A Novel Cooperative Mechanism for Cognitive
Radio Networks
Haobing Wang
Dept. of Electronic Engineering
Shanghai Jiao Tong University
Abstract—As the demand on frequency spectrum grows
rapidly, cognitive radio technique becomes a hot research problem and a promising approach to increase the efficiency of
spectrum utilization. In this paper, we consider the problem
of spectrum sharing among one primary user and multiple
secondary users and secondary user’s power control by a
cooperating method, where the primary user select a proper
set of secondary users to serve as the cooperative relays for
its transmission; and in return lease portion of channel access
time to the selected secondary users for their own transmission.
Primary user should broadcast the portion of channel access time
it will leave for secondary users, the selected secondary users
have the right to decide their power level used to help primary
user’s transmission in order to achieve proportional access time
to the channel. We assume that the primary user and secondary
users are rational and selfish, i.e., they only aim at maximizing
their own utility. As secondary users’ utility is in term of their
own transmission rate and the power cost for primary user’s
transmission, so secondary users will choose an optimal power
level to meet the tradeoff between transmission rate and power
cost. And primary user will choose a proper portion of channel
access time for secondary users to attract secondary users to
employ higher power level and maintain enough transmission
time at the same time. We can formulate this model as a game
between primary user and secondary users; and it is proved to
converge to a unique optimal equilibrium. By employing a proper
power control updating algorithm, we can achieve the optimal
equilibrium point, which will be shown in the numeric results.
I. I NTRODUCTION
With the dramatic development of the mobile telecommunication industry, the rapidly growing demand for radio spectrum
is becoming a serious problem we must face. Frequency spectrum, which is scarce resource for wireless communications,
may be congested by diverse users and applications in the
next generation wireless network. Traditionally, the resources
of frequency is allocated in the way of granting organizations
licenses to use certain frequencies; but this method will result
in a large portion of the scarce spectrum remaining unutilized.
Moreover, the spectrum utilization varies significantly with
time and location[1].
The inefficient usage of limited frequency spectrum can be
improved by using the cognitive radio(CR) technology, which
can enable wireless devices to utilize the spectrum adaptively
and efficiently. Cognitive radio is a special type of software
defined radio which is able to estimate the communication parameters and operate opportunistically to make use of unused
spectrum. By using frequency spectrum in this opportunistic
way, secondary user can sense which portion of the spectrum
are available, select the best-condition channel, coordinate
access to spectrum channels with other secondary users and
quit the channel when primary user appears.
As cognitive radio dynamically interact with each other so
that the outcome depend not only on their own action but also
on other users’ actions. So the game theory is naturally fit for
the cognitive radio system design. This paper applies game
theory approach to power control in cooperative cognitive
radio networks.
Recently, cooperative mechanism becomes an interesting
topic in cognitive radio research. In [10], the author propose
a very interesting novel cooperative network called cooperative cognitive radio network(CCRN). In this system model,
primary user can involve secondary users as the cooperative
relays and in return, the secondary users achieve the opportunity to access the wireless channel for their own data
transmission and the transmission time is related to its payment
to primary user. The author prove the existence of the unique
Nash equilibrium, and propose a traversal algorithm to achieve
the user’s optimal strategy. However, the secondary user’s
cooperative power is set to be fixed, which is not efficient;
also, the secondary users own transmission time is not related
to its contribution to primary users transmission, which is not
reasonable. Based on this observation, we propose a novel
cooperative mechanism to overcome the limitation.
In this paper, we make an effort in employing power
control into cooperative cognitive radio network. The main
contribution of the paper are as follows:
• We formulate the problem as a game between one primary user and N secondary users, where primary user
select a proper set of secondary users to serve as the
cooperative relays for its transmission; and in return lease
portion of channel access time to the selected secondary
users for their own transmission. Primary user’s strategy
is to select proper secondary users and to determine the
portion of channel access time left for secondary users’
transmission. Secondary users’ strategy is to choose the
power level used to help primary user’s transmission.
• We define utility functions of primary and secondary
users. Primary user’s utility function is related to its
transmission rate. While secondary user’s utility function
is related to their own transmission rate and the cost of
power, and secondary user’s access time is related to the
effort it makes when it acts as cooperative relay, i.e. the
power level it uses to help primary user’s transmission.
We prove that there is a unique Nash equilibrium in the
network, and work out corresponding strategy of primary
user and secondary users.
• We proposed a updating algorithm to select available
secondary users and achieve the optimal strategy of
primary and secondary users.
