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In-Depth Information
The first and second moments are given by
T
μ
−
t
=
(3.244)
2
and
2
T
2
t
2
=
)
,
(3.245)
(μ
−
2
)(μ
−
3
so the variance of the hyperbolic distribution is
μ
−
1
2
t
2
2
T
2
τ
≡
−
t
=
.
(3.246)
2
(μ
−
2
)
(μ
−
3
)
Equation (
3.246
) implies that the width of the distribution density
ψ(
t
)
, denoted by the
τ
μ
standard deviation
is in the interval (
3.242
).
Thus, we can define the critical events as those whose waiting-time distribution den-
sity has a divergent width. In addition, the crucial events with
, is divergent when
μ<
2 are characterized
by the divergent mean value
. We shall see that these events do not satisfy the
ergodicity condition and consequently are the least ordinary in the more general class
of crucial events.
t
=∞
3.5.3
Various Poisson processes
Up to this point we have adopted the jargon of the engineering literature and called
g
)
a failure rate, but in keeping with our intention of making the presentation as discipline-
rich as possible we replace the term
failure
with the term
event
. We will have occasion
to return to the nomenclature of failures, but the general discussion is much broader. In
fact much of what we focus our attention on is a special class of events called
crucial
events
.
Imagine that, when an event occurs, its occurrence time is recorded and immedi-
ately afterward we are ready to record the occurrence time of a subsequent event.
The engineering perspective of failure and the process of repair immediately after fail-
ure is especially convenient when we want to design a procedure to generate a time
series. Imagine that the repair process is instantaneous and that the
restoration
or
repair
effectiveness
is given by a quantity
q
in the interval 0
(
t
Renewal theory [
14
]
is based on the assumption that the repair is instantaneous and perfect. This means
that after repair the machine is “as good as new.” Kijma and Sumita [
26
] adopted a
generalized-renewal-process
(GRP) perspective based on the notion of
virtual age
:
≤
q
≤
1
.
A
n
=
qS
n
,
(3.247)
where
A
n
is the web's age before the
n
th repair,
S
n
is the web's age after the
n
th repair
and
q
is the restoration factor.
Ordinary renewal theory is recovered by setting the restoration factor to zero
q
0.
In this case the virtual age of the machine immediately after a repair is zero. The
machine is “as good as new” or
brand new
. This condition is illustrated in Figure
3.10
,
=