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

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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 \(q\in [0,1]\). 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.

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Notes

  1. The existence of the second derivative of function H(x) follows from Theorem 1, which is necessary for existence of the derivative \(\partial _{q}w(x,q)\) and, therefore, for representing the g-renewal equation in the form (33).

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Acknowledgements

The authors are grateful for financial support to the following Mexican Institutions: CONACYT, SNI and PROMEP.

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Authors and Affiliations

  1. PGIIE, Tecnologico Nacional de Mexico, Instituto Tecnologico de Morelia, 58120, Morelia, Mich., Mexico

    Serguei Maximov

  2. Facultad de Químico Farmacobiología, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mich., Mexico

    Consuelo de J. Cortes-Penagos

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  1. Serguei Maximov
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  2. Consuelo de J. Cortes-Penagos
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Correspondence to Serguei Maximov.

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Maximov, S., Cortes-Penagos, C.J. A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions. Math Meth Oper Res 92, 311–341 (2020). https://doi.org/10.1007/s00186-020-00715-9

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  • DOI: https://doi.org/10.1007/s00186-020-00715-9

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