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Figure 3.10.
Virtual age versus real age for varying restoration values, from [
44
]. Adopted with permission.
from which we see that the virtual age tends to increase with time but jumps to the van-
ishing value when an event occurs. With increasing time the virtual age may become
much smaller and in some cases infinitesimally smaller than the real time. If the repair is
not perfect, see the condition 0
<
<
1 of Figure
3.10
, the virtual age tends to increase
with time. Finally, if the repair is very bad, the restoration factor is
q
q
=
1, and the
network is restored to the “same as old” condition.
In order to describe this virtual age mathematically let's go back to the general rate
given by (
3.233
). The condition
q
τ
1
,the
time at which the first event occurs, the repair process generates a new machine, whose
failure rate
g
=
0 corresponds to assuming that at time time
(
t
)
, is given by
−
τ
1
)
]
β
g
(
t
)
=
r
0
[
1
+
r
1
(
t
(3.248)
τ
1
≤
<τ
1
+
τ
2
. The next event occurs at time
τ
1
+
τ
2
and after this second repair
for
t
(
)
the rate
g
t
ages according to
−
τ
1
−
τ
2
)
]
β
g
(
t
)
=
r
0
[
1
+
r
1
(
t
(3.249)
and so on. In the limiting case
q
is independent
of the event occurrence, and the virtual age of the machine coincides with real time.
In general the condition
q
=
1in(
3.247
), the time evolution of
g
(
t
)
1 is indistinguishable from a
non-homogeneous Poisson
process
(NHPP). In the literature of anomalous statistical physics an attractive example
of NHPP is given by the superstatistics of Beck [
6
]. To generate superstatistics, imagine
that the probability of the time-independent rate
g
is given by a distribution function
(
=
).
ψ(
)
g
In this case we can express the distribution density
t
for time-independent
g
as
∞
0
(
ge
−
gt
dg
ψ(
)
=
)
.
t
g
(3.250)
Let us assign to
(
g
)
the analytic form
T
μ
−
1
(μ
−
g
μ
−
2
e
−
gT
(
g
)
≡
.
(3.251)
1
)
