Information Technology Reference
In-Depth Information
This implies that
w
kn
U
n
(
a
)
U
n
(
a
)
ʦ
(
a
)
U
k
(
a
)
−
ʦ
(
a
)
=
−
U
k
(
a
)
+
−
sp
k
n
∈
N
w
kn
U
n
(
a
)
U
n
(
a
)
+
−
sp
k
s
k
n
∈
N
\
N
w
kn
U
n
(
a
)
U
n
(
a
)
,
U
k
(
a
)
=
−
U
k
(
a
)
+
−
s
k
n
∈
N
which completes the proof.
Proof of Theorem
4.2
As mentioned, the system state of the spectrum access Markov chain is defined as
the channel selection profile
a
ʘ
of all users. Since it is possible to get from any
state to any other state within finite steps of transition, the spectrum access Markov
chain is hence irreducible and has a stationary distribution.
We then show that the Markov chain is time reversible by showing that the
distribution in (
4.9
) satisfies the following detailed balance equations:
q
a
q
a
,
a
=
q
a
q
a
,
a
,
∈
a
,
a
∀
∈
ʘ.
(4.21)
To see this, we consider the following two cases:
1) If
a
/
∈
ʔ
a
,wehave
q
a
,
a
=
q
a
,
a
=
0 and the Eq. (
4.21
) holds.
2) If
a
∈
ʔ
a
, according to (
4.9
) and (
4.12
), we have
˄
n
|
M
n
|
exp (
ʸʦ
(
a
))
a
∈
ʘ
exp (
ʸʦ
(
q
a
q
a
,
a
=
a
))
exp
ʸS
n
(
a
n
,
a
−
n
)
×
exp
ʸS
n
(
a
n
,
a
−
n
)
,exp
(ʸS
n
(
a
n
,
a
−
n
)
)
max
{
}
exp
ʸ
ʦ
(
a
)
S
n
(
a
n
,
a
−
n
)
+
˄
n
|
M
n
|
a
∈
ʘ
exp (
ʸʦ
(
=
a
))
ˆ
1
×
exp
ʸS
n
(
a
n
,
a
−
n
)
,exp
(ʸS
n
(
a
n
,
a
−
n
)
)
,
max
{
}
and similarly,
exp
ʸ
ʦ
(
a
)
S
n
(
a
n
,
a
−
n
)
a
∈
ʘ
exp (
ʸʦ
(
+
˄
n
|
M
n
|
q
a
q
a
,
a
=
a
))
ˆ
1
exp
ʸS
n
(
a
n
,
a
−
n
)
,exp
(ʸS
n
(
a
n
,
a
−
n
)
)
}
×
.
max
{
