Abstract
A linear \(m\) -consecutive- \(k\) -out-of- \(n:F(G)\) system with sparse \(d\) consists of \(n\) components arranged in a line. The system fails (works) if and if there are at least \(m\) non overlapping runs of \(k\) consecutive failed (working) components with sparse \(d\) . In this paper, we introduce a linear weighted \(m\) -consecutive- \(k\) -out-of- \(n:F(G)\) system with sparse \(d\) consisting of weighted components. Such a system can find many applications in practice. We consider the situation where the system components are non-homogeneous Markov-dependent, and we derive closed-form formulas for the system reliability, the marginal reliability importance, and the joint reliability importance using conditional probability generating function method. We present numerical examples to illustrate the use of formulas.
