Reliability for sparsely connected consecutive-k systems

We introduce the system of consecutive failures with sparse d which is a natural extension of consecutive-k systems. Then a series of generalizations of consecutive- k systems are discussed, such as consecutive-k-out-n:F systems with sparse d, M consecutive-k-out-of-n:F systems with sparse d, and (n,f,k):F systems with sparse d. We present the formulation for the system reliability of these generalized consecutive-k systems with various component settings in terms of the finite Markov Chain imbedding idea, along with two numerical examples.