IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i16p3568-d1219492.html
   My bibliography  Save this article

Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information

Author

Listed:
  • Jewgeni H. Dshalalow

    (Department of Mathematical Sciences, College of Engineering and Science, Florida Institute of Technology, Melbourne, FL 32901, USA)

  • Hend Aljahani

    (Department of Mathematical Sciences, College of Engineering and Science, Florida Institute of Technology, Melbourne, FL 32901, USA)

Abstract

We study a reliability system subject to occasional random shocks of random magnitudes W 0 , W 1 , W 2 , … occurring at times τ 0 , τ 1 , τ 2 , … . Any such shock is harmless or critical dependent on W k ≤ H or W k > H , given a fixed threshold H . It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ , so that when δ T crosses some sustainability threshold D at time T , the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ 1 , τ 2 , … when the shocks are registered. Thus, if crossing of D by δ occurs at time T ∈ τ k , τ k + 1 , only at time τ k + 1 , can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the N th critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus ( D -operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T ). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.

Suggested Citation

  • Jewgeni H. Dshalalow & Hend Aljahani, 2023. "Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information," Mathematics, MDPI, vol. 11(16), pages 1-27, August.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3568-:d:1219492
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/16/3568/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/16/3568/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Coskun Kus & Altan Tuncel & Serkan Eryilmaz, 2022. "Assessment of Shock Models for a Particular Class of Intershock Time Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 213-231, March.
    2. Fermín Mallor & Javier Santos, 2003. "Reliability of systems subject to shocks with a stochastic dependence for the damages," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(2), pages 427-444, December.
    3. Jewgeni H. Dshalalow & Ryan T. White, 2022. "Fluctuation Analysis of a Soft-Extreme Shock Reliability Model," Mathematics, MDPI, vol. 10(18), pages 1-33, September.
    4. Jewgeni H. Dshalalow, 1997. "On the level crossing of multi-dimensional delayed renewal processes," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-7, January.
    5. Ali Doostmoradi & Mohammad Reza Akhoond & Mohammad Reza Zadkarami, 2023. "Reliability of a system under a new mixed shock model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(1), pages 156-169, January.
    6. Mohammad Hossein Poursaeed, 2021. "A run shock-erosion model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(5), pages 1228-1239, March.
    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. Jewgeni H. Dshalalow & Ryan T. White, 2022. "Fluctuation Analysis of a Soft-Extreme Shock Reliability Model," Mathematics, MDPI, vol. 10(18), pages 1-33, September.
    2. Lyu, Hao & Qu, Hongchen & Yang, Zaiyou & Ma, Li & Lu, Bing & Pecht, Michael, 2023. "Reliability analysis of dependent competing failure processes with time-varying δ shock model," Reliability Engineering and System Safety, Elsevier, vol. 229(C).
    3. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2018. "Constructing a Markov process for modelling a reliability system under multiple failures and replacements," Reliability Engineering and System Safety, Elsevier, vol. 173(C), pages 34-47.
    4. 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.
    5. Wei, Xiaohua & Bai, Sijun & Wu, Bei, 2023. "A novel shock-dependent preventive maintenance policy for degraded systems subject to dynamic environments and N-critical shocks," Reliability Engineering and System Safety, Elsevier, vol. 239(C).
    6. Zhao, Xian & Guo, Xiaoxin & Wang, Xiaoyue, 2018. "Reliability and maintenance policies for a two-stage shock model with self-healing mechanism," Reliability Engineering and System Safety, Elsevier, vol. 172(C), pages 185-194.
    7. Jewgeni H. Dshalalow & Ryan T. White, 2021. "Current Trends in Random Walks on Random Lattices," Mathematics, MDPI, vol. 9(10), pages 1-38, May.
    8. Stathis Chadjiconstantinidis & Altan Tuncel & Serkan Eryilmaz, 2023. "Α new mixed δ-shock model with a change in shock distribution," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 491-509, October.
    9. Jewgeni H. Dshalalow & Ahmed Merie & Ryan T. White, 2020. "Fluctuation Analysis in Parallel Queues with Hysteretic Control," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 295-327, March.
    10. Sophie Mercier & Hai Ha Pham, 2016. "A Random Shock Model with Mixed Effect, Including Competing Soft and Sudden Failures, and Dependence," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 377-400, June.
    11. Xiaoyue Wang & Ru Ning & Xian Zhao, 2023. "Generalized mixed shock model for multi-component systems in the shock environment with a change point," Journal of Risk and Reliability, , vol. 237(4), pages 619-635, August.
    12. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2016. "A warmstandby system under shocks and repair governed by MAPs," Reliability Engineering and System Safety, Elsevier, vol. 152(C), pages 331-338.
    13. María Luz Gámiz & Delia Montoro-Cazorla & María del Carmen Segovia-García & Rafael Pérez-Ocón, 2022. "MoMA Algorithm: A Bottom-Up Modeling Procedure for a Modular System under Environmental Conditions," Mathematics, MDPI, vol. 10(19), pages 1-19, September.
    14. Chadjiconstantinidis, Stathis & Eryilmaz, Serkan, 2023. "Reliability of a mixed δ-shock model with a random change point in shock magnitude distribution and an optimal replacement policy," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    15. Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On the general $$\delta $$ δ -shock model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 994-1029, December.
    16. Eryilmaz, Serkan, 2015. "Discrete time shock models involving runs," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 93-100.
    17. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2015. "A shock and wear model with dependence between the interarrival failures," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 339-352.
    18. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2014. "A redundant n-system under shocks and repairs following Markovian arrival processes," Reliability Engineering and System Safety, Elsevier, vol. 130(C), pages 69-75.
    19. Eryilmaz, Serkan & Unlu, Kamil Demirberk, 2023. "A new generalized δ-shock model and its application to 1-out-of-(m+1):G cold standby system," Reliability Engineering and System Safety, Elsevier, vol. 234(C).

    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:16:p:3568-:d:1219492. 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.