Digital Signal Processing Reference
In-Depth Information
λ
2
λ
1
+
λ
2
=
k
(
s
1
, s
2
)
V
2
(
s
1
, s
2
)
V
1
(
s
1
, s
2
)
λ
1
FIgure 11.5
Stability region for given
s
1
(γ
1
) and
s
2
(γ
2
).
The region given above takes on the shape illustrated in Figure 11.5, where
V
1
(
s
1
,
s
2
) =
(
V
1,
x
(
s
1
,
s
2
),
k
(
s
1
,
s
2
) -
V
1,
x
(
s
1
,
s
2
)) and
V
2
(
s
1
,
s
2
) = (
k
(
s
1
,
s
2
) -
V
2,
y
(
s
1
,
s
2
),
V
2,
y
(
s
1
,
s
2
)). The maximum
service rate of user 1 is achieved by setting
q
1
= 1 and
q
2
= 0 (i.e.,
user 1 always transmits
from the source buffer, and user 2 always transmits from the cooperative buffer), and simi-
larly for user 2. By taking the union of the pentagons in Figure 11.5 over all
s
1
,
s
2
, we obtain
the fully loaded region for the cooperative channel-aware system.
11.3.6 Performance Comparisons
In
Figure 11.6
, we plot the boundaries of three regions: (1) the stability region of the
conventional slotted ALOHA system, (2) the inner bound of the cooperative system, and
(3) the inner bound of the cooperative system with channel-aware transmission control.
We consider, as an example, the reception model where
=
,
1
0
if
otherwise
γγ
≥
th
Ψ()
γ
,
,
which says that the packet is perfectly received if the channel state exceeds a certain
threshold γ
th
and fails otherwise. For the case with channel awareness, we assume that
the channel-aware transmission control functions
s
i
(·), for all
i
, are step functions such
that the transmission probability is 1 if the local channel state exceeds a certain value,
and 0 otherwise. This was shown to be optimal in [13, 32].
Figure 11.6(a) is plotted for the case where ψ
1
=
e
γ
1
[Ψ(γ
1
)] = 0.1 and ψ
2
=
e
γ
2
[Ψ(γ
2
)] =
0.1, and Figure 11.6(b) is plotted for the case where ψ
1
= 0.2 and ψ
2
= 0.8. As shown in the
figures, cooperation enlarges the stability region, especially when one of the users has a
much better channel than the other. With cooperation, the user that on average experi-
ences a bad channel can utilize help from its partner to increase the throughput of its
own packets. When both users experience bad channels, little advantage can be gained
through pure cooperation, as shown in Figure 11.6(a). However, a significant gain is
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