Information Technology Reference
In-Depth Information
There is a singularity in (
3.236
)at
η
=−
1 that is vitiated by direct integration of
(
3.222
) to yield
exp
r
0
r
1
(
t
)
=
−
ln
(
1
+
r
1
t
)
exp
ln
1
t
r
0
r
1
r
0
r
1
=
−
r
1
+
+
ln
(
r
1
)
,
which after some algebra simplifies to the hyperbolic distribution
T
T
μ
−
1
(
t
)
=
.
(3.237)
+
t
The new parameters are expressed in terms of the old as
r
0
r
1
1
r
1
.
μ
=
1
+
and
T
=
(3.238)
The corresponding distribution density,
, is easily obtained from the derivative to
be given by the hyperbolic distribution density
ψ(
t
)
T
μ
−
1
ψ(
t
)
=
(μ
−
1
)
)
μ
.
(3.239)
(
T
+
t
The singularity at
1 is the origin of the hyperbolic distribution density. It is
important to notice that the conventional Poisson case can be derived from the gen-
eral age-specific failure rate by setting
r
1
=
η
=−
1, and we run an algorithm
corresponding to the prescription of (
3.233
), we probably obtain first the times
t
0
.
When
η
=−
T
,
which have a larger probability, and only later do the times
t
T
appear. However,
the renewal character of the process requires that a very long time may appear after a
very short one, and the illusion of Poisson statistics is due to the truncation produced by
adopting a short data sequence.
The properties of the hyperbolic distribution density are in themselves very inter-
esting. For example, the power spectrum
S
(
f
)
of such hyperbolic sequences has been
shown to be
1
f
η
,
S
(
f
)
∝
(3.240)
where
η
=
3
−
μ.
(3.241)
Consequently, this spectrum diverges for
f
→
0 when
μ
falls in the interval
1
<μ<
3
.
(3.242)
μ
For this reason, the events described by (
3.239
) with
in the interval (
3.242
) are called
critical events.
Another remarkable property of these events is determined by the central
moments
t
n
defined by
∞
t
n
=
t
n
ψ(
t
)
dt
.
(3.243)
0
