4 Reliability Theory
129
In the inferential scenario, we thus observe several failures from multiple levels
of stress. Usually, the least amount of observations comes from the level with
working stress conditions, because the devices tested on this level are assumed
to deteriorate by the slowest rate. All the observations are then joined together in
by single likelihood function for the model parameters; the model parameters are
inferred, and the desired lifetime distribution for the working stress level is derived
from them.
Accelerated life testing is thus among the methods which enabled practical
testing of highly reliable components. It is dependent on modelling the underlying
physical process which leads to the acceleration of the deterioration process.
Furthermore, the model of acceleration is also inferred and may be utilised in design
optimisation where we could investigate how the design parameters influence the
working conditions of the device and, therefore, its reliability.
We will show how the ALT can be formulated and solved for two commonly
used transformation models based on independent observations.
The Arrhenius law originated for describing how the transition rates changes
for chemical reactions based on the environmental conditions. It may be used as
a transformation model if we include an assumption that the lifetime distributions
on each level are exponential, with constant failure rate, because it prescribes the
relation directly between the failure rates on different levels. Let us parametrise the
exponential PDF with its failure rate so that
f (t|λ) = λ exp(−λt).
Then we can link the mean times to failure at different levels by the Arrhenius law.
μ(V ) = C exp
B
V
,
where C, B are model parameters which need to be estimated; V is a physical
observation describing the stress upon the component (e.g. electric potential,
temperature, . . . ); and μ(V ) is the mean TTF at level V . The failure rate on each
level can thus be obtained by taking the reciprocal value λ V =
1
μ(V ) . Since we can
now precisely specify the distribution at each level conditional on the knowledge of
model parameters B, C, we can also construct the likelihood function for the model
parameters B, C conditional on the observed values. That will take the form (with
the independency assumption)
L(B, C;
t,
v) :=
V
i
f (t V ,i |λ V ),
where t V ,i denotes i-th observation on level V , f (.|λ V ) is the lifetime pdf on
level V ,
t are the observed failure times and
v the respective levels on which the
observation had been made (Fig. 4.6).
129
In the inferential scenario, we thus observe several failures from multiple levels
of stress. Usually, the least amount of observations comes from the level with
working stress conditions, because the devices tested on this level are assumed
to deteriorate by the slowest rate. All the observations are then joined together in
by single likelihood function for the model parameters; the model parameters are
inferred, and the desired lifetime distribution for the working stress level is derived
from them.
Accelerated life testing is thus among the methods which enabled practical
testing of highly reliable components. It is dependent on modelling the underlying
physical process which leads to the acceleration of the deterioration process.
Furthermore, the model of acceleration is also inferred and may be utilised in design
optimisation where we could investigate how the design parameters influence the
working conditions of the device and, therefore, its reliability.
We will show how the ALT can be formulated and solved for two commonly
used transformation models based on independent observations.
The Arrhenius law originated for describing how the transition rates changes
for chemical reactions based on the environmental conditions. It may be used as
a transformation model if we include an assumption that the lifetime distributions
on each level are exponential, with constant failure rate, because it prescribes the
relation directly between the failure rates on different levels. Let us parametrise the
exponential PDF with its failure rate so that
f (t|λ) = λ exp(−λt).
Then we can link the mean times to failure at different levels by the Arrhenius law.
μ(V ) = C exp
B
V
,
where C, B are model parameters which need to be estimated; V is a physical
observation describing the stress upon the component (e.g. electric potential,
temperature, . . . ); and μ(V ) is the mean TTF at level V . The failure rate on each
level can thus be obtained by taking the reciprocal value λ V =
1
μ(V ) . Since we can
now precisely specify the distribution at each level conditional on the knowledge of
model parameters B, C, we can also construct the likelihood function for the model
parameters B, C conditional on the observed values. That will take the form (with
the independency assumption)
L(B, C;
t,
v) :=
V
i
f (t V ,i |λ V ),
where t V ,i denotes i-th observation on level V , f (.|λ V ) is the lifetime pdf on
level V ,
t are the observed failure times and
v the respective levels on which the
observation had been made (Fig. 4.6).