• Numeric analysis shows that under our framework, the
performance of primary and secondary user is better than
before, and secondary user with better channel condition
will achieve higher utility.
The remainder of the paper organized as follows. Related
works are reviewed in section II. In Section III, we introduce
our system model. The utility function and the Nash equilibrium analysis are presented in Section IV. In Section V, we
give the implementation protocol and algorithm of this model.
We use simulation results to discuss the performance of this
model in Section VI and the conclusion is presented in Section
VII.
•
II. R ELATED W ORK
Cognitive radio is a novel approach for improving the
utilization of a spectrum. In [2], an introduction to cognitive
radio techniques is provided, three fundamental cognitive tasks
and the emergent behavior of cognitive radio are discussed. A
survey of cognitive radio is presented in [1], issues on spectrum management, spectrum mobility and spectrum sharing
and related research challenges are discussed. A survey of
dynamic spectrum access techniques is provided in [3] where
three dynamic spectrum access models and related technical
challenges are discussed.
Game theory has been considered as an effective and reasonable method to study the behavior of users in cognitive networks, where different users competes for the same resource.
The problem of spectrum sharing in cognitive radio networks
can be formulated as various game models[26]. A single-stage
cooperative dynamic spectrum sharing game is presented in [6]
to analyze multiuser orthogonal frequency-division multipleaccess systems. Multi-stage dynamic game and auction game
are adopted in [7] for spectrum sharing and power allocation.
Also, [5] provides a game theoretical overview of dynamic
spectrum sharing from several aspects: analysis of network
users behaviors, efficient dynamic distributed design, and
optimality analysis.
Cooperative mechanism is also an interesting topic in
cognitive radio research. As far as we know, there are two
work focusing on using game theory to analyze the performance of cooperation in cognitive radio networks: [9] and
[10]. In [9], primary users lease their spectrum to secondary
users for a fraction of time and in exchange, they get the
cooperative transmission power from secondary users. [10]
propose a cooperative mechanism where primary user can
involve secondary users as the cooperative relays and in return,
the secondary users achieve the opportunity to access the
wireless channel for their own data transmission, the secondary
users’ transmission time is related to its payment to primary
user. However, in [10], secondary user’s cooperative power
is set to be fixed, which is not efficient. In [9], the author
adopted Decode-and-Forward (DF) as cooperation protocol,
it is hard to get the analytical solution of the secondary
users’ optimal cooperative power; so the author only propose
a traversal algorithm to get the optimal cooperative power
for secondary users, the cost will be intolerable when the
number of secondary user is large. In our model, we design
a power control mechanism for secondary users, and we
adopt Amplify-and-Forward (AF) protocol as our cooperation
protocol, which will make it feasible to achieve the analytical
solution of the secondary users’ optimal cooperative power.
Also, there is a limitation in both [9] and [10]: the secondary
user’s utility function does not concern about the secondary
users’ contribution to primary user’s transmission. So we adopt
time-division multiplexing access (TDMA) as the secondary
users access model and divide the time between multiple
secondary users according to their contribution to primary
user’s transmission. Our model is more rational because the
secondary users’ utility is positively related to the contribution
they make to primary user’s transmission, the incentive of
secondary user taking part in the cooperation will be increased,
which is beneficial for the whole network; also, it’s fairer that
the secondary user who contributes more to the primary user
gets more revenue.
III. S YSTEM M ODEL AND U TILITY F UNCTION
A. System Model
In this section, we describe the model of our utility-based
cooperative cognitive radio network and the basic parameters
in this system. The system is sketched in Figure 1.
We consider one primary user and multiple secondary
users operating in the same area. In this network, no traffic
requirement is imposed. So primary and secondary users all
do their best to transmit data as much as possible. The primary
transmitter select a set of secondary users S as relays in its data
transmission, and in return, the selected secondary users can
access the channel when primary users are not active, which is
announced by primary user at the beginning of a transmission
slot. The selected secondary users have to choose a power
level to cooperate with the primary link, and the access time
is related to its power level.
