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

Serguei Maximov () and Consuelo de J. Cortes-Penagos ()
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Serguei Maximov: PGIIE, Tecnologico Nacional de Mexico, Instituto Tecnologico de Morelia
Consuelo de J. Cortes-Penagos: Universidad Michoacana de San Nicolás de Hidalgo

Mathematical Methods of Operations Research, 2020, vol. 92, issue 2, No 4, 341 pages

Abstract: 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.

Keywords: G-renewal process; Optimal maintenance; Volterra integral equation; Asymptotic solution; 62N05; 45D05; 34M30 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s00186-020-00715-9

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