Reliability and Birnbaum Importance for Sparsely Connected Circular Consecutive- Systems


A consecutive-$k$ -out-of-$ n $: $ rm F ( rm G )$ system with sparse $ d $ consists of $ n $ components ordered in a line or a circle, whereas the system fails (works) iff, there exist a minimum of $ k $ consecutive failed (working) parts with sparse $ d $ for $ 0 le d le n - k $. During this paper, a circular consecutive-$ k $-out-of-$ n $ system with sparse $ d $ is considered. Some equations for system reliability and Birnbaum importance are derived by suggests that of the finite Markov chain imbedding approach. Then the Birnbaum importance of parts is compared within the situations where the system is beneath an IID model, and where one among the components is known to be failed, respectively. Finally, some numerical examples are followed to illustrate the results obtained within the paper.

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