Reliability analysis of repairable systems considering unobserved heterogeneity and competing risks

Abstract

In repairable systems, the failure process defined by the failure intensity function can be impacted by three crucial characteristics: the type of repair performed after the failures occur, the underlying cause of the failure (competing risks) and the presence of unobservable factors acting on each failure time or on the system as a whole (unobserved heterogeneity). In particular, unobserved heterogeneity can be modeled by frailty models, well known in the reliability literature. However, the majority of existing studies in this domain assume only minimal repairs after failures, which is a highly restrictive assumption and not always applicable. There is, therefore, a theoretical gap to be explored encompassing frailty models that consider systems subject to both perfect and imperfect repairs and still subject to competing risks. The primary objective of this work is to present new parametric univariate and shared frailty models for multiple repairable systems, considering different types of repairs and a competing risks framework. These proposed models extend and generalize those already existing in the literature, as they account for perfect repairs and all possible failure memories within both the ARA$_m$ and ARI$_m$ classes of imperfect repairs. In this sense, they are models capable of simultaneously identifying the effect of the repairs actions and the presence of unobserved heterogeneity among failure times or among the systems under analysis. This characteristic holds substantial relevance in real-world situations, as a deeper understanding of the system's failure process can lead to improved preventive maintenance policies and reduced repair-related costs. In all proposed models, we assume that the initial failure intensity follows a Power Law Process and that the parametric frailty terms associated with failure times or systems follow a Gamma distribution. We employ a frequentist approach to construct the likelihood function for each model and suggest numerical methods for obtaining maximum likelihood estimators and their corresponding asymptotic confidence intervals. Additionally, we propose the use of Bayesian methodologies based on Markov Chain Monte Carlo algorithms as an alternative to the frequentist method. Simulation studies are conducted for each proposed model, and, finally, the methods presented are applied to real datasets.