Component importance for multi-state system lifetimes with renewal functions

Importance measures are widely used to characterize the roles of components in systems. The system lifetime can be divided into different life stages. Traditionally, importance measures do not consider the possible effect of the expected number of component failures over a system's lifetime and over different life stages, which, however, has a great effect on the system performance changes, and should therefore be taken into consideration. This paper extends the integrated importance measure (IIM) from unit time to system lifetime, and to different life stages. Based on the renewal functions of components, this measure can evaluate the changes of the system performance due to component failures. This generalization of the IIM describes which component is the most important to improve the performance of the system during the system lifetime and at different life stages. An example of the application of an oil transportation system is presented to illustrate the use of the generalized IIM.