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.