
3354 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 7, SEPTEMBER 2014
where 1
m
(t)=1 if and only if PU m transmits in slot t.Itcan
be found that if queue X
m
(t) is stable for all m, the constraint
on collisions will be satisfied.
Since our objective is to maximize the weighted sum and
stability of all the queues simultaneously, we use Lyapunov
function L(t) and Lyapunov drift Δ(t) to bound all queues,
which are defined as follows.
If Q(t)=[U
1
(t),..., U
L
(t),X
1
(t),...,X
M
(t)] denotes
the collection of both virtual and actual queues, the Lyapunov
function can be expressed as
L (Q(t))
Δ
=
1
2
L
l=1
U
2
l
(t)+
M
m=1
X
2
m
(t)
(12)
and the Lyapunov drift is
Δ(t)
Δ
= E {L (Q(t + 1)) − L (Q(t + 1))} . (13)
Then, we have the following properties.
Lemma 1:
Δ(t) ≤ B + E
L
l=1
U
l
(t)(R
l
(t) − b
l
(t))
+ E
M
m=1
X
m
(t)(C
m
(t) − ρ
m
1
m
(t))
(14)
where B satisfies
B ≥
1
2
M
m=1
E
(ρ
m
1
m
(t) − C
m
(t))
2
+
1
2
L
l=1
E
R
2
l
(t)+b
2
l
(t)
−
L
l=1
E {b
l
(t)R
l
(t)} . (15)
Proof: The proof can be found in Appendix A.
Then, we try to find a proper value of B to bound the right-
hand side (RHS) of (14). The key is to find an upper bound of
L
l=1
b
2
l
(t). Recall t hat
L
l=1
b
2
l
(t)=
L
l=1
N
n=1
p
ln
M
m=1
μ
nm
(t)S
m
(t)
2
. (16)
According to Cauchy–Schwarz inequality, the following in-
equality holds:
N
n=1
p
ln
M
m=1
μ
nm
(t)S
m
(t)
2
≤
N
n=1
p
2
ln
⎛
⎝
N
n=1
M
m=1
μ
nm
(t)S
m
(t)
2
⎞
⎠
. (17)
Therefore, by combining (17) and (7), we can obtain
L
l=1
b
2
l
(t) ≤
L
l=1
N
n=1
p
ln
⎛
⎝
N
n=1
M
m=1
μ
nm
(t)S
m
(t)
2
⎞
⎠
≤ N
⎛
⎝
N
n=1
M
m=1
μ
nm
(t)S
m
(t)
2
⎞
⎠
≤ N
2
. (18)
Then, the RHS of (15) can be bounded by
RHS of (15) ≤
LA
2
max
+ N
2
+ M +
M
m=1
ρ
2
m
2
. (19)
Then, we define B
Δ
=(1/2)(LA
2
max
+N
2
+M +
M
m=1
ρ
2
m
).
The reward function is defined as VE{
L
l=1
θ
l
R
l
(t)},
where V is a control parameter to deal with the tradeoff between
the system throughput and delay, which will be explained later.
By subtracting the reward function from (14), we obtain
Δ(t) − VE
L
l=1
θ
l
R
l
(t)
≤ B + E
L
l=1
U
l
(t)(R
l
(t) − b
l
(t))
+ E
M
m=1
X
m
(t)(C
m
(t) − ρ
m
1
m
(t))
− VE
L
l=1
θ
l
R
l
(t)
. (20)
D. Opportunistic C-Additive Approximation Algorithm
Our algorithm aims at finding a proper μ and R(t)=[R
1
(t),
R
2
(t),...,R
L
(t)] to stabilize all queues and maximize the
network throughput. The RHS of (20) can be rewritten as
RHS of (20)=
L
l=1
(U
l
(t) − Vθ
l
) R
l
(t)
+
n, m
μ
nm
(t)
X
m
(t)(1 − P
m
(t)) − P
m
(t)
L
l=1
U
l
(t)p
ln
.
(21)
Then, the problem can be divided into two parts.
1) Rate control:
Minimize :
L
l=1
R
l
(t)(U
l
(t) − Vθ
l
) . (22)
The rate control scheme for the static single-hop sec-
ondary network. The rate control problem can be solved
in a distributed manner by choosing R
l
(t)=A
l
(t) if
(U
l
(t) − Vθ
l
) < 0; otherwise, R
l
(t)=0.
2) Channel Allocation:
Maximize :
n, m
μ
nm
(t)
P
m
(t)
L
l=1
U
l
(t)p
ln
− X
m
(t)(1 − P
m
(t))
.
(23)
The channel allocation scheme for the static single-hop
secondary network. The channel allocation problem can
be also solved using greedy maximal weight scheduling
in [17] with some modification on the weights of the
bipartite graph.