Linear two-dimensional consecutive k-type systems in multi-state case

In the context of consecutive k -type systems, multi-state system models are only considered in the one-dimensional case and not in the two-dimensional case due to the complexity involved. In this paper, we consider several linear two-dimensional consecutive k -type systems in the multi-state case for the first time, as generalization of consecutive k -out-of- n systems and l -consecutive- k -out-of- n systems without/with overlapping. These systems include multi-state linear connected-(k , r)-out-of-(m, n): G systems, multi-state linear connected-(k , r)-or-(r , k)-out-of-(m, n): G systems, multi-state linear l -connected-(k , r)-out-of-(m, n): G systems without/with overlapping, and multi-state linear l -connected-(k , r)-or-(r , k)-out-of-(m, n): G systems without/with overlapping. We then derive their reliability functions by using the finite Markov chain imbedding approach (FMCIA) in a new way. We also present several examples to illustrate all the results developed here.