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namely the survival probability related to a task of priority x . The dependence of
ψ
and
on the priority index x is due to the fact that the waiting time depends on the priority
level.
The average waiting time of a task with priority x is obtained by averaging over the
time t weighted with the priority-indexed waiting-time density
ψ(
,
)
x
t
,giving
t
1
τ(
x
) = (
x
)
t
[
1
(
x
) ]
.
(6.29)
t
=
1
In the continuous-time notation the sum is replaced with an integral and we obtain
e ( x ) t
τ(
x
) =
tdt
ψ(
x
,
t
) =
tdt
(
x
)
,
(6.30)
0
0
so that introducing the new integration variable
θ =
t
(
x
)
allows us to write
0 θ
1
(
1
(
e θ =
τ(
x
) =
d
θ
) .
(6.31)
x
)
x
Therefore the average waiting time of a task of priority x is inversely related to the
probability that a task with that priority is carried out. Moreover, from the form of the
probability of a task of priority x being chosen ( 6.25 ) we have the inverse power-law
form for the average waiting time
1
x γ .
τ(
x
)
(6.32)
According to ( 6.32 ) the probability density of waiting a time
τ
,
ψ(τ)
, is related to the
probability
π(
x
)
of selecting a given line of Figure 6.4 by means of the equality of
probabilities
π(
x
)
dx
= ψ(τ)
d
τ.
(6.33)
Here we select the priority by randomly selecting a line in Figure 6.4 and therefore
choose
π(
x
) =
1 to obtain
.
dx
d
ψ(τ) =
(6.34)
τ
Consequently, from ( 6.32 )wehave
1
ψ(τ)
,
(6.35)
1
+
1
τ
so that the properties in the Einstein and Darwin correspondence in Figure 6.3 can be
explained by setting
γ =
2
.
(6.36)
We note that this inverse power law ( 6.32 ) with an index of 3/2 was obtained by Barabási
[ 8 ] with the deterministic protocol p
γ =∞ .
Nothing in the above argument determines the roles of the arrival parameter
=
1 and is recovered by setting
λ
and
the servicing parameter
in the post-office lines of Figure 6.4 . Using queuing theory
Abate and Whitt [ 1 ] proved that the average waiting time is given by
ν
 
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