Information Technology Reference
In-Depth Information
From this definition it is evident that
(
t
)
=
P
(
k
=
0
;
t
).
(5.53)
It is interesting to note that when the time denotes the lifetime of a manufactured object
then
is the prob-
ability of failure. This terminology has been extended beyond renewal theory and the
predicted lifetimes of manufactured objects to other, more inclusive contexts, such as
the failure of complex webs, and was taken up in our discussion of random walks.
The expected number of events
N
(
t
)
is called the survival probability and correspondingly 1
−
(
t
)
(
t
)
is called the renewal function
m
(
t
)
and is
defined by
∞
m
(
t
)
=
N
(
t
)
=
P
(
t
k
≤
t
),
(5.54)
k
=
1
where
t
k
is the time of occurrence of the
k
th event. The renewal function can also be
expressed as the convolution
t
0
ψ(
t
)
[
t
)
]
dt
,
m
(
t
)
=
t
−
1
−
m
(
(5.55)
whose Laplace transform is
ψ(
u
)
m
(
u
)
=
u
1
)
.
(5.56)
−
ψ(
u
The Laplace transform of the waiting-time distribution is given by
ψ(
u
(
u
)
=
1
−
u
)
(5.57)
and, using the Laplace transform of the MLF,
t
k
β
E
(
k
)
β
u
−
(λ
)
β
;
u
β
−
1
u
β
+
λ
β
k
+
1
;
k
!
LT
t
=
u
>λ,
(5.58)
we obtain for the waiting-time distribution function with
k
=
0
u
β
u
β
+
λ
β
.
ψ(
u
)
=
1
−
(5.59)
Consequently, the Laplace transform of the renewal function is
)
=
λ
β
u
β
+
1
m
(
u
(5.60)
and in the time domain
)
β
(λ
t
m
(
t
)
=
+
β)
.
(5.61)
(
1
Thus the renewal function increases with time more slowly than a Poisson process
(
β
=
1
)
.
