4 Reliability Theory
125
arbitrary decomposition of the space of component states {0, 1} N into disjoint sets
D 1 , . . . , D k . The probability of event S can then be expressed as
P (S) =
i
P (S|D i )P (D i ).
(4.3)
The signatures we describe in this section both rely on this formula but differ in the
choice of the underlying decomposition.
The original system signatures were proposed by Samaniego [20] and celebrated
successful applications in the system reliability analysis and system structure
optimisation. On the other hand, they could only be applied to systems with
components with independent identically distributed (i.i.d.) lifetimes, which is
overly restrictive for many practical scenarios, since most of the systems are
composed of heterogeneous components. This limitation was overcome by the
introduction of survival signature [7] by Coolen and Coolen-Maturi which allows
us to model systems with multiple types of components.
4.3.4.1 System Signature
System signature is introduced for systems composed of components with i.i.d.
TTF. The i.i.d. requirement either restricts us to analyse systems consisting of
multiple instances of the same component or to systems for which we assume
that the other components are totally reliable (cannot fail). Nevertheless, many
practical systems can still be analysed using this methodology, like traffic networks,
telecommunication networks, computer components . . .
The system signature is defined as a discrete probability vector q 1 , . . . , q N ,
where q i denotes the probability that the i-th component failure will result in the
failure of the system. The expression system reliability can be simplified into
P (TTF sys > t) =
N
i=1
q i P (TTF (i:N) > t),
where TTF (i:N) denotes the ith order statistic (a random variable describing the
probability distribution of ith failure time in the sample of size N). In the i.i.d. case,
P (TTF (i:N) > t) =
N
r=N −i+1
N
r
[1 − F (t)]
r
[F (t)]
N −r ,
where F is the common CDF for the component lifetimes.
Samaniego has shown that the system signature may serve as a way of comparing
systems. He provides theorems about how different stochastic orderings of system
125
arbitrary decomposition of the space of component states {0, 1} N into disjoint sets
D 1 , . . . , D k . The probability of event S can then be expressed as
P (S) =
i
P (S|D i )P (D i ).
(4.3)
The signatures we describe in this section both rely on this formula but differ in the
choice of the underlying decomposition.
The original system signatures were proposed by Samaniego [20] and celebrated
successful applications in the system reliability analysis and system structure
optimisation. On the other hand, they could only be applied to systems with
components with independent identically distributed (i.i.d.) lifetimes, which is
overly restrictive for many practical scenarios, since most of the systems are
composed of heterogeneous components. This limitation was overcome by the
introduction of survival signature [7] by Coolen and Coolen-Maturi which allows
us to model systems with multiple types of components.
4.3.4.1 System Signature
System signature is introduced for systems composed of components with i.i.d.
TTF. The i.i.d. requirement either restricts us to analyse systems consisting of
multiple instances of the same component or to systems for which we assume
that the other components are totally reliable (cannot fail). Nevertheless, many
practical systems can still be analysed using this methodology, like traffic networks,
telecommunication networks, computer components . . .
The system signature is defined as a discrete probability vector q 1 , . . . , q N ,
where q i denotes the probability that the i-th component failure will result in the
failure of the system. The expression system reliability can be simplified into
P (TTF sys > t) =
N
i=1
q i P (TTF (i:N) > t),
where TTF (i:N) denotes the ith order statistic (a random variable describing the
probability distribution of ith failure time in the sample of size N). In the i.i.d. case,
P (TTF (i:N) > t) =
N
r=N −i+1
N
r
[1 − F (t)]
r
[F (t)]
N −r ,
where F is the common CDF for the component lifetimes.
Samaniego has shown that the system signature may serve as a way of comparing
systems. He provides theorems about how different stochastic orderings of system