There are several measures of the importance of a component within a s
ystem in determining system reliability; these measures include struct
ural importance, marginal reliability-importance, joint reliability-im
portance, and link importance. These measures are reviewed and then re
lated to each other. Extensions are proposed to each importance measur
e to cover reliability models in which the components have two failure
-modes rather than the conventional one failure-mode; these extensions
have properties similar to those in the one failure-mode model.