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so that the length of time spent within the unit interval is
1
γ
τ
=−
ln
ξ,
(3.189)
and, since
ξ
is a random variable, the sojourn time
τ
is also a random variable. The
waiting- or sojourn-time distribution function
ψ(τ)
is consequently determined by
ψ(τ)
d
τ
=
p
(ξ)
d
ξ
(3.190)
and by assumption the random distribution of initial conditions determined by back
injection is uniform, so that
p
(ξ)
=
1. Thus, (
3.190
) can be solved to determine the
waiting-time distribution
=
γ
d
ξ
e
−
γτ
,
ψ(τ)
=
(3.191)
d
τ
which is an exponential probability distribution.
It is worth spending a little time discussing the properties of Poisson processes since
they are used so often in modeling statistical phenomena in the physical, social and life
sciences. Moreover, it is by modifying these properties that complexity is made mani-
fest in a variety of the webs we discuss. Consider the notion of renewal defined by the
requirement that the variables {
τ
1
,
τ
2
,
...
} are independent identically distributed (iid)
random variables, all with the same probability density function
. The character-
istic function corresponding to the occurrence of
n
events in the time interval (0
ψ(τ)
,
t
)is
given by the product of the probabilities of
n
independent events,
ψ
n
(
u
)
=
exp
[−
u
(τ
1
+
τ
2
+···+
τ
n
)
]
τ
,
(3.192)
where the brackets with a subscript
denote an average over the distribution times for
the
n
variables. Note that this technique is used to generate the time series
t
1
=
τ
1
,
τ
t
2
=
τ
1
+
τ
2
and so on. The iid statistical assumption implies that the probability density
factors into a product of
n
identical terms, and from (
3.192
) we obtain
n
ψ(
n
e
−
u
τ
k
ψ
n
(
u
)
=
τ
=
u
)
.
(3.193)
k
=
1
Assume that the statistics for a single event is exponential so that the Laplace transform
of the waiting-time distribution given by (
3.191
)is
γ
γ
+
ψ(
u
)
=
u
.
(3.194)
Inserting this into (
3.193
) and taking the inverse Laplace transform yields
n
−
1
)
=
γ(γ
t
)
e
−
γ
t
ψ
n
(
t
.
(3.195)
(
)
n
In the renewal-theory [
14
] and queueing-theory [
20
] literature (
3.195
) goes by the name
of the Erlang distribution. Note that it is
not
the Poisson distribution.
Consider the occurrence of events in our dynamical equation to be a counting process
{
N
(
t
),
t
≥
0
}
, where
N
(
t
)
is the total number of events up to and including time
t
,
