Reliabilities of Some Multistate Consecutive-k Systems.

In this paper, we consider four different kinds of multistate consecutive- ${\boldsymbol{k}}$ systems, namely a multistate linear ${\boldsymbol{m}}$ -consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system, a multistate linear consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system with sparse ${\boldsymbol{d}}$ , a multistate linear ${\boldsymbol{m}}$ -consecutive- ${\boldsymbol{k}}$ -out-of- $n$ : $G$ system with sparse ${\boldsymbol{d}}$ , and a multistate linear $ < n,{\boldsymbol{f}},{\boldsymbol{k}} > :G$ system. The reliability formulas for these systems are derived by using the finite Markov chain imbedding approach. Unlike some previous research, the vectors ${\boldsymbol{m}},{\boldsymbol{k}},{\boldsymbol{d}}$ , and ${\boldsymbol{f}}$ do not need to be monotonic in this paper. Some illustrative examples are also presented to show how the reliabilities of the four multistate consecutive-k systems are calculated. [ABSTRACT FROM AUTHOR]