Reliability of Simple 3-Dimnesional Consecutive k-Out-of-n: F Systems

Problem statement: In this study, the reliability of some special cases of a 3-dimnesional consecutive k-out-of-n: F system and k-within consecutive (2,2,2) -out-of- (m,2,2): F system were discussed. Approach: The 3 dimensional systems is a generalization of 1 and 2 dimensional, systems. Projections from 3 dimensional systems to 2 or 1 dimension as special cases is very helpful to find the reliability of these 3 dimensional systems. Results: Many reliability expressions for the 3-dimensional consecutive (k₁,k₂,k₃) -out-of- (m,n,𝓁): F system for some special values of k₁, k₂, k₃, m, n, 𝓁 are presented. Further, the reliability of k-within (2, 2, 2)-out-of-(m,2,2): F system using Markov chains is considered. Conclusion/Recommendations: In general, it is difficult to find the reliability of (k₁,k₂,k₃) -out-of- (m,n,𝓁): F system, we studied special cases of this system and recommend generalizing the result for any value of k₁, k₂, k₃, m, n, 𝓁. This study presented the reliability formulas of simple 3D systems using results of consecutive k-out-of-n: F systems and 2D consecutive k-out-of-n: F systems and Markov chains.