Evaluation of joint reliability of linear two-dimensional consecutive k-type systems in a zigzag structure

In this paper, we study in detail several linear two-dimensional consecutive $k$-type systems in a zigzag structure that include linear connected-$(k,r)$-out-of-$(m,n)$: F system, linear $l$-connected-$(k,r)$-out-of-$(m,n)$: F system without/with overlapping and their counterparts with $(k,r)$ replaced by $(k,r)$-or-$(r,k)$, when they have shared components. Joint reliability functions of these systems are derived with the use of finite Markov chain imbedding approach (FMCIA). Numerical examples are then provided for illustrating the computational process of the method developed here and its efficiency. Finally, some possible applications and generalizations of the established results are discussed.