Birnbaum Importance for Linear Consecutive- $k$ -out-of- $n$ Systems With Sparse

Since Birnbaum importance was introduced in 1969, there have been more than twenty kinds of importance measures so far. Among the various measures, Birnbaum importance plays an extremely important role because many importance measures have been defined under its illumination and have relationships with it. A lot of work has been done for Birnbaum importance in consecutive- k systems since the systems were introduced. Because the problems in practice are increasingly complicated, in 2007, Zhao proposed consecutive- k systems with sparse d, which is an extension of the current consecutive- k systems. In this paper, we study Birnbaum importance for linear consecutive- k-out-of- n systems with sparse d. Some equations on Birnbaum importance are proposed. With these equations, the ranking of components in the system on the basis of Birnbaum importance is given; and then some patterns of ranking are presented. Finally, two numerical examples are given to illustrate the results obtained in this paper.