Generalized renewal process for repairable systems based on finite Weibull mixture

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m-fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.