Question

Show that k-out-of-n : G system is a coherent system hence show that series system is also a coherent system.

Answer 1

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Example 1.1. Consider the (C, 2, 3 : F ) system whose lifetime is given by

T2 | 3:F = min(max(T1, T2), max(T2, T3)).

The order statistic representation of T2 | 3:F for each possible ordering is as follows:

T1 < T2 < T3 ⇒ T2 | 3:F = T2:3,

T1 < T3 < T2 ⇒ T2 | 3:F = T3:3,

T2 < T1 < T3 ⇒ T2 | 3:F = T2:3,

T2 < T3 < T1 ⇒ T2 | 3:F = T2:3,

T3 < T1 < T2 ⇒ T2 | 3:F = T3:3,

T3 < T2 < T1 ⇒ T2 | 3:F = T2:3

Answer 2

A linear consecutive-k-out-of-n : F system consists of n linearly ordered components such

that the system fails if and only if at least k consecutive components fail. A linear consecutive-

k-out-of-n : G system, on the other hand, consists of n linearly ordered components such that

the system functions if and only if at least k consecutive components function. We denote

the consecutive-k-out-of-n : F and consecutive-k-out-of-n : G systems by (C, k, n : F ) and

(C, k, n : G), respectively.

Obviously, the series system is represented by (C, n, n : G) = (C, 1, n : F ) and the parallel

system by (C, n, n : F ) = (C, 1, n : G). A (C, k, n : F ) system is the dual of a (C, k, n : G)

system. The definition of dual systems can be found in Barlow and Proschan (1975, p. 12).

Let Xi denote the state of component i (Xi = 0 if component i has failed and Xi = 1 if

component i is working).

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