Reliability importance of a component is a quantitative measure of the importance of the individual component in contributing to system reliability.In this paper, an appropriate Markov chain imbedding technique is employed to obtain the reliability of an multi-state m-consecutive-at least-k-out-of-n: F systems when the system components are independently functioning with not necessarily equal reliability. Finally, an illustrative is given example.

Problem statements: In this research, the researcher aimed to discuses the system failure probability of the model 2-within-consecutive (2, 2) out of (n, m) system for special values of m. Approach: The basic idea for evaluating the failure probability was the usage of the number of configuration of k (k = 2, 3, 4) parallel columns each contained n components in a 2×2-matrix. Results: The equation for the linear k-within (r, s) out of (n, m) system were reached. In this study the failure probability of 2-within-consecutive (2, 2) out of (n, m) system for m = 2, 3, 4. Conclusion/Recommendations: In general, it was difficult to evaluate the failure probability in the two-dimensional reliability structures such as the linear k-within (r, s) out of (n, m) system. The researcher established the failure probability and then the reliability of three special cases. It was recommended to generalize the results for any values of k, r, s and m