Signatures of indirect majority systems

Abstract

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

Applications of average and projected systems to the study of coherent systems

“…Note that s = p + p (2,1,3) + p (3,1,2) 3 . Now, the average system lifetime T is the mixed system obtained from a uniform stochastic mixture of these three systems or, equivalently, it is the mixed system lifetime defined by T = X 1:3 with probability 1/3 and T = X 2:3 with probability 2/3.…”

“…When the component lifetimes are independent and identically distributed (IID) with common absolutely continuous reliability function F , the system reliability function is a mixture of the reliability functions of the order statistics associated to F with the coefficients in the mixture forming the signature vector (see [2]). Properties and applications of signatures are given in [3][4][5][6][7]. This representation was extended to the case of coherent systems with exchangeable component lifetimes by Navarro and Rychlik [8] (absolutely continuous case) and Navarro et al [9].…”

The Kruskal-Katona Theorem and a Characterization of System Signatures

“…The component X σ (l) giving place to the lth failure belongs then to both a minimal cut set and a minimal path set. Since σ indicates the order in which the components fail, the component X σ (l) is the common element to the minimal cut and path sets that appear in the first l and last n − l + 1 positions of the vector (X σ (1) , . .…”

Reliability of coherent systems with a single cold standby component

“…. , n. [5] where r i (n) denotes thenumber of path sets including i working components in a coherent system. Since signature has vital importance to investigate the behavior of the system lifetime, recently many developments have been made in this area, see [6][7][8][9][10][11][12][13].…”