Characterizations of the Relative Behavior of Two Systems via Properties of Their Signature Vectors

Abstract

The signature of a system of components with independent and identically distributed (iid) lifetimes is a probability vector whose ith element represents the probability that the ith component failure causes the system to fail. (1985) introduced the concept and used it to characterize the class of systems that have increasing failure rates (IFR) when the components are iid IFR. (1999) showed that when signatures are viewed as discrete probability distributions, the stochastic, hazard rate or likelihood ratio ordering of two signature vectors implies the same ordering of the lifetimes of the corresponding systems in iid components. In this paper, these latter results are extended in a variety of ways. For example, conditions on system signatures are identified that are not only sufficient for such orderings of lifetimes to hold, but are also necessary. More generally, given any two coherent systems whose iid components have survival functions S i (t) and failure rates r i (t), respectively, for i=1,2, the number and locations of crossings of the systems’ survival functions or failure rates in (0, ∞) can be fully specified in terms of the two system signatures. One is thus able to deduce how these systems compare to each other in real time, in contrast to the asymptotic comparisons one finds in the literature.