Optimal Consecutive- k −Out−of− n Systems under a Fixed Budget

A consecutive- k −out−of− n system is a graph with n vertices where the system fails if and only if some path of k consecutive vertices all fail. The assumption is made that q i , is the failing probability of the ith vertex and the states of the vertices are statistically independent. The problem is to find the best system, i.e., a set { q 1 , …, q n }, and its assignment to the vertices of the system graph that maximizes the reliability of the system, under the constraint that the product of q i 's is a constant. We solve the problem for all lines and some cycles. We then extend our results to trees and show that for given n and k the best tree is a star and the worst is a line.

[1]  Gerald J. Lieberman,et al.  On the Consecutive-k-of-n:F System , 1982, IEEE Transactions on Reliability.

[2]  Shun-Chen Niu,et al.  Reliability of Consecutive-k-out-of-n:F System , 1981, IEEE Transactions on Reliability.

[3]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[4]  Stavros Papastavridis,et al.  Fast Recursive Algorithm to Evaluate the Reliability of a Circular Consecutive-k-out-of-n:F System , 1987, IEEE Transactions on Reliability.

[5]  Ding-Zhu Du,et al.  Reliabilities of Consecutive-2 Graphs , 1987 .

[6]  F. Hwang,et al.  Fast Solutions for Consecutive-k-out-of-n: F System , 1982, IEEE Transactions on Reliability.

[7]  D. Malon,et al.  Optimal Consecutive-2-out-of-n:F Component Sequencing , 1984, IEEE Transactions on Reliability.

[8]  M. Santha,et al.  Consecutive-2 systems on trees , 1987, Probability in the Engineering and Informational Sciences.

[9]  F. K. Hwang,et al.  Optimal Consecutive-2-Out-of-n Systems , 1986, Math. Oper. Res..

[10]  Stavros Papastavridis,et al.  Exact Reliability Formulas for Linear & Circular Consecutive-k-out-of-n:F Systems , 1985, IEEE Transactions on Reliability.

[11]  F. Hwang,et al.  Simplified reliabilities for consecutive- k -out-of- n systems , 1986 .

[12]  Ding-Zhu Du,et al.  Optimal assignments for consecutive-2 graphs , 1987 .

[13]  D. Z. Du,et al.  Optimal consecutive-2 systems of lines and cycles , 1985, Networks.

[14]  Dalen T. Chiang,et al.  Relayed Communication via Consecutive-k-out-of-n: F System , 1986, IEEE Transactions on Reliability.

[15]  J. George Shanthikumar,et al.  Recursive Algorithm to Evaluate the Reliability of a Consecutive-k-out-of-n:F System , 1982, IEEE Transactions on Reliability.