Reliability estimation of a system subject to condition monitoring with two dependent failure modes

A new competing risk model is proposed to calculate the Conditional Mean Residual Life (CMRL) and Conditional Reliability Function (CRF) of a system subject to two dependent failure modes, namely, degradation failure and catastrophic failure. The degradation process can be represented by a three-state continuous-time stochastic process having a healthy state, a warning state, and a failure state. The system is subject to condition monitoring at regular sampling times that provides partial information about the system is working state and only the failure state is observable. To model the dependency between two failure modes, it is assumed that the joint distribution of the time to catastrophic failure and sojourn time in the healthy state follow Marshal-Olkin bivariate exponential distributions. The Expectation-Maximization algorithm is developed to estimate the model's parameters and the explicit formulas for the CRF and CMRL are derived in terms of the posterior probability that the system is in the warning state. A comparison with a previously published model is provided to illustrate the effectiveness of the proposed model using real data.