This article discusses the stochastic behavior and reliability properties for the inactivity times of failed components in coherent systems under double monitoring. A mixture representation of reliability function is obtained for the inactivity times of failed components, and some stochastic comparison results are also established. Furthermore, some sufficient conditions are developed in terms of the aging properties of the inactivity times of failed components. Finally, some numerical examples are presented to illustrate the theoretical results.

In this manuscript, we gain a mixture representation for reliability function of the residual lifetime of unfailed components in a coherent system under periodical inspections, given that the number of failed components before time $t_1$ is $r(\ge 0)$, but the system is still operating at time $t_1$, and the system eventually failed at time $t_2(>t_1)$. Some aging properties and stochastic orders of the residual lifetime on survival components are also established. Finally, some numerical examples and graphs are given in order to confirm the theoretical results.

Industrial systems function in dynamic settings characterized by changing conditions. This manuscript develops a reliability model and a condition-based maintenance approach for a single-component system with three states (normal, defective, and failed). The system is prone to failures resulting from both internal degradation and external shocks, and these shocks can cause abrupt rises in degradation. Internal degradation is simulated through a Wiener process that features elevated drift and diffusion coefficients when the system is in the defective state. External shocks follow a non-homogeneous Poisson process and are classified as safe, damaging, and fatal shocks. A preventive degradation threshold is defined according to the state of the system, and particle swarm optimization is utilized to determine the inspection interval and the optimal preventive threshold. Finally, the proposed maintenance model is verified using a numerical example.