Joint reliability of linear two-dimensional consecutive k -type systems with shared components

In this paper, linear two-dimensional consecutive k -type systems with shared components are considered for the first time. This includes the cases of linear connected- ( k , r ) -out-of- ( m , n ) : F systems, linear connected- ( k , r ) -or- ( r , k ) -out-of- ( m , n ) : F systems, linear l -connected- ( k , r ) -out-of- ( m , n ) : F systems without/with overlapping, and linear l -connected- ( k , r ) -or- ( r , k ) -out-of- ( m , n ) : F systems without/with overlapping. Using Kronecker product, finite Markov chain imbedding approach (FMCIA) is applied in a new way to derive the joint reliability functions of these two-dimensional consecutive k -type systems with shared components. Their accuracy and computational efficiency are illustrated with the use of some numerical examples. Finally, some concluding remarks are provided.