In our model, one time slot is generally divided into two
parts: a fraction α of the slot is used for primary user’s
transmission, while the rest 1 − α unit time of slot is used
for secondary users’ transmission. In the former fraction of
slot, the first 12 α fraction is used for the transmission from
primary transmitter PT to secondary transmitters STs (Figure
1(a)), the second 12 α fraction is used for STs to transmit the
received data to primary receiver PR (Figure 1(b)). In the latter
fraction of slot, the selected secondary users access the channel
in time-division multiplexing access (TDMA) mode (Figure
1(c)). The access time secondary user i obtains is proportional
to the contribution it makes in the cooperative process, which
SR4
SR4
ST4
PR
ST2
PT
PR
ST1
ST1
ST3
SR1
PT
PR
ST1
SR3
SR1
ST3
¢
¢
ST5
¢
¢
¢
¢
¢
SR1
SR5
ST5
ST5
¢
SR3
SR5
SR5
¢
SR2
ST2
PT
SR3
ST4
SR2
ST2
ST3
SR4
ST4
SR2
¢
¢
(b)
(a)
¢
(c)
Fig. 1. System model: (a) in first 12 α fraction of a time slot, PT transmits data to PR and STs; (b) in second 12 α fraction of a time slot, PT and STs transmit
data to PR cooperatively; (c) in third 1 − α fraction of a time slot, secondary users transmit their own data in TDMA mode.(0 ≤ α ≤ 1)
is related to the cooperative power level Pi it chooses:
Pi Gi,P GP,i
j∈S (Pj Gj,P GP,j )
ti = (1 − α) ∑
(1)
We assume that the channel condition between two nodes
is invariable, and they are modeled as independent Gaussian
random variables; GP denotes the channel gain between PT
and PR; GP,i denotes the channel gain between PT and STi ;
Gi,P denotes the channel gain between STi and PR; Gi
denotes the channel gain between STi and secondary receiver
SRi ; and σ 2 is the noise variance. It is also assumed that
primary user transmitting at fixed power level P0 , and the
power level secondary users use for their own transmission is
fixed at PS .
In our model, the amplify-and-forward(AF) cooperation
protocol[19] is employed to implement the cooperation between primary user and secondary users.
First, considering the signal-to-noise ratio(SNR) at PR
caused by PT’s direct transmission, we can have:
ΓP =
P0 GP
σ2
(2)
So the relayed SNR of secondary user i can be represented
as follows:
Pi P0 Gi,P GP,i
(6)
Γi = 2
σ (Pi Gi,P + P0 GP,i + σ 2 )
Assuming that the network we discuss is energy constrained.
Then, the primary user’s transmission rate with relays’ help
can be presented as follows:
∑
RP = W log2 (1 + ΓP +
Γi )
i∈S
∑
P0 GP
Pi P0 Gi,P GP,i
= W log2 (1 +
+
)
σ2
σ 2 (Pi Gi,P + P0 GP,i + σ 2 )
i∈S
(7)
The transmission rate of secondary user i can also be
achieved in similar way:
PS Gi
), i ∈ S
(8)
σ2
For simplicity, W will be set to be 1 in the following
discussion.
Ri = W log2 (1 +
B. Utility Function
Then, we consider about secondary users’ contribution to
primary user’s transmission. With secondary user i being primary user’s relay, we assume that Xi is the PT’s transmission
signal to secondary user i, Yi is the received signal at STi , and
Zi is the received signal at P R from STi . So we can have:
√
Yi = P0 GP,i Xi + ηP,i
(3)
In this section, we define the utility function for both
primary user and secondary users, which is the basis of further
game analysis.
Primary user’s only goal is maximizing its transmission rate.
So the utility function could be defined as follow:
where ηP,i ∼ N (0, σ ).
where α is the portion of a slot primary user remain for its
own data transmission. RP is the achievable transmission rate
with secondary users’ help.
As primary user’s strategy is to choose a value for α and
a set of secondary users S as its relays. It is obvious that
α must be set properly. If α is too high, then the time
remained for secondary users’ transmission will be too short
that secondary users will make less effort in cooperation, i.e.
2
Zi =
√
Pi Gi,P
Yi
+ ηi,P
|Yi |
where ηi,P ∼ N (0, σ 2 ). Substituting (3) into (4):
√
√
Pi Gi,P ( P0 GP,i Xi + ηP,i )
√
Zi =
+ ηi,P
P0 GP,i + σ 2
(4)
(5)
UP = αRP (α)
(9)
Pi will be relatively small; so the primary transmission rate
may decrease. On the other hand, if α is set to be too low,
the utility of primary user will be unnecessarily low. So there
is a tradeoff on α. Then the optimization problem for primary
user could be formulated as follows:
max UP = αRP (α), 1 ≥ α ≥ 0.
