EL IDEAL DE BELLEZA SUPERIOR COMO PRINCIPIO DE LA AUTONOMÍA POÉTICA

Samaniego (2007) introduced the concept of signature and provided a very good overview of this novel method for describing a system, while Coolen and Coolen- Maturi (2012) proposed several extensions relating to signature. Aslett (2012) devel- oped a computer module based on the statistical programming language R to calcu- late the system signature, which is especially useful in systems with large numbers of components.

In order to present a definition of signature, there are some concepts that should be defined first, including system structure function, coherent system, minimal paths, minimal cuts and the reliability of a coherent system.

1.3.1

System structure function

For a system with m components, let xi be the state of the ith component for i =

1,2, ..., m where xi = 1 if it is working and xi = 0 if it is not working. The vector

x= (x1, x2, ..., xm)∈(0,1)m is called the state vector. The system structure function

φ(x) can be written as

φ(x) =

 



1 if the system is working 0 if the system is not working

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and the most common examples to illustrate the system structure function are series and parallel systems. The series system works only if every component is working, in which case the structure function of a series system can be written as

φ(x) = min(x1, x2, ..., xm) = m

Y

i=1

xi. (1.3.2)

Conversely, the parallel system works as long as at least one component is working, in which case the structure function for a parallel system can be written as

φ(x) = max(x1, x2, ..., xm) = 1− m Y i=1 (1₋xi). (1.3.3)

1.3.2

Coherent system

A system is coherent if and only if every component is relevant and the structure function representing the system is monotone, Samaniego (2007).

The first condition refers to a system of order m components with a state vector (x1, ..., xi−1, a, xi+1..., xm) wherea∈ {0,1}. Theith component is said to be irrelevant

if:

φ(x1, ..., xi−1,1, xi+1..., xm) =φ(x1, ..., xi−1,0, xi+1..., xm)

for all possible state vectors. If a component is not irrelevant, then it is defined to be a relevant component.

The second condition is the monotone structure function. The structure function

φ(.) of an order m system is said to be monotone if

x_≤y_⇒φ(x)_≤φ(y)

wherex, y_{∈ {}0,1_}m_{and the inequality on the left is taken element-wise. In particular,}

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1.3.3

Minimal paths and minimal cut sets

For a coherent system, a set of components P is said to be a path set if the system works whenever all the components in the set P work. If no proper subset of P is a path set, then P is said to be a minimal path set. The algebraic union of all minimal path sets is the set of all the system’s components.

However, a set of componentsC is said to be a cut set if the system fails whenever all the components in the set C fail. If no proper subset of C is a cut set, then C is said to be a minimal cut set. The algebraic union of all minimal cut sets is the set of all the system’s components.

1.3.4

System signature definition

Consider a coherent system with m independent and identically distributed com- ponents. Let Ts > 0 be the random failure time of the system and Ti:m the ith

order statistic of the m random component failure times for i = 1,2, ..., m, where

T1:m ≤ T2:m ≤ ...≤ Tm:m. The signature of the system is the m-dimensional proba-

bility vector S = (s1, s2, ..., sm) with elements

si =P(Ts =Ti:m) (1.3.4)

so the signature is the probability that the system failure occurs at the moment of the

ith component failure: Samaniego (2007); Coolen and Coolen-Maturi (2012). Since

Ts resides in the set {T1:m, T2:m, ..., Tm:m}with probability one, it follows that si ≥0

for all i and Pm

i=1si = 1.

Computing the signature is dependent on the number of components in the system and the system structure. For example, a series system fails when the first component fails, so the signature vector for a series system can be written asS = (1,0, ...,0), while

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2 3

1

Figure 1.7: System with 3 independent and identically distributed components.

a parallel system fails whenever all the system components fail, so the signature vector for a parallel system can be written asS = (0,0, ...,1). For other system signatures, let us consider an example of a system with three independent and identically distributed components as pictured in Figure 1.7. The failure times of these three components can be ordered in 3! = 6 arrangements.

Table 1.1: Ordered component failure times for a system with 3 i.i.d.components. ordered component failure times order statistic equal to system failure timeTs

T1 < T2 < T3 T2:3 T1 < T3 < T2 T2:3 T2 < T1 < T3 T2:3 T2 < T3 < T1 T3:3 T3 < T1 < T2 T2:3 T3 < T2 < T1 T3:3

For this system, we can note that there are only two minimal cut sets, _{1,2_} and

{1,3_}. The smallest minimal cut set has two members, which means that the sys- tem will not fail when the first component fails for all system components. The system will fail when the second component fails if the ordered component fail- ure time takes any one of the minimal cut sets _{1,2_} and _{1,3_} (note we do not have any minimal cut set as a subset of those sets). Then the system signature is

19 2 1 5 4 3

Figure 1.8: System with 5 independent and identically distributed components.

S = (0 6,

4 6,

2

6)=(0,0.66,0.33). For another example, let us consider a system with five independent and identically distributed components as pictured in Figure 1.8. The failure times of this system’s components can be ordered in 5!=120 ways, and it has only two minimal cut sets, _{1,2_} and _{3,4,5_}. The system signature vector is

S = (₁₂₀0 ,₁₂₀12,₁₂₀36,₁₂₀72,₁₂₀0 )= (0,0.1,0.3,0.6,0).

1.3.5

Signature and system reliability

Samaniego (2007) introduced a very useful theorem to compute the reliability function for any coherent system with m independent and identically distributed components with a continuous lifetime distribution.

Theorem 1.3.1. (Samaniego, 2007) Let T1, T1, ..., Tm be the i.i.d. component life-

times of an order m component coherent system with signature S. Let Ts be the

system lifetime. Then

P(Ts> t) = m X i=1 si i−1 X j=0 m j [F(t)]j_[R(t)]m−j _(1.3.5)

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where F(t) and R(t) are the failure function and reliability function of system com-

ponents.

This theorem can also be written:

P(Ts > t) = m X i=1 si m X j=m−i+1 m j [F(t)]m−j_[R(t)]j_{; (1.3.6)}

see Coolen and Coolen-Maturi (2012) and Aslett et al. (2014).