Bivariate failure modeling

Issues related to the construction of bivariate reliability models and their application to maintenance planning are discussed. The distinction between bivariate failure models and models of first passage time to a failure threshold clarifies the motivation for the development of bivariate models. The authors present a taxonomy of model classes and identify two classes as their focus. The model classes examined here are those in which the two variables are related by a stochastic function and those in which the variables are simply correlated. Examples of the models of each of the two classes are defined. The general approach to model formulation is explained so that the reader may construct alternate forms. The sometimes subtle aspects of model analysis are discussed with particular emphasis on the interpretation of bivariate failure probabilities and on the calculating of numerical results. Associated issues related to the construction of renewal models that can be used in maintenance planning are also discussed. For each of the topics addressed, example calculations are provided or else the technical roadblocks to continuing analysis are identified. In the authors' view, the types of bivariate models described here provide a new way to study the reliability of equipment for which univariate measures are incomplete. Thus, a new area of reliability research is identified. The definitions they offer may be modified, and the approach to model formulation they present may be used to define other models. They raise several open questions concerning the model construction and analysis. Both conceptual definitions and analytical methods warrant further exploration.