(10)
Secondary user should consider not only the transmission
rate it can achieve, but also the energy cost in helping primary
user’s transmission. So secondary users’ goal is maximizing
its transmission rate under a reasonable energy cost. Then the
utility function of secondary user i could be defined to be
its achievable transmission rate minus the its energy cost in
helping primary transmission:
1
Ui = wi Ri ti − αPi
2
Pi Gi,P GP,i
1
= (1 − α)wi ∑
Ri − αPi
(P
G
G
)
2
j j,P P,j
j∈S
(11)
where wi is the equivalent revenue per unit transmission rate
contributes to the overall utility for secondary user i.
Secondary users’ strategy is choosing proper Pi , so the
optimization problem for secondary user i could be formulated
as follows:
Pi Gi,P GP,i
1
max Ui = (1 − α)wi ∑
Ri − αPi ,
2
(12)
j∈S (Pj Gj,P GP,j )
By solving equation (13), we can have:
(√
)
1
1
−
α
Pi∗ =
2wi
Gi,P GP,i ARi − A
Gi,P GP,i
α
(14)
1−α
if A < 2wi
Gi,P GP,i Ri
α
∑
whereA = j∈S,j̸=i (Pj Gj,P GP,j )
Theorem 2: The Nash equilibriun of the noncooperation
power level selection game is unique.
Proof. See Appendix B.
Then we solve the equation set (13) for each secondary
user i ∈ S, we can get the unique Nash equilibrium of the
noncooperative power level selection game:
N0 − 1
2(1 − α)(N0 − 1)
(B −
)/B 2 (15)
αGi,P GP,i
wi Gi,P GP,i Ri
∑
where B = j∈S wj Gj,P1GP,j Rj , N0 is the total number of
selected secondary users.
From equation (15), the constraint in equation (14) can be
presented as follows:
Pi∗ =
B>
N0 − 1
wi Gi,P GP,i Ri
(16)
which will be used to select proper relay set S.
{Pi } ≥ 0, i ∈ S.
B. Primary User optimal Strategy Selection
IV. G AME T HEORY A NALYSIS
Based on the utility function we define above, we will
analyze the game to obtain solutions to the game outcomes.
And the existence and uniqueness of the optimal equilibrium
will be proved.
A. Secondary User Power Control Strategy Selection
Given the relay set S and slot division parameter by primary
user, the selected secondary users’ problem is to choose a
optimal cooperative power level to maximize its utility. So
there is a noncooperative power level selection game G =
[S, {Ti }, {Ui ()}], where S is the selected relay set; Ti is the
strategy set of secondary user i, and Ui () is the utility funtion
of secondary user i. Each secondary user i must select its
strategy within the strategy space T = [Ti ]i∈S : Pi ≥ 0 to
maximize its utility function ui (Pi , P−i ).
Theorem 1: A Nash equilibrium exists in the noncooperation power level selection game, G = [S, {Ti }, {Ui ()}].
Proof. See Appendix A.
By Theorem 1, we know that a Nash equilibrium exists in
the noncooperative game and Ui (P) is concave in Pi . So we
can get the best-response function when the first derivative of
Ui with respect to Pi equals to 0.
∑
Gi,P GP,i j∈S,j̸=i (Pj Gj,P GP,j )
1
∂Ui
∑
= (1 − α)wi
Ri − α
2
∂Pi
( j∈S (Pj Gj,P GP,j ))
2
=0
(13)
As is referred above, the optimization problem for primary
user is presented in equation (10). So we substitute (15) into
(9):
(
)
P0 GP ∑
Pi∗ P0 Gi,P GP,i
+
,
UP = α log2 1 +
σ2
σ 2 (Pi∗ Gi,P + P0 GP,i + σ 2 )
i∈S
1≥α≥0
(17)
Primary user’s object is maximizing its utility by selecting
optimal parameter α∗ . So by the first order optimality condition, we can have:
∂UP
∂RP (α)
= RP (α) + α
=0
∂α
∂α
(18)
Substituting (7) into (18) and solve it, we can get the optimal
α∗ ,
α∗ = α∗ (σ 2 , {wi }, {Gi }, {Gi,P }, {GP,i }), i ∈ S
′
′
(19)
′
For ∀0 ≤ α ≤ 1, Up (α ) ≤ Up (α∗ ); for ∀Pi ≥
′
0, Ui (P1∗ , P2∗ , ..., Pi , ..., PN0∗ ) ≤ Ui (P1∗ , P2∗ , ..., Pi∗ , ..., PN0∗ ).
