Reliability analysis of shock-based deterioration using phase-type distributions

Abstract

This paper presents a model to estimate the lifetime of degrading infrastructure systems subject to shocks based on the family of Phase-type (PH) distributions. In particular, the paper focuses on damage accumulation when both the inter-arrival time of shocks and their sizes are random. PH distributions are applied to approximate any probability distribution with positive support; furthermore, their matrix-geometric properties allow to handle problems involving the calculation of convolutions (e.g., sum of shock sizes). The proposed PH shock model relaxes the identically distributed assumption for the inter-arrival times and/or shock sizes. Besides, the model provides easy-to-evaluate expressions for important reliability quantities such as the density function and the moments of the lifetime, and the mean and moments of the cumulative shock deterioration at any time. In order to fit data or theoretical distributions to PH, the paper compares and discusses two PH fitting algorithms: the Moment Matching (MM) and the Expectation Maximization (EM) methods in terms of accuracy, computational efficiency and the available information of the random variables to fit. Then, it provides an algorithm for the reliability estimation of infrastructures along with a study of its accuracy and efficiency; the results show acceptable execution times for most practical applications. Finally, the use of PH to handle degradation is illustrated with several examples of engineering interest; i.e., deterioration due to crack growth, corrosion, aftershocks sequences, among others.