Reliability Evaluation of Large-Scale Systems With Identical Units

The reliability assessment of a large-scale system that considers its units´ degradation is challenging due to the resulting dimensionality problem. We propose a new methodology that allows us to overcome difficulties in analyzing large-scale system dynamics, and devise analytical methods for finding the multivariate distribution of the dynamically changing system condition. When each unit´s degradation condition can be classified into a finite number of states, and the transition distribution from one state to another is known, we obtain the asymptotic distribution of the number of units at each degradation state using fluid and diffusion limits. Specifically, we use a uniform acceleration technique, and obtain the time-varying mean vector and the covariance matrix of the number of units at multiple degradation states. When a state transition follows a non-Markovian deterioration process, we integrate phase-type distribution approximations with the fluid and diffusion limits. We show that, with any transition time distributions, the distribution of the number of units at multiple degradation conditions can be approximated by the multivariate Gaussian distribution as the total number of units gets large. The analytical results enable us to perform probabilistic assessment of the system condition during the system´s service life. Our numerical studies suggest that the proposed methods can accurately characterize the stochastic evolution of the system condition over time.