So we finally get the optimal parameter α∗ and the corresponding optimal cooperative power level Pi∗ . The solution (15) and
(19) is a Stackelberg equilibrium, which will be proved in the
following subsection.
C. Proof of the Equilibrium
In this subsection, we will prove that the solution (15) and
(19) is a Stackelberg equilibrium.
Property 1: For ∀i ∈ S, the optimal power level Pi∗ is
decreasing with α.
Proof. See Appendix C.
Property 2: Primary user’s utility funtion UP is concave in
α.
Proof. See Appendix D.
Theorem 3: Pi (i ∈ S) and α solved in (15) and (19) is a
Stackelberg equilibrium for the model in this paper.
Proof. See Appendix E.
V. I MPLEMENTATION P ROTOCOL
In this section, we will propose a cooperation protocol.
In our model, we assume that the channels are stable and
the channel condition are detected by corresponding terminal:
Gi,P is detected by PR, GP,i is detected by STi and Gi is
detected by SRi . Then the channel condition and other necessary parameter will be sent to PT. Based on these information,
primary user can select a proper set of secondary users as its
cooperative relays and calculate the optimal portion parameter
α∗ . At the beginning of transmission, primary user will set α
to be 1, then it will decrease α gradually until UP achives
its maximal value, i.e., α reaches its optimal
value. Also,
∑
primary user will broadcast the value of j∈S wj Gj,P1GP,j Rj
to secondary users, so that secondary users can calculate their
optimal power level corresponding to received value of α. In
the following subsections, we will discuss the relay selection
method and the updating algorithm.
A. Relay Selection
Since all the analysis above is based on an assumption that
the relay set S has already been selected, we will discuss the
relay selection method in this subsection.
In order to accomodate the possible users in good position in the network, we select secondary users who satisfies
the selection criteria (16). Without loss of generality, we
assume that w1 G1,P GP,1 R1 ≥ w2 G2,P GP,2 R2 ≥ ... ≥
wNS GNS ,P GP,NS RNS , where NS is the number of secondary
user exists in the network. Then we will select the secondary
users as relay in the way shown in Table A.
A.
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
Relay Selection Algorithm
put all the secondary users in the network into relay set S
arrange the secondary users in this order: w1 G1,P GP,1 R1
w2 G2,P GP,2 R2 ≥ ... ≥ wNS GNS ,P GP,NS RNS
for t = 0 to NS − 1 step by 1 do
N0 = NS − t
∑
N0 −1
1
f lag =
j∈S w G
G
R − w
G
G
R
j
j,P
P,j
j
N0
N0 ,P
if f lag ≤ 0 then
remove secondary user N0 from relay set S
else
break
end if
end for
P,N0
≥
N0
Theorem 4: In the network we define in this paper, supposing that w1 G1,P GP,1 R1 ≥ w2 G2,P GP,2 R2 ≥ ... ≥
wNS GNS ,P GP,NS RNS and there exists a possible relay set
S0 = {i|1 ≤ i ≤ NR }, i.e. secondary user i(1 ≤ i ≤ NR ) is
′
′
in S0 and satisfy the selection criteria (16). For ∀S ⊆ S0 , S
is also a possible relay set.
Proof. See Appendix F.
It is shown by Theorem 4 that the algorithm in Table A is
an effective way to select possible secondary users in good
position as relay.
B. Updating Algorithm for Primary User
From the explanation above, we know that there exists a
Stackelberg equilibrium. Then we will propose an updating
algorithm to achieve the Stackelberg equilibrium in this subsection.
In our model, as the selected secondary users must adapt
their strategy according to primary user’s strategy, so primary
user can be regarded as ”leader” and the selected secondary
users can be regarded as ”follower”. We only need to design
an updating algorithm for primary user to achieve its optimal
parameter α∗ , then according to Theorem 3, each selected
secondary users will achive its own optimal power level Pi∗ .
By Property 2, we know that Primary user’s utility funtion
UP is concave in α. So if we set α to be 1 at the beginning,
then decrease α gradually. By the concavity of UP , we can
finally achieve the optimal α∗ . This is reasonable as primary
user remain the whole time slot for its own transmission at
the beginning, then primary user will increase the time left for
secondary users’ transmission and corresponsively the selected
secondary users will increase their cooperative power level Pi ,
which will increase primary user’s transmission rate, by this
way, primary user will maximize its utility. So we define an
updating algorithm as follows: at the beginning, we set α to
be 1. In the loop, we calculate primary user’s utility according
to current value of α and record it; then we compare primary
user’s utility with its last recorded value, if primary user’s
utility is still increasing, i.e., we doesn’t achive the optimal
α∗ yet, so α will be lower by a very small value δ; if primary
user’s utility is not increasing, that means we have achieved
the optimal α∗ last loop, so we should set α to be α + δ,
which is the optimal α∗ ; then we stop updating α. The whole
updating process is presented in Table B.
