# Journal of Mathematics and Statistics

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

M. Gharib, E. M. El-Sayed and I.I.H. Nashwan

DOI : 10.3844/jmssp.2010.261.264

Journal of Mathematics and Statistics

Volume 6, Issue 3

Pages 261-264

### Abstract

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.