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.

代表一般更新过程的Kijima 1型维护模型是可靠性理论中最重要的模型之一。g更新方程是Kijima理论的核心，它是第二类Volterra积分方程。尽管已对这些方程式进行了深入研究，但尚未获得g更新方程的闭式解。尽管先前已经开发了几种近似g更新函数的半经验技术，但目前仍在关注用于解决这一类广泛基础分布问题的分析方法。在本文中，对于具有不递减危险函数的分布以及所有恢复因子\（q \ in [0,1] \）的值，获得了g更新率的长时间渐近线。。将获得的分析结果与两种类型基础分布的数值解进行比较，显示出良好的渐近匹配。考虑维修成本与恢复系数的函数关系，将获得的近似g更新率用于维修优化。为了说明我们的结果的有效性，进行了几个数值示例。