IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i23p4754-d1287242.html
   My bibliography  Save this article

On the Positive Recurrence of Finite Regenerative Stochastic Models

Author

Listed:
  • Evsey Morozov

    (Institute of Applied Mathematical Research, Karelian Research Centre, Russian Academy of Sciences, 185035 Petrozavodsk, Russia
    Department of Applied Mathematics and Informatics, Yaroslav-the-Wise Novgorod State University, 173020 Veliky Novgorod, Russia
    Department of Applied Mathematics and Cybernetics, Petrozavodsk State University, 185910 Petrozavodsk, Russia
    These authors contributed equally to this work.)

  • Vladimir Rykov

    (Department of Applied Mathematics and Computer Modelling, National University of Oil and Gas (Gubkin University), 119991 Moscow, Russia
    Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

We consider a general approach to establish the positive recurrence (stability) of regenerative stochastic systems. The approach is based on the renewal theory and a characterization of the remaining renewal time of the embedded renewal process generated by regeneration. We discuss how this analysis is simplified for some classes of the stochastic systems. The general approach is then illustrated by the stability analysis of a k -out-of- n repairable system containing n unreliable components with exponential lifetimes. Then we extend the stability analysis to the system with non-exponential lifetimes.

Suggested Citation

  • Evsey Morozov & Vladimir Rykov, 2023. "On the Positive Recurrence of Finite Regenerative Stochastic Models," Mathematics, MDPI, vol. 11(23), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4754-:d:1287242
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/23/4754/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/23/4754/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Karl Sigman, 1990. "One-Dependent Regenerative Processes and Queues in Continuous Time," Mathematics of Operations Research, INFORMS, vol. 15(1), pages 175-189, February.
    2. Yonit Barron & Uri Yechiali, 2017. "Generalized control-limit preventive repair policies for deteriorating cold and warm standby Markovian systems," IISE Transactions, Taylor & Francis Journals, vol. 49(11), pages 1031-1049, November.
    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. Mansour Shrahili & Mohamed Kayid, 2023. "Stochastic Orderings of the Idle Time of Inactive Standby Systems," Mathematics, MDPI, vol. 11(20), pages 1-21, October.
    2. Chen, Wu-Lin & Wang, Kuo-Hsiung, 2018. "Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy," Reliability Engineering and System Safety, Elsevier, vol. 180(C), pages 476-486.
    3. Alsmeyer, Gerold, 1996. "Superposed continuous renewal processes A Markov renewal approach," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 311-322, February.
    4. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Research Memorandum 736, Tilburg University, School of Economics and Management.
    5. Wu, Hui & Li, Yan-Fu & Bérenguer, Christophe, 2020. "Optimal inspection and maintenance for a repairable k-out-of-n: G warm standby system," Reliability Engineering and System Safety, Elsevier, vol. 193(C).
    6. Nieuwenhuis, G., 1996. "Ergodicity Conditions and Cesaro Limit Results for Marked Point Processes," Other publications TiSEM e9963ce3-420e-49b5-88cc-9, Tilburg University, School of Economics and Management.
    7. Halim Damerdji & Peter W. Glynn, 1998. "Limit Theory for Performance Modeling of Future Event Set Algorithms," Management Science, INFORMS, vol. 44(12-Part-1), pages 1709-1722, December.
    8. Bardhan, Indrajit & Sigman, Karl, 1995. "Stationary regimes for inventory processes," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 77-86, March.
    9. Irina Peshkova & Evsey Morozov & Michele Pagano, 2024. "Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes," Mathematics, MDPI, vol. 12(14), pages 1-22, July.
    10. Bazsa-Oldenkamp, E.M. & den Iseger, P., 2003. "A two level decentralized distribution system with compound renewal demand," Econometric Institute Research Papers EI 2002-45, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    11. Liang, Zhenglin & Parlikad, Ajith Kumar, 2020. "Predictive group maintenance for multi-system multi-component networks," Reliability Engineering and System Safety, Elsevier, vol. 195(C).
    12. Joby K. Jose & M. Drisya, 2020. "Time-dependent stress–strength reliability models based on phase type distribution," Computational Statistics, Springer, vol. 35(3), pages 1345-1371, September.
    13. Konstantin Avrachenkov & Evsey Morozov, 2014. "Stability analysis of GI/GI/c/K retrial queue with constant retrial rate," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(3), pages 273-291, June.
    14. Bazsa-Oldenkamp, E.M. & den Iseger, P., 2002. "A two level decentralized distribution system with compound renewal demand," Econometric Institute Research Papers EI 2002-45, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    15. Bazsa-Oldenkamp, E.M. & den Iseger, P., 2003. "Wide sense one-dependent processes with embedded Harris chains and their applications in inventory management," Econometric Institute Research Papers EI 2002-44, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    16. Peter Glynn & Yanlin Qu, 2023. "On a New Characterization of Harris Recurrence for Markov Chains and Processes," Mathematics, MDPI, vol. 11(9), pages 1-7, May.
    17. Xiaowei Zhang & Peter W. Glynn, 2018. "Affine Jump-Diffusions: Stochastic Stability and Limit Theorems," Papers 1811.00122, arXiv.org.
    18. Gao, Shan & Wang, Jinting, 2021. "Reliability and availability analysis of a retrial system with mixed standbys and an unreliable repair facility," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    19. Glynn, Peter & Sigman, Karl, 1998. "Independent sampling of a stochastic process," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 151-164, June.
    20. E. Morozov & B. Steyaert, 2013. "Stability analysis of a two-station cascade queueing network," Annals of Operations Research, Springer, vol. 202(1), pages 135-160, 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:11:y:2023:i:23:p:4754-:d:1287242. 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.