Properties of a multivariate survival distribution generated by a Weibull and inverse-Gaussian mixture

The authors consider the influence of the work environment on a system of nonrenewable components. The failure times for the components are Weibull distributed and the work environment has an inverse Gaussian distribution. A multivariate Weibull and inverse Gaussian mixture distribution is derived. Several pertinent properties for this multivariate distribution are discussed that shed some light on the nature of the distribution. The authors account for the operating environment and its changing nature by averaging over a parameter corresponding to the environment. The distribution is applied to find the mean number of components working at some mission time and the reliability for k-out-of-n components