Some properties of a classification system for multivariate life distributions

Abstract

A good classification system for multivariate life distributions can play a major role in system analysis during the early stage of product design. The problem of multivariate classification has drawn the attention of the researchers, and various classes are now available in the literature. D. Roy (1994) proposed a unified classification system for the multivariate life laws. It not only retains the much-desired chain of implication but generalizes quite a few characterization results relating to various life distribution classes. This paper examines this classification system of Roy and establishes its importance by presenting some fundamental properties of the same for use during early stages of product planning. These properties are aimed at retaining the same classification results under deletion, addition, and scaling of components and subsystems of a product during the early stages of design. A multivariate series combination has been examined from the viewpoint of closure under multivariate IFR, IFRA, and NBU classes. The major use of a classification system is to provide reliability bounds under breaking the product into meaningful s-independent subsystems and their reintegration. This paper ensures that this happens in the multivariate setup so that close reliability bounds can be obtained in place of a complicated analysis. A demonstration example is presented which describes the calculation of reliability bounds in a general way, covering both series and parallel combinations and a complex dependent setup.