Information Technology Reference
In-Depth Information
Fig. 3.6
Impact of number of
users
1.4
NCG
SGUM−ER
SGUM−real
NUM
1.3
1.2
1.1
1
10
11
12
13
14
15
16
17
18
19
20
Number of users
∂V
(
p
1
,
p
2
)
∂p
1
In particular, we have
p
SNE
1
≥
p
S
1
<
0 when
p
1
≥
p
S
1
. Since
,
V
(
p
1
,
p
2
)is
p
S
1
. Using Lemma
3.2.
,
p
SNE
1
decreasing in
p
1
when
p
1
≥
is decreasing in
s
12
, and
hence
V
(
p
SNE
1
,
p
SNE
2
) is increasing in
s
12
since
p
SNE
is independent of
s
12
. Similarly,
2
we can show that
V
(
p
SNE
1
,
p
SNE
2
) is increasing in
s
21
.
Remark 3.3.
Similar to the SGUM-based random access control game, for the two-
user SGUM-based power control game, each user's strategy at the SNE is also a
dominant strategy. As a user's social tie level with the other increases, the user's
transmit power at the SNE decreases, and the social welfare increases. Therefore, as
the social tie level increases, a user's SNE strategy migrates from its NE strategy for
a standard NCG to its social optimal strategy for NUM (as illustrated in Fig.
3.5
).
3.3.3
Numerical Results
We consider
N
users each of which is a link consisting of a transmitter and a receiver.
Each transmitter or receiver is randomly located in a square area with side length
500 m. Under the physical interference model, we assume that the channel condition
of a link (communication or interference link) only depends on the path loss effect
with path loss factor 3. We assume that the transmit power of each link is 1 W and
the noise power at each receiver is 0.1 W.
Figure
3.6
shows the normalized social welfare for a varying number of users. We
can see that the SGUM solution for the ER model based social graph can achieve a
performance gain up to 23 % over the NCG solution, and its performance loss from
the NUM solution is at most 10 %. The SGUM solution for the real data based social
graph can achieve a performance gain up to 15 %.
3.4
Summary
In this chapter, we study the SGUM-based random access control and power control.
For the SGUM-based random access control game, we derive the unique SNE. For
the SGUM-based power control game, we show that it is a supermodular game and
