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In-Depth Information
1
η
=
1
τ
0
=
−
√
r
(6.37)
ν(
1
)
from (
6.23
), where
≡
λ
r
ν
.
(6.38)
We note that in the case
r
>
1 we can express the length
l
of a line as
l
(
t
)
=
(λ
−
ν)
t
.
(6.39)
In fact, there are more customers arriving than there are being served. On the other hand
it is evident that due to the stochastic nature of the process, when
, the line will
temporarily become large and from time to time it will shrink back to the condition
l
λ
=
ν
0 is the time that
a customer has to wait to be served. Earlier we determined that the time interval between
two consecutive returns to the origin is given by
=
0. The distance between two consecutive returns to the origin
l
=
ψ(τ)
∝
τ
−
3
/
2
. This is the prediction
of (
6.23
) when
r
=
1, in which case the average waiting time diverges as seen in (
6.37
).
1, the arrival rate of the tasks is smaller than the execution rate. Thus, very
frequently the line will vanish and the task will be executed almost immediately on the
arrival of a customer. We see that in the case
r
When
r
<
→
0 the waiting time becomes identical
to the execution time.
When
r
1, the arrival rate of the tasks is greater than the execution rate. Thus, very
frequently the line becomes infinitely long and the task will not be executed almost
immediately on the arrival of a customer. We see that in the case
r
>
→∞
there are
customers who are never served or letters that go unanswered.
What about the index 5/2 of the inverse power law of (
6.22
)? Techniques have been
developed to obtain analytic solutions for priority-based queuing models that are biased,
but the arguments depend on phase-space methods for the development of probabil-
ity densities, particularly the first-passage-time probability density. In the last section
we discussed a model of the time intervals between letter responses of Darwin and
Einstein developed by Barabási. We now generalize that formalism to determine the
first-passage-time pdf. With this pdf in hand we apply (
3.183
) to the first-passage-time
pdf for a diffusion process with drift
(
q
0
)
=
ν>
a
0
(6.40)
and strength of the fluctuations
b
(
q
0
)
=
D
,
(6.41)
resulting in the second-order differential equation of motion with constant coefficients
W
+
ν
∂
W
∂
2
D
∂
u W
q
0
−
=
0
.
(6.42)
q
0
∂
The general solution to this equation satisfying the vanishing boundary condition as
q
0
→∞
is
