On the importance measures of Markov chain imbeddable reliability systems: some advances for k-out-of-n: G and consecutive-k-out-of-n: G structures

In the present paper, we study the Birnbaum's Reliability Importance measures of the components for Markov Chain Imbeddable structures. A generating function approach is presented for determining the Birnbaum's importance measure for a reliability system consisting of independent and identically distributed components. The general setup is implemented for two specific members of the class of Markov Chain Imbeddable structures. The resulting recurrences for computing the corresponding important measures are established. More precisely, the consecutive-k-out-of-n: G system and the k-out-of-n: G system are under investigation. For the first-mentioned structure a recursive scheme (accompanied with the necessary set of initial conditions) for the Birnbaum's importance measure of its components is introduced, while for the latter one an explicit formula for determining the corresponding quantities is proved. A numerical investigation is carried out by the aid of the theoretical outcomes of the manuscript and some concluding remarks about the design parameters of the underlying systems are also discussed.