4 Reliability Theory
113
at preventing them. Reliability analysis is used here for quantitative assessment of
whether such a measure is adequate, but the underlying interpretation of probability
is purely epistemic.
In this chapter, we will describe the basic question of reliability theory—
predicting an occurrence of an event—with some further discussion on the engineering applications. Our choice of modelling tool is the probability theory, which has
been standard in this field, although the magnitude of the uncertainties involved in
some of the applications is better captured by imprecise probability models, which
offer more degrees of freedom in modelling the available information (see Chap. 2).
The issues will mostly be demonstrated on evaluating the probability that a system
will complete its designated task once built and put into operation. The actual
meaning behind the assigned numerical probabilities varies among applications
and is, therefore, upon the particular analysts to translate it for their situation.
For simplicity, we may, hereon, assume the frequency interpretation of probability
measures; thus the probability of system functioning will mean that if we were to
test “infinitely” many instances of the same system, reliability tells us what fraction
of them will be functional. Nevertheless, if any quantity in a model is uncertain in an
epistemic sense, the overall model inherits this interpretation and can further only
be used to describe our degree of belief in the occurrence of an event.
In the second section, we will introduce the basic terminology and interpretation
of the quantities used in reliability theory. In the third section, we will show how
a problem can be decomposed into smaller parts when viewed as a system of
components. In the fourth section, we will show some applications of statistics in
reliability theory, hence how to answer some of the reliability-related questions on
the basis of available observations. In the fifth section, we will show some methods
for increasing reliability of systems by adding redundant components and how
the system maintenance and other policies enable us to operate systems over long
periods of time.
4.2 Mathematical Theory of Reliability
4.2.1 Structural Reliability
Our general aim is to construct a system which will function as desired. Once put
into operation, the system will occupy a specific state x ∈ Ω X . Suppose that in
the set Ω X , we may further distinguish states which we label as being desirable,
Ω M ⊂ Ω X , to represent what we actually mean by if a system functions. This
might represent that the stresses on a bridge are smaller than its resistance so it will
not collapse or that two planes pass at safe distance and will not crash, etc. But
since the system is subjected to interaction with the real world, hence inherits its
intrinsic uncertainties, our knowledge about the actual state will also be uncertain.
Say we model it by a random variable X obtaining values in Ω X . Now, instead
113
at preventing them. Reliability analysis is used here for quantitative assessment of
whether such a measure is adequate, but the underlying interpretation of probability
is purely epistemic.
In this chapter, we will describe the basic question of reliability theory—
predicting an occurrence of an event—with some further discussion on the engineering applications. Our choice of modelling tool is the probability theory, which has
been standard in this field, although the magnitude of the uncertainties involved in
some of the applications is better captured by imprecise probability models, which
offer more degrees of freedom in modelling the available information (see Chap. 2).
The issues will mostly be demonstrated on evaluating the probability that a system
will complete its designated task once built and put into operation. The actual
meaning behind the assigned numerical probabilities varies among applications
and is, therefore, upon the particular analysts to translate it for their situation.
For simplicity, we may, hereon, assume the frequency interpretation of probability
measures; thus the probability of system functioning will mean that if we were to
test “infinitely” many instances of the same system, reliability tells us what fraction
of them will be functional. Nevertheless, if any quantity in a model is uncertain in an
epistemic sense, the overall model inherits this interpretation and can further only
be used to describe our degree of belief in the occurrence of an event.
In the second section, we will introduce the basic terminology and interpretation
of the quantities used in reliability theory. In the third section, we will show how
a problem can be decomposed into smaller parts when viewed as a system of
components. In the fourth section, we will show some applications of statistics in
reliability theory, hence how to answer some of the reliability-related questions on
the basis of available observations. In the fifth section, we will show some methods
for increasing reliability of systems by adding redundant components and how
the system maintenance and other policies enable us to operate systems over long
periods of time.
4.2 Mathematical Theory of Reliability
4.2.1 Structural Reliability
Our general aim is to construct a system which will function as desired. Once put
into operation, the system will occupy a specific state x ∈ Ω X . Suppose that in
the set Ω X , we may further distinguish states which we label as being desirable,
Ω M ⊂ Ω X , to represent what we actually mean by if a system functions. This
might represent that the stresses on a bridge are smaller than its resistance so it will
not collapse or that two planes pass at safe distance and will not crash, etc. But
since the system is subjected to interaction with the real world, hence inherits its
intrinsic uncertainties, our knowledge about the actual state will also be uncertain.
Say we model it by a random variable X obtaining values in Ω X . Now, instead