IDEAS home Printed from https://ideas.repec.org/a/bit/bsrysr/v4y2013i1p65-75.html
   My bibliography  Save this article

A Simple Discrete Approximation for the Renewal Function

Author

Listed:
  • Brezavšček Alenka

    (University of Maribor, Faculty of Organizational Sciences, Kranj, Slovenia)

Abstract

Background: The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analysed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists.

Suggested Citation

  • Brezavšček Alenka, 2013. "A Simple Discrete Approximation for the Renewal Function," Business Systems Research, Sciendo, vol. 4(1), pages 65-75, March.
    • Handle: RePEc:bit:bsrysr:v:4:y:2013:i:1:p:65-75
    DOI: 10.2478/bsrj-2013-0006
    as

    Download full text from publisher

    File URL: https://doi.org/10.2478/bsrj-2013-0006
    Download Restriction: no
    File URL: https://libkey.io/10.2478/bsrj-2013-0006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Jiang, R., 2008. "A Gamma–normal series truncation approximation for computing the Weibull renewal function," Reliability Engineering and System Safety, Elsevier, vol. 93(4), pages 616-626.
    2. van Noortwijk, J.M. & van der Weide, J.A.M., 2008. "Applications to continuous-time processes of computational techniques for discrete-time renewal processes," Reliability Engineering and System Safety, Elsevier, vol. 93(12), pages 1853-1860.
    3. Jiang, R., 2010. "A simple approximation for the renewal function with an increasing failure rate," Reliability Engineering and System Safety, Elsevier, vol. 95(9), pages 963-969.
    4. Hu, Xiaomi, 2006. "Approximation of partial distribution in renewal function calculation," Computational Statistics & Data Analysis, Elsevier, vol. 50(6), pages 1615-1624, 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. Jiang, R., 2020. "A novel two-fold sectional approximation of renewal function and its applications," Reliability Engineering and System Safety, Elsevier, vol. 193(C).
    2. Gómez Fernández, Juan F. & Márquez, Adolfo Crespo & López-Campos, Mónica A., 2016. "Customer-oriented risk assessment in network utilities," Reliability Engineering and System Safety, Elsevier, vol. 147(C), pages 72-83.
    3. Mitra Fouladirad & Antoine Grall, 2015. "Monitoring and condition-based maintenance with abrupt change in a system’s deterioration rate," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(12), pages 2183-2194, September.
    4. Asadi, Majid, 2023. "On a parametric model for the mean number of system repairs with applications," Reliability Engineering and System Safety, Elsevier, vol. 234(C).
    5. Parsa, Hossein & Jin, Mingzhou, 2013. "An improved approximation for the renewal function and its integral with an application in two-echelon inventory management," International Journal of Production Economics, Elsevier, vol. 146(1), pages 142-152.
    6. Jiang, R., 2010. "A simple approximation for the renewal function with an increasing failure rate," Reliability Engineering and System Safety, Elsevier, vol. 95(9), pages 963-969.
    7. Stathis Chadjiconstantinidis, 2023. "Sequences of Improved Two-Sided Bounds for the Renewal Function and the Solutions of Renewal-Type Equations," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-31, June.
    8. R. Jiang, 2022. "Two approximations of renewal function for any arbitrary lifetime distribution," Annals of Operations Research, Springer, vol. 311(1), pages 151-165, April.
    9. Wu, Shaomin, 2014. "Warranty return policies for products with unknown claim causes and their optimisation," International Journal of Production Economics, Elsevier, vol. 156(C), pages 52-61.
    10. Wang, Zhaoqiang & Hu, Changhua & Wang, Wenbin & Zhou, Zhijie & Si, Xiaosheng, 2014. "A case study of remaining storage life prediction using stochastic filtering with the influence of condition monitoring," Reliability Engineering and System Safety, Elsevier, vol. 132(C), pages 186-195.
    11. Khac Tuan Huynh & Anne Barros & Christophe Bérenguer, 2012. "Adaptive condition-based maintenance decision framework for deteriorating systems operating under variable environment and uncertain condition monitoring," Journal of Risk and Reliability, , vol. 226(6), pages 602-623, December.
    12. Jiang, R., 2009. "An accurate approximate solution of optimal sequential age replacement policy for a finite-time horizon," Reliability Engineering and System Safety, Elsevier, vol. 94(8), pages 1245-1250.

    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:bit:bsrysr:v:4:y:2013:i:1:p:65-75. 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: Peter Golla (email available below). General contact details of provider: https://www.sciendo.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.