A Multivariate Cumulative Damage Shock Model with Block Preventive Maintenance

Abstract

This paper is concerned with the dependence structure of a multi-component, cumulative damage shock model, in which a system of components is subject to correlated shock damages and group repairs. A component fails when its cumulative damage exceeds a given threshold, and is then replaced by a new component. We incorporate a block maintenance schedule that simultaneously performs imperfect repairs at various components according to a predetermined time table. Using the coupling method, we show that if the multivariate shock damage and imperfect repair processes are, respectively, stochastically larger and smaller, than the multivariate failure process (i.e., the failure counts of various components accumulated over time) is stochastically larger. This result allows us to show that if both correlated shock damages and group imperfect repairs at various components are more orthant (supermodular) dependent, then the failure counts of various components are also more orthant (supermodular) dependent. An application to a multi-component system subject to simultaneous shock damages and simultaneous repairs is also given.