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The first and second moments are given by
T
μ
t
=
(3.244)
2
and
2 T 2
t 2
=
) ,
(3.245)
2
)(μ
3
so the variance of the hyperbolic distribution is
μ
1
2
t 2
2
T 2
τ
t
=
.
(3.246)
2
2
)
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 ,
=
 
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