B.
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
Updating Algorithm
f lag = 0 /*flag identifies whether optimal α∗ is achieved */
record(0) = 0 /*array used for recording primary user’s utility */
α=1
for t = 1 to L do /*L is a number large enough*/
calculate UP according to (17)
record(t + 1) = UP
if UP > q(t)&&f lag == 0 then
α = α − δ /*δ is a number small enough*/
elseif UP < q(t) then
α=α+δ
f lag = 1
end if
end for
It is obvious that we can finally achieve a value that lies in
interval (α∗ − δ, α∗ + δ), when δ is small enough, we could
approximately regarded the achieved value as α∗ .
0.5
18
17
0
8
2
PT
7
3
16
PR
1
4
6
Utility of PU
9
15
14
5
13
10
−0.5
0
0.5
Fig. 2.
12
0.8
1
Topology of the system
0.85
Fig. 3.
0.9
α
1
Optimal α of primary user
1
VI. S IMULATION R ESULTS A ND A NALYSIS
Utilities of SUs
3
Powers of SUs
In this subsection, we consider a geometrical model where
PT is located at coordination (0,0), PR is located at coordination (1,0), and secondary tranmitters are located randomly
on a square centering on (0.5,0) with side length D = 1;
and secondary receivers are located randomly on a unit square
centering on corresponding secondary transmitter. The propagation loss factor is set to be 2. 10 secondary users are
supposed to exist in the network, wi are all set to be 10. The
noise level is σ 2 = 10−4 and updating step δ = 10−5 .
Figure 2 is the topology of the network. The green squares
represent SRs and the blue circles represent STs, a pair of
ST and SR is connected by a green line. STs, which are
selected as relay, are identified by solid circles; while other
STs are identified by hollow circles. The secondary users are
arranged by the order: w1 G1,P GP,1 R1 ≥ w2 G2,P GP,2 R2 ≥
... ≥ w10 G10,P GP,10 R10 . In Figure 2, 6 secondary users are
chosen by primary user as relays. There are two factors that
affect the relay selection: location of ST and the distance
between ST and SR. The location of STi will affects the
channel condition between STi and primary users, i.e., the
value of GP,i and Gi,P . As a secondary user with higher GP,i
and Gi,P will better help primary user’s transmission, so it is
beneficial for to select such secondary users as relays. Also,
the distance between STi and SRi will affect the value of Gi .
As a secondary user with higher Gi will get more revenue from
the cooperation with the same power cost, so such secondary
user will have more incentive to take part in the cooperation,
which is beneficial for primary user. So we can find that all
the selected secondary users in Figure 2 are in good position
and the distance between transmitter and receiver are not too
long.
Figure 3 unveils the utility of primary user according
to α varies from 0 to 1. From Figure 3, we know that
α∗ = 0.96 and the corresponding optimal primary user’s utility
UP (α∗ ) = 16.43. When α = 1, i.e., primary user remains the
whole time slot for its own transmission, UP (1) = 13.2879;
0.95
2
0.5
1
0
1 2 3 4 5 6 7 8 9 10
SU index
Fig. 4.
0
1 2 3 4 5 6 7 8 9 10
SU index
Secondary users’ optimal power and utility
as secondary user will have no incentive to take part in the
cooperation, so UP (1) will be less than UP (α∗ ). According to
simulation results, the improvement on primary user’s utility
is 20% - 35%. Therefore, this result shows that it is beneficial
to implement the cooperation with parameter α to be α∗ .
To study the behavior of secondary users, we focus on the
cooperative power they choose and the revenue they get. In
Figure 4, the secondary user with better channel condition
offers higher cooperative power and get more revenue. According to equation (1) and (9), the secondary user with better
channel condition will get more revenue from the cooperation
with the same power cost, so it will tend to offer higher
cooperative power. In return, owing to the higher cooperative
power and higher channel gain, the secondary user with good
channel condition will get more revenue, i.e., higher utility. It
is also reasonable that the secondary users who are not selected
offers no power and their utility is 0.
