Information Technology Reference
In-Depth Information
3.5.2
Possible forms of failure rates
The simplest possible choice for the age-specific failure rate is a time-independent
constant
g
(
t
)
=
γ.
(3.225)
A constant failure rate implies that the rate of failure does not depend on the age of the
process and consequently the survival probability is the simple exponential
e
−
γ
t
(
t
)
=
(3.226)
and, from the derivative relation, the probability density is
e
−
γ
t
ψ(
t
)
=
γ
.
(3.227)
Here we see that the exponential probability density implies that the rate of failure for,
say, a machine is the same for each time interval. One can also conclude from this
argument that the number of failures in a given time interval has Poisson statistics.
It should be reemphasized that the concept of an age-specific failure rate has been
developed having in mind engineering problems [
9
] and maintenance [
19
]. The concept
is not limited to this class of applications but can also be applied to living networks,
including human beings, under the name of mortality risk [
18
]. One of the more popular
formulas for the mortality risk was proposed by Gompertz [
17
]:
Ae
α
t
g
(
t
)
=
.
(3.228)
The exponential risk generates a survival probability that may look strange,
exp
A
e
α
t
(
t
)
=
α
(
1
−
)
,
(3.229)
so that the rate of decrease of the survival probability is doubly exponentially fast. This
peculiar formula captures the rise in mortality, the loss of survivability, in a great variety
of species [
17
].
Another well-known proposal for the rate was given by Weibull [
10
]:
t
β
−
1
(
)
=
αβ
g
t
(3.230)
with
α,β >
0. In this case of an algebraic age-specific failure rate we obtain
t
β
]
,
(
t
)
=
exp
[−
α
(3.231)
so that the Weibull probability density is
t
β
−
1
exp
t
β
]
.
ψ(
t
)
=
αβ
[−
α
(3.232)
It is important to notice that, for
1, the mortality risk decreases rather than increases
as a function of time. This may be a proper description of infant mortality risk [
10
].
There are cases in which
g
β<
(
t
)
may decrease at short times and then increase at long
times.
In the materials-science literature the Weibull distribution characterizes the influence
of cracks and imperfections in a piece of material on the overall response to stress and
