Information Technology Reference
In-Depth Information
The probability that
N
(
t
)
=
k
is determined by the convolution equation
t
0
(
t
)ψ
k
(
t
)
dt
,
P
(
N
(
t
)
=
k
)
=
t
−
(5.62)
t
) and
which is the product of the probability that
k
events occur in the interval (0
,
no events occur in the time interval (
t
,
t
). The Laplace transform of
P
(
N
(
t
)
=
k
)
is
given by
1
k
u
β
u
β
+
λ
β
u
β
−
1
u
β
+
λ
β
P
k
(
)
=
ψ(
k
(
u
u
)
u
)
=
−
β
u
β
−
1
u
β
+
λ
β
k
+
1
k
λ
=
(5.63)
and the inverse Laplace transform is
k
β
−
(λ
)
β
,
)
=
(λ
t
)
P
(
N
(
t
)
=
k
E
t
(5.64)
β
k
!
which is a clear generalization of the Poisson distribution with parameter
λ
t
and is
called the
-
fractional Poisson distribution
by Mainardi
et al.
[
15
]. Of course (
5.64
)
becomes an ordinary Poisson distribution when
β
β
=
1
.
5.2.2
The Weibull distribution
In an earlier section we asserted without proof that the MLF was a stretched exponential
at early times and an inverse power law at late times. It is rather easy to prove these
asymptotic forms. Consider first the Laplace transform of the MLF,
u
β
−
1
u
β
+
λ
β
.
E
β
(
u
)
=
(5.65)
The short-time limit
t
→
0 of the MLF is obtained from the
u
→∞
limit of (
5.65
)by
means of a Taylor expansion,
k
u
k
β
∞
1
u
→∞
E
lim
u
β
(
u
)
=
k
=
0
(
−
1
)
,
(5.66)
whose inverse Laplace transform is
∞
k
β
k
(λ
t
)
1
λ
e
−
(λ
t
)
β
W
(
t
)
=
0
(
−
1
)
)
=
;
t
.
(5.67)
(
k
k
=
The subscript W on the survival probability is used because the pdf
d
W
(
t
)
=
βλ
β
t
β
−
1
e
−
(λ
t
)
β
ψ
W
(
t
)
=−
(5.68)
dt
is the Weibull distribution (WD).
The WDwas empirically developed byWeibull in 1939 [
28
] to determine the distribu-
tion of crack sizes in materials and consequently in this context the independent variable
