4 Reliability Theory
115
previous subsection, making the system state a function of both t and d. We will
omit the design parameter for the rest of this section.
Much interest in reliability theory lies in modelling this deterioration process
[14, Sec. 6]. An intuitive way is to consider that a device depletes some intrinsic
resources (or equivalently that it is accumulating a “wear out”) and define the
failure state as such with these resources depleted. Let us consider a non-decreasing
function H (t), which models a cumulative depletion of some inner resource, and
H 0 the amount of this resource available to the device. Then we consider the device
functional at time t, if it has not yet depleted its inner resources, i.e. if H (t) < H 0 .
Equivalently, we can interpret this as the device not having reached a critical amount
of “wear out”, e.g. accumulation of sediments or overall mass loss due to abrasion.
In order to provide an assessment of reliability, we need to model H (t) and H 0 ,
which will generally both be uncertain. If these models are available (e.g. based
on physics models), we can, again, employ probability theory to assess probability
of the event of interest. These models may be available for specific problems
(crack formation and abrasion, see [22] for more). If they are not available for the
investigated system, reliability theory aims to provide ways of constructing them by
the methods of statistical inference (some examples are shown in Sect. 4.4 and more
can be found in any statistics textbook, e.g. [6]).
Let us consider a common scenario for a new device put into operation. If we
assume that the device is functioning at time t = 0 (we try to assure this by
post-production testing, but it is possible to generalise the methods for cases with socalled hidden failures), we can model the time to failure (TTF), the time when the
device depletes its inner resources-thus it fails, as a non-negative random variable.
As a random variable, the TTF can be described by its cumulative distribution
function (CDF) F (t) or, more commonly in the reliability theory context, its
survival function R(t).
F TTF (t) = P r(TTF < t), R TTF (t) = 1 − F TTF (t).
The advantage of modelling the TTF lies in the straightforward specification of
the probability that a device will be operational over the mission time T M ,
P r(device fulfils its mission) = P r(TTF > T M ) = R(T M ).
The most commonly used distribution to describe the TTF is the exponential
one. This model assumes that the failures occur at random, regardless of how long
the device has already been operational and what was its operational history. For
the exponential distribution, R(t) = exp(−λt), for a failure rate parameter λ.
The exponential distribution is often used just for its mathematical convenience,
although there are situations where its usage is justified, e.g. for modelling devices
during the stable life phase (Fig. 4.1). Other distributions, which provide more
flexible modelling options, are, e.g. Weibull, Cauchy, Log-normal or Gamma
distributions. More about the basic mathematical models can be found in any
introductory text in reliability theory, e.g. in [14, Ch. 3].
115
previous subsection, making the system state a function of both t and d. We will
omit the design parameter for the rest of this section.
Much interest in reliability theory lies in modelling this deterioration process
[14, Sec. 6]. An intuitive way is to consider that a device depletes some intrinsic
resources (or equivalently that it is accumulating a “wear out”) and define the
failure state as such with these resources depleted. Let us consider a non-decreasing
function H (t), which models a cumulative depletion of some inner resource, and
H 0 the amount of this resource available to the device. Then we consider the device
functional at time t, if it has not yet depleted its inner resources, i.e. if H (t) < H 0 .
Equivalently, we can interpret this as the device not having reached a critical amount
of “wear out”, e.g. accumulation of sediments or overall mass loss due to abrasion.
In order to provide an assessment of reliability, we need to model H (t) and H 0 ,
which will generally both be uncertain. If these models are available (e.g. based
on physics models), we can, again, employ probability theory to assess probability
of the event of interest. These models may be available for specific problems
(crack formation and abrasion, see [22] for more). If they are not available for the
investigated system, reliability theory aims to provide ways of constructing them by
the methods of statistical inference (some examples are shown in Sect. 4.4 and more
can be found in any statistics textbook, e.g. [6]).
Let us consider a common scenario for a new device put into operation. If we
assume that the device is functioning at time t = 0 (we try to assure this by
post-production testing, but it is possible to generalise the methods for cases with socalled hidden failures), we can model the time to failure (TTF), the time when the
device depletes its inner resources-thus it fails, as a non-negative random variable.
As a random variable, the TTF can be described by its cumulative distribution
function (CDF) F (t) or, more commonly in the reliability theory context, its
survival function R(t).
F TTF (t) = P r(TTF < t), R TTF (t) = 1 − F TTF (t).
The advantage of modelling the TTF lies in the straightforward specification of
the probability that a device will be operational over the mission time T M ,
P r(device fulfils its mission) = P r(TTF > T M ) = R(T M ).
The most commonly used distribution to describe the TTF is the exponential
one. This model assumes that the failures occur at random, regardless of how long
the device has already been operational and what was its operational history. For
the exponential distribution, R(t) = exp(−λt), for a failure rate parameter λ.
The exponential distribution is often used just for its mathematical convenience,
although there are situations where its usage is justified, e.g. for modelling devices
during the stable life phase (Fig. 4.1). Other distributions, which provide more
flexible modelling options, are, e.g. Weibull, Cauchy, Log-normal or Gamma
distributions. More about the basic mathematical models can be found in any
introductory text in reliability theory, e.g. in [14, Ch. 3].