Joint reliability importance of k-out-of-n systems

Joint Reliability Importance (JRI) is investigated to provide information on the type and the degree of interactions between two components by identifying the sign and the size of it. In specific, JRI is analyzed for k-out-of-n systems with (i) independently identically distributed (IID) components, (ii) non-identical and independent components and (iii) pair-wise dependent components. The closed form solution of JRI is derived for IID components. By using the solution, the variations of JRI are investigated with respect to the level of redundancy (n/k). In non-identical and independent case, the point of sign change of JRI is obtained. It is also shown that the sign of JRI can be determined by the relationships between the Schur-convexity (concavity) and the JRI. For dependent case, only the pair-wise dependence is considered. As a result, the error caused by assuming statistical independence between pair-wise dependent components is shown to be measured by their covariance and JRI.