Then we compare the impact of different relay selection on
the system. In this simulation, according to subsection V-A, the
number of possible relay which satisfies the selection criteria
(16) is 6. So we choose first N (1 ≤ N ≤ 6) secondary
17
Optimal α
Optimal Utility
1
1
2 3 4 5 6
Number of Relays
15
13
1
2 3 4 5 6
Number of Relays
Fig. 5. Primary users’ utility and optimal α with different number of relays
users as relays. Figure 5 shows that with the number of
relay increasing, the utility of primary user increases and
the the optimal α∗ decreases. It is reasonable because if the
number of relay increases, the competition among secondary
users will be intenser, so secondary user will tend to offer
higher cooperative power, which is beneficial for primary
user’s transmission; this also means primary user will get more
revenue by release more time to secondary users in order to
intensify their competition. We also observe that when the
number of relay is 1, the optimal α∗ is equal to 1. As there is
only one relay in the network, the relay will have no incentive
to offer cooperative power because there is no competition
to access to the channel; so in return, primary user will not
release transmission time to the relay.
VII. C ONCLUSION
In this paper, we propose an power control mechanism in a
cooperative radio network, where primary user select proper
secondary users as cooperative relay and in return, leases
portion of time slot to secondary users for their transmission.
In the power control mechanism, secondary users’ access
opportunity is positive related to its cooperative power level,
which motivates the cooperation. We prove that a unique Nash
equilibrium exists by game analysis, and we propose an evolutional approach to the Nash equilibrium. Also, we propose
a relay selection method to accomodate all the possible users
in the network. Numerical results show that the evolutional
approach is feasible, and primary user will select secondary
users at good position as its relay and secondary users with
better channel condition will achieve higher utility.
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A PPENDIX
A.Proof of Theorem 1
Proposition 1: A Nash equlibrium exists in game G =
[S, {Ti }, {Ui ()}], if for all i ∈ S:
1)Ti is a nonempty, convex, and compact subset of some
Euclidean space RN .
2)Ui (P) is continuous in P and concave in Pi .[21]
It is obvious that the strategy space T defined in (IV-A) is
a nonempty, convex, and compact subset of Euclidean space
RN . From equation (11), we can see that Ui (P) is continuous
in P. Then the only problem left for us is proving that Ui (P)
is concave in Pi . So we take the secondary derivative of Ui
with respect to Pi :
∑
G2i,P G2P,i j∈S,j̸=i (Pj Gj,P GP,j )
∂ 2 Ui
∑
= −2(1 − α)wi
Ri
( j∈S (Pj Gj,P GP,j ))3
∂Pi 2
<0
(20)
As the secondary derivative of Ui with respect to Pi is
always less than 0, so Ui (P) is concave in Pi . Therefore,
a Nash equilibrium exists in the noncooperation power level
selection game.
As for scalability, for all β > 1, we can have:
βI(P) − I(βP)
)
(√
β
1−α
=
2wi
Gi,P GP,i ARi − A
Gi,P GP,i
α
(√
)
1
1−α
(24)
−
2βwi
Gi,P GP,i ARi − βA
Gi,P GP,i
α
√ √
β− β
1−α
=
2wi
Gi,P GP,i ARi
Gi,P GP,i
α
>0
The positivity, monotonicity and scalablity of Ii (P) are
proved, so I(P) is a standard funtion. Therefore, the Nash
equilibriun of the noncooperation power level selection game
is unique.
B.Proof of Theorem 2
C.Proof of Property 1
By Theorem 1, it is shown that a Nash equilibrium exists in
the noncooperative game. So the Nash equilibrium P has to
satisfiy P = I(P) = (I1 (P), I1 (P), ..., IN (P)), where Ii (P)
is the best-responce function of secondary user i given the
other selected secondary user’s power level strategy P−i .
From [22], we learn that if I(P) is a standard funtion, then
the fixed point P = I(P) is unique, i.e., the Nash equilibrium
the noncooperation game is unique.
Definition 1: A function I(P) is standard if for all P ≥ 0
the following properties are satisfied[22]:
From (15), we can take the first order derivative of Pi∗ with
respect to α:
•
•
•
Positivity I(P) > 0
′
′
Monotonicity If P ≥ P , then I(P) ≥ I(P ).
Scalability For all β > 1, βI(P) > I(βP).
whereA =
function is:
Ii (P) =
∑
1−α
Gi,P GP,i Ri
α
j∈S,j̸=i (Pj Gj,P GP,j ).
