This article proposes an integrated approach for reliability modeling and maintenance scheduling of repairable parallel systems subject to hidden failures. The system consists of heterogeneous redundant subsystems whose failures are detected only by inspections. Inspections at periodic times reveal the components state, and repair actions are decided by the excursion of a basic state process describing the total number of failed components (deterioration level) in each subsystem. Using the standard renewal arguments, this article aims at minimizing the average cost rate by the joint determination of the optimal inspection interval $\tau$ and maintenance parameters $\kappa$ (partial repair threshold) and $\ell$ (preventive replacement threshold). We illustrate the procedure for the case as the components’ lifetimes conform to the Weibull distribution. The proposed model and the response of the optimal solutions to the model’s parameters are illustrated through numerical examples. The approach outlined here allows exploring different scenarios and further extensions.

In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.