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A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions

Author

Listed:
  • Serguei Maximov

    (PGIIE, Tecnologico Nacional de Mexico, Instituto Tecnologico de Morelia)

  • Consuelo de J. Cortes-Penagos

    (Universidad Michoacana de San Nicolás de Hidalgo)

Abstract

The Kijima’s type 1 maintenance model, representing the general renewal process, is one of the most important in the reliability theory. The g-renewal equation is central in Kijima’s theory and it is a Volterra integral equation of the second kind. Although these equations are well-studied, a closed-form solution to the g-renewal equation has not yet been obtained. Despite the fact that several semi-empirical techniques to approximate the g-renewal function have been previously developed, analytical approaches to solve this equation for a wide class of underlying distributions is still of current interest. In this paper, a long-time asymptotic for the g-renewal rate is obtained for distributions with nondecreasing hazard functions and for all values of the restoration factor $$q\in [0,1]$$ q ∈ [ 0 , 1 ] . The obtained analytical result is compared with the numerical solutions for two types of underlying distributions, showing a good asymptotic match. The obtained approximate g-renewal rate is employed for maintenance optimization, considering the repair cost as a function of the restoration factor. Several numerical examples are performed in order to show the efficiency of our results.

Suggested Citation

  • Serguei Maximov & Consuelo de J. Cortes-Penagos, 2020. "A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 311-341, October.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:2:d:10.1007_s00186-020-00715-9
    DOI: 10.1007/s00186-020-00715-9
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    References listed on IDEAS

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