Importance of components in kk-out-of-nn system with components having random weights

The purpose of this paper is to study the component importance in kk-out-of-nn system with components having random weights. The system works if at least kk components work and the total weight of all working components is above the required level cc. A new measure of component importance, called weighted importance, for such a system is introduced and its relation with Birnbaum reliability measure of importance is studied. It is shown that the larger weight stochastically implies the larger weighted measure of importance. It is also shown that if two components are identically distributed then the larger weight stochastically implies the larger Birnbaum reliability measure of importance. Finally, it is proved that if the weight of a component is stochastically increased, then the weighted and Birnbaum reliability measures of importance are increased accordingly.