Digital Signal Processing Reference
In-Depth Information
an independent limit on each individual user's transmission rate. In reality,
therefore, if
R
is the peak rate of the link onto which traffic from multiple
users is multiplexed, the sum of the user transmission rates is bounded by
R
and each user's transmission rate is bounded by an even smaller quantity,
L
(
L
<
R
). In view of the above discussion, a more realistic On/Off model can
be obtained by letting the rewarding process
G
n
in Equation 6.36 be cutoff-
Pareto distributed, i.e., the pdf (probability density function) of
G
n
is given
by
K
L
α
(
α
)
=
(
α
)(
−
(
−
))
+
δ(
−
)
f
L
x
;
,K
f
x
;
,K
1
u
x
L
x
L
(6.38)
where
f
(.)
denotes the Pareto density function (refer to Equation 6.20),
u
(.)
is the unit step function,
is the Dirac function, and
L
represents a limit
imposed to the random variable. It can be easily verified that the integral of
f
L
δ(.)
(
α
)
−∞
∞
is one. The existence of these two rate
limits,
L
and
R
, was shown in References 34 and 35 to result in the distinctive
two slope behavior of the LLCD of the overall traffic.
34
The LLCD of synthe-
sized traffic based on this model and that of real traffic are shown in Figure 6.6.
The synthesized traffic was constructed as a superposition of 50 On/Off pro-
cesses with cutoff Pareto distributed rates according to
f
10
4
.
5
x
;
,K
taken for
x
between
to
10
1
.
78
(
x
;1
.
13
,
50
∗
)
.
10
0
10
−
1
10
−
2
10
−
3
10
−
4
10
−
5
Real Traffic
Synthesized Traffic
10
−
6
10
0
10
1
10
2
10
3
10
4
10
5
log
10
(x)
FIGURE 6.6
LLCD of real traffic and synthesized traffic as superposition of 50 On/Off processes with cutoff
Pareto distributed rates.
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