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Application of Decomposable Semi-Regenerative Processes to the Study of k -out-of- n Systems

Author

Listed:
  • Vladimir Rykov

    (Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., 117197 Moscow, Russia
    Department of Applied Mathematics and Computer Modelling, Gubkin Oil & Gas Russian State University (Gubkin University), 65 Leninsky Prospect, 119991 Moscow, Russia
    Institute for Transmission Information Problems (Named after A.A. Kharkevich) RAS, Bolshoy Karetny, 19, GSP-4, 127051 Moscow, Russia)

  • Nika Ivanova

    (Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., 117197 Moscow, Russia
    V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Str., 117997 Moscow, Russia)

  • Dmitry Kozyrev

    (Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., 117197 Moscow, Russia
    V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Str., 117997 Moscow, Russia)

Abstract

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k -out-of- n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k -out-of- n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system.

Suggested Citation

  • Vladimir Rykov & Nika Ivanova & Dmitry Kozyrev, 2021. "Application of Decomposable Semi-Regenerative Processes to the Study of k -out-of- n Systems," Mathematics, MDPI, vol. 9(16), pages 1-23, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1933-:d:614215
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    References listed on IDEAS

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    1. Yin, Juan & Cui, Lirong, 2021. "Reliability for consecutive-k-out-of-n: F systems with shared components between adjacent subsystems," Reliability Engineering and System Safety, Elsevier, vol. 210(C).
    2. Levitin, Gregory & Xing, Liudong & Dai, Yuanshun, 2021. "Optimal operation and maintenance scheduling in m-out-of-n standby systems with reusable elements," Reliability Engineering and System Safety, Elsevier, vol. 211(C).
    3. Srinivas R. Chakravarthy & A. Krishnamoorthy & P. V. Ushakumari, 2001. "A k -out-of- n reliability system with an unreliable server and phase type repairs and services: the ( N , T ) policy," International Journal of Stochastic Analysis, Hindawi, vol. 14, pages 1-20, January.
    4. Zdeněk Kala, 2020. "Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques," Sustainability, MDPI, vol. 12(11), pages 1-19, June.
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    Cited by:

    1. Nika Ivanova, 2023. "On Importance of Sensitivity Analysis on an Example of a k -out-of- n System," Mathematics, MDPI, vol. 11(5), pages 1-18, February.
    2. Vladimir Rykov & Olga Kochueva & Elvira Zaripova, 2023. "Renewable k -Out-of- n System with the Component-Wise Strategy of Preventive System Maintenance," Mathematics, MDPI, vol. 11(9), pages 1-21, May.

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