Information Technology Reference
In-Depth Information
Figure 6.4.
This graph illustrates the priority-task model proposed by Barabási [
8
]. We interpret the
illustration as a post office. There are
N
lines corresponding to
L
windows,
N
=
1
/
L
.The
variable
x
moving within the interval [0, 1] indicates the priority level. The closer the window to
x
=
λ
ν
we select one
customer and satisfy his (her) request. The faces at the beginning of each line indicate clerks in
action with an efficiency that increases as we move from 0 to 1.
1 the higher the priority. The customers arrive at the rate
. With a rate
Thus, we convert a problem of the human mind, with its prioritizing of tasks, into an
issue of social organization. In fact, the clerks of the post office have a different effi-
ciency and the smiling face of Figure
6.4
closest to the maximum priority
x
=
1isthe
most efficient clerk in the office.
A merely random choice corresponds to
γ
=
0in(
6.25
), whereas setting
γ
=
1is
equivalent to always selecting the customer in line closest to
x
=
1. Note that here we
set
. Thus, there is no delay caused by more customers arriving per unit time than
customers being satisfied by the clerks of this post office per unit time.
The probability that a task with priority
x
waits a time interval
ν>λ
τ
before execution is
]
τ
−
1
ψ(
x
,τ)
=
[1
−
(
x
)
(
x
).
(6.26)
Note that we have again adopted the notation of the waiting-time distribution and used
the product of the probability of not having a priority
x
in each of the first
1 time
intervals with that of having a priority
x
in the last interval. This is a discrete version of
the following:
τ
−
d
(
x
,τ)
ψ(
x
,τ)
≈
exp
[−
(
x
)τ
]
(
x
)
=−
,
(6.27)
d
τ
which is the definition of the waiting-time distribution density in terms of the time
derivative of the survival probability. Here we have generalized the survival probability
such that the rate carries the priority index
(
x
,τ)
=
exp
[−
(
x
)τ
]
,
(6.28)
