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

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. 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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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 (2020). https://doi.org/10.1007/s00186-020-00715-9

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Keywords

  • G-renewal process
  • Optimal maintenance
  • Volterra integral equation
  • Asymptotic solution

Mathematics Subject Classification

  • 62N05
  • 45D05
  • 34M30