Chapter 6 - Multivariate Reliability Concepts

Abstract

There are many complex devices and systems whose functioning depend on several components that may be dependent or independent. In such a scenario, the lifetime is composed of the lifetimes of components that may depend on different physical properties. In such a case, the study of system reliability can be facilitated only through the joint distribution of the component lifetimes. Thus, there is a need to extend various reliability concepts to higher dimensions. In this chapter, we discuss various multivariate reliability concepts. The development of concepts depends on the manner in which a univariate notion is generalized to suit the multivariate case. So, there are several possible definitions for a particular notion. We give five types of definitions of the multivariate hazard rate, namely, the scalar hazard rate, the conditional hazard rate, the vector hazard rate, the total hazard rate and an alternative conditional hazard rate. Each rate is defined and interpreted and the formulas for determining the life distribution from them are presented. Various properties of each of the rates are also discussed. Similarly, three alternative definitions of multivariate mean residual life, their properties including the relationships with hazard rates and expressions for the survival function and appropriate hazard rates in terms of mean residual life are also given. The variance-covariance matrix of the bivariate residual life is defined. As its elements, the variances and covariance of residual lives are discussed. Identities connecting the variance and covariance functions with other reliability functions form the discussion that follows. The reliability functions in reversed time are then considered by extending the univariate concepts to different forms of multivariate reversed hazard rate, mean residual life, variance and covariance residual life. In modelling multivariate data, the nature of dependence between the constituent variables play an important role. Therefore, we discuss various time-dependent measures of association and their relationship with different reliability functions. This chapter then concludes with the study of three different forms of bivariate equilibrium distributions. In each case, we show how the reliability functions of the baseline model is related to those of the corresponding equilibrium forms.