. Wu, W., Tang, Y., Yu, M., Jiang, Y.: Computation and profit analysis of a \(k\)-out-of-\(n\): G repairable system under \(N\)-policy with multiple vacations and one replaceable repair facility. RAIRO-Oper. Res. 49(4), 717–734 (2015). https://doi.org/10.1051/ro/2015001

    Article  MathSciNet  MATH  Google Scholar 

  15. Kuo, W., Zuo, M.: Optimal Reliability Modeling: Principles and Applications. Wiley, New York (2003)

    Google Scholar 

  16. Houankpo, H.G.K., Kozyrev, D.: Mathematical and simulation model for reliability analysis of a heterogeneous redundant data transmission system. Mathematics 9, 2884 (2021). https://doi.org/10.3390/math9222884

    Article  Google Scholar 

  17. Ivanova, N.: Modeling and simulation of reliability function of a \(k\)-out-of-\(n\): F system. Commun. Comput. Inf. Sci. 1337 (2020). https://doi.org/10.1007/978-3-030-66242-4_22

  18. Kala, Z.: New importance measures based on failure probability in global sensitivity analysis of reliability. Mathematics 9, 2425 (2021). https://doi.org/10.3390/math9192425

    Article  Google Scholar 

  19. Cox, D.R.: The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Math. Proc. Cambridge Philos. Soc. 51(3), 433–441 (1955). https://doi.org/10.1017/S0305004100030437

    Article  MathSciNet  MATH  Google Scholar 

  20. Kalashnikov, V.: Method of supplementary variables. In: Mathematical Methods in Queuing Theory. Mathematics and Its Applications, vol. 271. Springer, Dordrecht (1994). https://doi.org/10.1007/978-94-017-2197-4_10

  21. Rykov, V.: On steady state probabilities of renewable system with Marshal–Olkin failure model. Stat. Pap. 59(4), 1577–1588 (2018). https://doi.org/10.1007/s00362-018-1037-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This publication has been supported by the RUDN University Scientific Projects Grant System, project No. P02 IF, theme No. 021930-2-000 (review and analytic results). The publication has been partially funded by RFBR according to the research project No. 20-01-00575A (problem setting) and RSF project No. 22-49-02023 (formal analysis, validation).

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation

    N. M. Ivanova

  2. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, Moscow, 117997, Russia

    N. M. Ivanova

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  1. N. M. Ivanova
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Correspondence to N. M. Ivanova .

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Editors and Affiliations

  1. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. Peoples' Friendship University of Russia, Moscow, Russia

    Konstantin E. Samouylov

  3. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Ivanova, N.M. (2022). On Steady State Reliability and Sensitivity Analysis of a k-out-of-n System Under Full Repair Scenario. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks: Control, Computation, Communications. DCCN 2022. Lecture Notes in Computer Science, vol 13766 . Springer, Cham. https://doi.org/10.1007/978-3-031-23207-7_33

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  • DOI: https://doi.org/10.1007/978-3-031-23207-7_33

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