Specifying interdependence in networked systems

Realistic assessments of the reliability of networked systems, series and parallel systems being special cases, require that we account for interdependence between the component life-lengths. The key to doing this is the specification and use of a suitable probability model in two or more dimensions. Consequently, several multivariate probabilistic models have been proposed in the literature. Many of these models have marginal distributions that are exponential; the ones by Gumbel, and by Marshall and Olkin being some of the earliest and the best known. The purpose of this paper is two fold: The first purpose is to articulate the nature of dependence encapsulated by such models, using a perspective which is best appreciated by a user. Specifically, we anchor on the bivariate case, and focus attention on the conditional mean as a measure of dependence. The second purpose, motivated by the first, is to introduce a new family of multivariate distributions with exponential marginals, whose conditional mean fills a void in the general forms of the conditional means of the available models. The method of "copulas" is used to generate this new family of distributions. Attention is focused on the case of exponential marginals, because the notion of "hazard potentials" enables us to use multivariate distributions with exponential marginals as a seed for generating multivariate distributions with marginals other than the exponential.