On the dual reliability systems of (n,f,k)(n,f,k) and 〈n,f,k〉〈n,f,k〉

The (n,f,k)(n,f,k)(〈n,f,k〉)(〈n,f,k〉) system consists of nn components ordered in a line or a circle and the system fails if and only if there exist at least ff failed components or (and) at least kk consecutive failed components. This type of models involves two common failure criteria of a system. In this paper, the duality of these systems is obtained and the proofs are given. We also obtain the recursive equations for the system reliability for these models. Furthermore, the finite Markov chain imbedding approach is employed to obtain some more compact reliability formulas for independent and Markov-dependent cases.