A Weibull Model with Time-Varying Scale Parameter for Modeling Failure Processes of Repairable Systems

Abstract

A lifetime distribution model is usually inappropriate for modeling the times between successive failures (TBF) of a repairable system since the TBFs are generally not independent and identically distributed. Traditional methods for modeling the failure process of repairable systems focus on fitting the mean cumulative function (MCF) to a parametric model such as the power-law model. However, the fitted model does not directly provide the distribution of the time to the next failure at a given system age or failure event. To address this issue, we propose a Weibull model with constant shape parameter and time-varying scale parameter for modeling TBFs of a repairable system in this paper. It includes the Weibull distribution as its special case. Three specific parametric models for the scale parameter are developed. The models are suitable for the situations where the system's MTBF can be monotonic or bathtub-shaped, and bounded. Such a model can be viewed as a Weibull process model. Potential applications of the models include modeling manufacturing defect occurrence processes and evaluating the effectiveness of maintenance. Three real-world examples are included to illustrate the appropriateness and usefulness of these models.