Information Technology Reference
In-Depth Information
1
ʸ
ln
|≤
ʦ
1
ʸ
ln
q
a
ʦ
(
a
)
≤
+
|
ʩ
+
|
ʩ
|
,
a
∈
ʩ
which completes the proof.
We next discuss the efficiency of the SNE by the distributed spectrum access
algorithm when
ʸ
is sufficiently large (i.e.,
ʸ
). Let
V
(
a
) be the total individual
utility received by all t
he
users under the channel selection profile
a
, i.e.,
V
(
a
)
ₒ∞
=
n
=
1
u
n
(
a
). We denote
a
as the NUM solution that maximizes the system-wide utility
(i.e.,
a
=
arg max
a
∈
ʩ
V
(
a
)) and
a
as the convergent SNE by the distributed spectrum
ˆ
access algorithm (i.e.,
arg max
a
∈
ʩ
ʦ
(
a
)). We then define the perfor
m
ance gap
ˁ
as the difference betwee
n
the total utility received at the NUM solution
a
and that
of the SNE
a
ˆ
=
a
, i.e.,
ˁ
ˆ
=
V
(
a
)
−
V
(
a
). We can show the following result.
ˆ
Theorem 4.4
The performance gap ˁ of the SNE by the distributed spectrum access
algorithm is at most
N
N
1
2
1
2
s
nm
)
P
m
d
−
ʱ
P
m
d
−
ʱ
(1
−
mn
+
mn
.
sp
n
p
n
sp
n
n
=
1
n
=
1
m
∈
N
m
∈
N
\
N
Proof
According to (
4.1
) and (
4.5
), we have that
N
N
N
P
m
d
−
ʱ
ˉ
a
n
V
(
a
)
=
u
n
(
a
)
=−
mn
I
{
a
n
=
a
m
}
−
p
n
n
=
1
n
=
1
n
=
1
m
∈
N
N
1
2
s
nm
)
P
m
d
−
ʱ
=
ʦ
(
a
)
−
(1
−
mn
I
{
a
n
=
a
m
}
sp
n
n
=
1
m
∈
N
N
1
2
P
m
d
−
ʱ
−
mn
I
{
a
n
=
a
m
}
.
p
n
sp
n
n
=
1
m
∈
N
\
N
We then have that
ˁ
=
V
(
a
)
−
V
(
a
)
=
ʦ
(
a
)
−
ʦ
(
a
)
N
mn
I
{
a
n
=
a
m
}
−
I
{
a
n
=
a
m
}
1
2
−
s
nm
)
P
m
d
−
ʱ
−
(1
sp
n
n
=
1
m
∈
N
N
mn
I
{
a
n
=
a
m
}
−
I
{
a
n
=
a
m
}
.
1
2
P
m
d
−
ʱ
−
(4.14)
p
n
sp
n
n
=
1
m
∈
N
\
N
Since
ʦ
(
a
)
ˆ
=
max
a
∈
ʩ
ʦ
(
a
)
≥
ʦ
(
a
) and
V
(
a
)
=
max
a
∈
ʩ
V
(
a
)
≥
V
(
a
), we know
ˆ
from (
4.14
) that
N
mn
I
{
a
n
=
a
m
}
−
I
{
a
n
=
a
m
}
1
2
s
nm
)
P
m
d
−
ʱ
≤
−
ˁ
(1
sp
n
