Information Technology Reference
In-Depth Information
The statistical analysis of the Darwin and Einstein correspondence motivated
Barabási [
8
] to look for a social rather than mathematical [
1
] explanation of these
results. Barabási proposed a model based on executing
L
distinct tasks, such as writing
a letter or sending an email. Each task has a random priority index
x
i
(
i
=
1
,...,
L
)
ρ(
)
extracted from a probability density function
x
independently of each other. The
≤
≤
dynamical rule is the following: with probability 0
p
1 the most urgent task is
selected, while with complementary probability 1
p
the selection of the task is
random. The selected task is executed and removed from the list. At this point the com-
pleted task is replaced by a new task with random priority extracted again from
−
ρ(
x
)
.
It is evident that when
p
0 the duration of the wait before the next letter in the list is
answered depends only on the total number of letters
L
and the time
=
τ
;
1
τ
1
L
e
−
τ/τ
0
(τ)
=
−
≈
,
(6.17)
with the parameter in the exponential determined by the total number of letters
1
L
.
τ
0
=
(6.18)
In the language of queuing theory, if there are
L
customers, each with the same
probability of being served, the waiting-time distribution density is the exponential
function
1
τ
0
e
−
τ/τ
0
ψ(τ)
=
.
(6.19)
This waiting-time distribution is consistent with certain results of queuing theory.
Abate and Whitt have shown [
1
] that the survival probability
(
t
)
is asymptotically
given by
exp
exp
t
0
)
]
≈
e
−
η
t
(
t
)
=
[−
η(
t
−
1
−
α
,
(6.20)
where the double exponential is the well-known Gumbel distribution for extreme events
[
20
]. This distribution is obtained when the data in the underlying process consist
of independent elements such as those given by a Poisson process. Another survival
probability is given by
−
α
t
3
/
2
e
−
η
t
(
t
)
≈
1
,
(6.21)
where the parameters are different from those in the first case. This second distribution
is of special interest because it yields the waiting-time distribution density
)
=
αη
3
2
α
t
3
/
2
e
−
η
t
t
5
/
2
e
−
η
t
ψ(
t
+
,
(6.22)
which in the long-time limit reduces to
const
t
3
/
2
e
−
η
t
ψ(
t
)
∝
.
(6.23)
Note that the truncated inverse power law is reminiscent of the waiting-time distribu-
tion density discussed earlier, where we found that the inverse power law, with
μ
=
1
.
5,
