4 Reliability Theory
119
The reliability of a system can also be expressed by the reliability function
h : [0, 1] N → [0, 1], which directly models the relation between a vector
representing probabilities that each individual component functions and probability
that the system functions. For example, for a serial system (all components have to
function to consider the system to be functioning) with N = 3 components with
p i := P r(X i = 1) being the reliability of component i, it holds that P r(X S = 1) =
h(p 1 , p 2 , p 3 ) = p 1 · p 2 · p 3 .
The structure function, as defined here, is dependent only on the current states
of the components and, thus, allows us to separate static structure dependencies
from temporal evolution of component states as described in Sect. 4.2.2. The same
applies for the reliability function, which only depends on the probability that
components function at a given time instance. Generalisations are possible, but the
actual mathematical model is dependent on the investigated scenario. Nevertheless,
even in our restricted case, the evaluation of a system reliability has exponential
complexity. It would require us to sum over all the elements of the state space
(∼ 2 N ). The reliability function is also exponentially complex to construct but may
be later used multiple times, e.g. for reconstruction of temporal evolution of system
state (the survival function) or in the problems of statistical inference, and make
these tasks tractable.
The structure function can be generally described by a table, prescribing the state
of the system to every possible configuration, but such a table would be impractical
to construct, work with and inspect for any system of realistic size, because the
number of rows grows exponentially with the number of components. There exist
several alternative ways to specify the structure function. These enable us to present
the structure function graphically which also allows us to analyse it qualitatively by
the tools and notions of the graph theory.
4.3.2 Graphical Models
4.3.2.1 Reliability Block Diagrams
Reliability block diagrams (RBDs) capture how the system components are
connected [17, Ch. 5]. They constitute a natural way for modelling systems whose
function is related to various kinds of transportation and communication (railroads,
computer networks, etc.) but can be generally used to depict any structure function.
For traffic networks, communication networks or also the power networks, an
RBD describes the network topology and allows us to easily construct structure
functions for classes of problems addressing the so-called k-terminal network
reliability. For these problems, we define that a system with N components functions
if the “k” pre-specified components are connected through nodes corresponding to
functioning components. But RBDs do not need to refer to anything physical and
can be used just as a graphical description of the system structure function.
119
The reliability of a system can also be expressed by the reliability function
h : [0, 1] N → [0, 1], which directly models the relation between a vector
representing probabilities that each individual component functions and probability
that the system functions. For example, for a serial system (all components have to
function to consider the system to be functioning) with N = 3 components with
p i := P r(X i = 1) being the reliability of component i, it holds that P r(X S = 1) =
h(p 1 , p 2 , p 3 ) = p 1 · p 2 · p 3 .
The structure function, as defined here, is dependent only on the current states
of the components and, thus, allows us to separate static structure dependencies
from temporal evolution of component states as described in Sect. 4.2.2. The same
applies for the reliability function, which only depends on the probability that
components function at a given time instance. Generalisations are possible, but the
actual mathematical model is dependent on the investigated scenario. Nevertheless,
even in our restricted case, the evaluation of a system reliability has exponential
complexity. It would require us to sum over all the elements of the state space
(∼ 2 N ). The reliability function is also exponentially complex to construct but may
be later used multiple times, e.g. for reconstruction of temporal evolution of system
state (the survival function) or in the problems of statistical inference, and make
these tasks tractable.
The structure function can be generally described by a table, prescribing the state
of the system to every possible configuration, but such a table would be impractical
to construct, work with and inspect for any system of realistic size, because the
number of rows grows exponentially with the number of components. There exist
several alternative ways to specify the structure function. These enable us to present
the structure function graphically which also allows us to analyse it qualitatively by
the tools and notions of the graph theory.
4.3.2 Graphical Models
4.3.2.1 Reliability Block Diagrams
Reliability block diagrams (RBDs) capture how the system components are
connected [17, Ch. 5]. They constitute a natural way for modelling systems whose
function is related to various kinds of transportation and communication (railroads,
computer networks, etc.) but can be generally used to depict any structure function.
For traffic networks, communication networks or also the power networks, an
RBD describes the network topology and allows us to easily construct structure
functions for classes of problems addressing the so-called k-terminal network
reliability. For these problems, we define that a system with N components functions
if the “k” pre-specified components are connected through nodes corresponding to
functioning components. But RBDs do not need to refer to anything physical and
can be used just as a graphical description of the system structure function.