Digital Signal Processing Reference
In-Depth Information
To define alternating On/Off sources, we first need to introduce the two-
stage alternating renewal processes. During an On-period, the source gener-
ates traffic at a (nominal) constant rate 1. During an Off-period, the source
remains silent, and no packets are sent. Let
X
0
,X
1
,X
2
,
...
be i.i.d nonnegative
random variables representing the lengths of the On-period and
Y
0
,Y
1
,Y
2
,
...
,
be i.i.d nonnegative random variables representing the length of Off-periods.
Each of the
X
and
Y
sequences is supposed to be i.i.d. The distribution of the
X
(denoted by
f
1
(
t
)
) and the
Y
(
f
0
(
t
)
)are heavy-tail distributed with indices
α
1
and
α
0
,respectively, where 1
<α
1
<
2 and 1
<α
0
<
2. Hence both distri-
butions have finite mean, which will be denoted by
µ
1
and
µ
0
,respectively.
The On/Off process can be represented by
∞
V
(
t
)
=
1
[
S
n
,S
n
+
X
n
+
1
)
(
t
)
,
t
≥
0
,
n
=
0
where
S
n
denotes the time of occurrence of the
k
th On period, and
n
S
n
=
S
0
+
T
i
,
n
≥
1;
i
=
1
1
[
s
1
,s
2
)
(
is the indicator function, which is defined as nonzero and equal to
one only for
t
t
)
∈
[
s
1
,s
2
)
, and
T
j
=
X
j
+
Y
j
. The distribution of
S
0
is adjusted
so that
V
is strict sense stationary (see Reference 33 for details).
The expected value of
V
(
t
)
(
t
)
equals
µ
1
/(µ
0
+
µ
1
)
, and its power spectral
density is
32
Re
[1
ω
−
2
2
−
Q
0
(
−
j
ω)
][1
−
Q
1
(
−
j
ω)
]
S
(ω)
=
E
{
V
(
t
)
}
δ(ω/
2
π)
+
µ
+
µ
1
−
Q
0
(
−
j
ω)
Q
1
(
−
j
ω)
0
1
(6.30)
where
Q
0
(
−
j
ω)
,
Q
1
(
−
j
ω)
are the Fourier transforms of
f
0
(
t
)
, and
f
1
(
t
)
,
respectively.
For the special case of Pareto-distributed On and Off durations, it was
shown in Reference 20 that for
ω
∼
0itholds
α
k
k
α
e
j
α
2
−
α)w
α
−
−
(
−
ω
)
=
(
−
α
ω.
1
Q
j
k
1
j
(6.31)
1
Inserting Equation 6.31 into Equation 6.30, ignoring the higher-order terms
and considering
0
+
,weobtain
ω
→
2
ω
α
1
−
2
ω
α
2
−
2
S
w
(ω)
∼
1
(
C
1
+
C
2
)
(6.32)
µ
+
µ
0
where
C
1
,C
2
are constants.
Equivalently, the autocorrelation function of the AFRP becomes
R
w
(τ )
∼
τ
−
(α
i
−
1
)
,
as
τ
→∞
(6.33)
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