This paper presents modified preventive maintenance policies for an operating system that works at random processing times and is imperfectly maintained. The system may suffer from one of the two types of failures based on a time-dependent imperfect maintenance mechanism: type-Ⅰ (minor) failure, which can be rectified by minimal repair, and type-Ⅱ (catastrophic) failure, which can be removed by corrective maintenance. When the system needs to be maintained, two policies "preventive maintenance-first (PMF) and preventive maintenance-last (PML)" may be applied. In each maintenance interval, before any type-Ⅱ failure occurs, the system is maintained at a planned time $ T $ or at the completion of a working time, whichever occurs first and last, which are called PMF and PML, respectively. After any maintenance activity, the system improves but its failure characteristic is also altered. At the $ N $-th maintenance, the system is replaced rather than maintained. For each policy, the optimal preventive maintenance schedule ($ T $, $ N $)$ ^{*} $ that minimizes the mean cost rate function is derived analytically and determined numerically in terms of its existence and uniqueness. The proposed models provide a general framework for analyzing the maintenance policies in reliability theory.

中文翻译：

---

具有通用的不完善维护模型的最佳预防性“先维护或后维护”策略

本文介绍了针对在随机处理时间下工作且维护不完善的操作系统的预防性维护策略。基于时间相关的不完善维护机制，系统可能会遭受两种类型的故障之一：可以通过最少的维修纠正的Ⅰ型（次要）故障，以及可以通过以下方式修复的Ⅱ型（灾难性）故障：通过纠正性维护移除。当需要维护系统时，可以应用两个策略“预防性维护优先（PMF）和预防性维护最后（PML）”。在每个维护间隔中，在任何II类故障发生之前，系统都会按计划的时间$ T $或在工作时间完成时进行维护，以先发生和后发生的时间为准，分别称为PMF和PML。进行任何维护活动后，系统得到改善，但其故障特性也发生了变化。在第N次维护时，将替换而不是维护系统。对于每个策略，通过分析得出其平均成本率函数最小的最优预防性维护计划（$ T $，$ N $）$ ^ {*} $，并根据其存在性和唯一性从数值上确定。该模型为可靠性理论中的维修策略分析提供了一个通用框架。