A stochastic regime switching model for the failure process of a repairable system

Abstract

This paper presents a stochastic model and estimation procedure for analyzing the failure process of a repairable system. We consider repairable systems whose successive interfailure times reveal a significant dependence while showing an insignificant trend. Neither the renewal process nor the non-homogeneous Poisson process are adequate for modeling such failure processes. Especially when the interfailure times show a cyclic pattern, we may consider a switching of the regimes (states) governing the lifetime distribution of the system. We propose a Markov switching model describing the failure process for such a case. The model postulates that a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain. Each of the distinct lifetime distributions represents a failure type that may change after successive repairs. Our model generalizes the mixture model by allowing the mixture probabilities to change during the transient period of the system. The model can capture the transient behavior of the system. The interfailure times constitute a set of incomplete data because the states are not explicitly identified. For the incomplete data, we propose a procedure for finding the maximum likelihood estimates of the model parameters by adopting the expectation and maximization principle. We also suggest a statistical method to determine the number of significant states. A Monte Carlo study is performed with two-parameter Weibull lifetime distributions. The results show consistency and good properties of the estimates. Some sets of Proschan's air conditioning unit data [ Technometrics, 1963, 5′ 375–383] are also analyzed and the results are discussed with respect to the number of significant states and the performance of the prediction.