IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i3p459-d336586.html
   My bibliography  Save this article

Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

Author

Listed:
  • Boyan Dimitrov

    (Department of Mathematics, Kettering University, Flint, MI 48504, USA)

  • Vladimir Rykov

    (Department of Applied Mathematics and Computer Modeling, Gubkin Russian State Oil and Gas University (Gubkin University), 119991 Moscow, Russia
    Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia)

  • Tatiana Milovanova

    (Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia)

Abstract

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements.

Suggested Citation

  • Boyan Dimitrov & Vladimir Rykov & Tatiana Milovanova, 2020. "Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis," Mathematics, MDPI, vol. 8(3), pages 1-12, March.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:459-:d:336586
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/3/459/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/8/3/459/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Li, Xiaohu & Pellerey, Franco, 2011. "Generalized Marshall-Olkin distributions and related bivariate aging properties," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1399-1409, November.
    2. Omey, E. & Willekens, E., 1989. "Abelian and Tauberian theorems for the Laplace transform of functions in several variables," Journal of Multivariate Analysis, Elsevier, vol. 30(2), pages 292-306, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pellerey, Franco & Shaked, Moshe & Yasaei Sekeh, Salimeh, 2012. "Comparisons of concordance in additive models," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 2059-2067.
    2. Jianhua Lin & Xiaohu Li, 2014. "Multivariate Generalized Marshall–Olkin Distributions and Copulas," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 53-78, March.
    3. Hyunju Lee & Ji Hwan Cha, 2021. "A general multivariate new better than used (MNBU) distribution and its properties," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(1), pages 27-46, January.
    4. Umberto Cherubini & Sabrina Mulinacci, 2021. "Hierarchical Archimedean Dependence in Common Shock Models," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 143-163, March.
    5. Hugo Brango & Angie Guerrero & Humberto Llinás, 2024. "Marshall–Olkin Bivariate Weibull Model with Modified Singularity (MOBW- μ ): A Study of Its Properties and Correlation Structure," Mathematics, MDPI, vol. 12(14), pages 1-16, July.
    6. Matthias Scherer & Henrik Sloot, 2019. "Exogenous shock models: analytical characterization and probabilistic construction," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(8), pages 931-959, November.
    7. Sabrina Mulinacci, 2022. "A Marshall-Olkin Type Multivariate Model with Underlying Dependent Shocks," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2455-2484, December.
    8. Gwo Dong Lin & Xiaoling Dou & Satoshi Kuriki, 2019. "The Bivariate Lack-of-Memory Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(2), pages 273-297, December.
    9. Li, Yang & Sun, Jianguo & Song, Shuguang, 2012. "Statistical analysis of bivariate failure time data with Marshall–Olkin Weibull models," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 2041-2050.
    10. Sabrina Mulinacci, 2018. "Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 205-236, March.
    11. Sloot Henrik, 2020. "The deFinetti representation of generalised Marshall–Olkin sequences," Dependence Modeling, De Gruyter, vol. 8(1), pages 107-118, January.
    12. Yang Lu, 2020. "The distribution of unobserved heterogeneity in competing risks models," Statistical Papers, Springer, vol. 61(2), pages 681-696, April.
    13. Somayeh Ashrafi & Majid Asadi & Razieh Rostami, 2024. "On preventive maintenance of k-out-of-n systems subject to fatal shocks," Journal of Risk and Reliability, , vol. 238(2), pages 291-303, April.
    14. Sabrina Mulinacci, 2017. "A systemic shock model for too big to fail financial institutions," Papers 1704.02160, arXiv.org, revised Apr 2017.
    15. Mercier, Sophie & Pham, Hai Ha, 2017. "A bivariate failure time model with random shocks and mixed effects," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 33-51.
    16. Pinto, Jayme & Kolev, Nikolai, 2015. "Sibuya-type bivariate lack of memory property," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 119-128.
    17. Yinping You & Xiaohu Li & Narayanaswamy Balakrishnan, 2014. "On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(8), pages 1041-1056, November.
    18. Mallor, F. & Omey, E. & Santos, J., 2007. "Multivariate weighted renewal functions," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 30-39, January.
    19. Gobbi, Fabio & Kolev, Nikolai & Mulinacci, Sabrina, 2021. "Ryu-type extended Marshall-Olkin model with implicit shocks and joint life insurance applications," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 342-358.
    20. Sloot Henrik, 2020. "The deFinetti representation of generalised Marshall–Olkin sequences," Dependence Modeling, De Gruyter, vol. 8(1), pages 107-118, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:459-:d:336586. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.