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cardiac beat intervals. Here again we have a competition between the extreme variability
of the heart rate and the long-time correlation necessary to keep the longest excur-
sions, those that could produce death, from occurring. Consequently, even though the
control cannot predict when the largest excursions will occur, it acts so as to statis-
tically suppress them, thereby anticipating the improbably long intervals in a healthy
heart.
3.5
Moving from Hamiltonians to events
In this section we offer some general advice on how to use events to describe web
dynamics as an alternative to the continuous Hamiltonian prescription. In the literature
on renewal processes such events correspond to failure processes. The renewal condition
is established imagining that immediately after a failure a machine is brought back to
the condition of being brand new; it is reconstructed. Thus, an event occurrence can be
interpreted also as an instantaneous rejuvenation. Note that the dynamical description
of such a renewal process would be strongly nonlinear. The event description has the
nonlinear dynamical structure built into it.
One area of investigation in which it is important to characterize the statistical
properties of things is operations research, which is a branch of engineering that quan-
tifies the behavior of everyday activities such as decision making. Suppose that you
have a car and are considering replacing it with a new one; this is not a decision
to make lightly. It is necessary to balance the cost of maintenance, the number of
remaining payments and the reliability of the present car against those properties of
the replacement vehicle. There are well-established procedures for doing such calcu-
lations that often involve probabilities, since we do not have complete information
about the future and we have a need to quantify the properties of interest. One such
property is the probability that the car will just fail tomorrow or the day after. So let
us examine the probability of failure, or its complement, the survival probability of
your car.
3.5.1
Age-specific failure rates
According to the renewal-theory literature [
14
]the
age-specific failure rate g
(
t
)
is
defined by
(
<τ
≤
+
|
<τ)
Pr
t
t
t
t
(
)
=
,
g
t
lim
(3.215)
t
t
→
0
where the numerator is the probability that the time of a failure
τ
lies in the interval
(
conditional on no failure occurring before time
t
. Thus, using the calculus of
conditional probabilities, we factor the joint probability of two events
A
and
B
occurring
as the product of the probability of event
A
occurring conditional on event
B
occurring
and the probability of event
B
occurring:
t
,
t
+
t
)