1
Gi,P GP,i
(√
2wi
(21)
So the best-responce
1−α
Gi,P GP,i ARi − A
α
We assume that M = 1 + P0σG2 P , Ci = P0 Gi,P GP,i ,
0 −1)
Di = σ 2 Gi,P , Ei = σ 2 (P0 GP,i + σ 2 ), Fi = G2(N
(B −
i,P GP,i
N0 −1
2
)/B
.
Then
we
can
rewrite
equation
(7)
and
(25)
wi Gi,P GP,i Ri
as follows:
∑ Ci Pi
RP = log2 (M +
)
(26)
Di Pi + Ei
i∈S
)
(22)
As for positivity. From equation(21) and equation (22), we
can have:
(√
)
1
1−α
Gi,P GP,i ARi − A
Ii (P) =
2wi
Gi,P GP,i
α
(√
)
(23)
1
>
A2 − A
Gi,P GP,i
=0
As
√ for monotonicity, Ii (P) is quadratic with respect
A. So Ii (P) is monotonically increasing if A ≤
1−α
w
i Gi,P GP,i Ri .
2α
to
This result make sense because if primary user remains
more time for its own transmission, i.e. increasing α; the
selected relays will have less incentive to take part in the
cooperative process, so they will decrease their cooperative
power Pi .
D.Proof of Property 2
Assuming that the constraint in equation (14) is satisfied,
A < 2wi
∂Pi∗
2(N0 − 1)
N0 − 1
=− 2
(B −
)/B 2 < 0 (25)
∂α
α Gi,P GP,i
wi Gi,P GP,i Ri
and
∂Pi∗
Fi
=− 2
∂α
α
(27)
From equation (26), (27) and (18), we can have:
∑ Ci Pi
α
∂UP
= log2 (M +
)+
×
∂α
Di Pi + Ei
ln 2
i∈S
∑
1
Ci Ei Fi
1
· (− 2 )
∑
Ci Pi
2
α
M + i∈S Di Pi +Ei i∈S (Di Pi + Ei )
∑ Ci Pi
1
)−
×
= log2 (M +
Di Pi + Ei
α ln 2
i∈S
∑
1
Ci Ei Fi
∑
Ci Pi
(Di Pi + Ei )2
M + i∈S D P +E
i
i
i
i∈S
(28)
Then, the secondary derivative of UP with respect to α can
be achieved:
1
1
∂ 2 UP
=− 2
∑
∂α2
α ln 2 M + i∈S
+
−
−
=−
−
∑
Ci Pi
Di Pi +Ei i∈S
∑
1
Ci Ei Fi
1
∑
C
P
i
i
ln 2 M + i∈S D P +E
(Di Pi + Ei )2
i i
i i∈S
∑
2
i Fi
( i∈S (DCi Pi E
2)
1
i +Ei )
∑
i
α ln 2 α2 (M + i∈S D CPi P+E
)2
i i
i
∑
2Ci Di Ei Fi
1
i∈S (Di Pi +Ei )3
∑
i
α ln 2 α2 (M + i∈S D CPi P+E
)
i i
i
∑
Ci Ei Fi
2
( i∈S (Di Pi +Ei )2 )
1
∑
i
2
α ln 2 α (M + i∈S D CPi P+E
)2
i i
i
∑
2Ci Di Ei Fi
1
i∈S (Di Pi +Ei )3
∑
i
α ln 2 α2 (M + i∈S D CPi P+E
)
i i
i
(29)
k∈S ′
≥
E.Proof of Theorem 3
When Primary user broadcast the parameter α, the selected secondary users will respond with a cooperative power
level Pi∗ as shown in (15). By Theorem 1 and Theorem 2,
we prove that {Pi∗ } is the unique Nash equilibriun of the
′
noncooperation power level selection game. So for ∀Pi ≥
′
0, Ui (P1∗ , P2∗ , ..., Pi , ..., PN0∗ ) ≤ Ui (P1∗ , P2∗ , ..., Pi∗ , ..., PN0∗ ),
i.e., {Pi∗ } is optimal response strategy for the selected secondary users. Also, by Property 2, primary user’s utility
funtion UP is concave in α; so primary user can always find
′
′
the optimal α∗ , i.e., for ∀α ∈ [0, 1], UP (α ) ≤ UP (α∗ ). So
Pi (i ∈ S) and α solved in (15) and (19) is a Stackelberg
equilibrium for the model in this paper.
F.Proof of Theorem 4
′
Since S ⊆ S0 , without loss of generality, we assume that
′
′
the number of secondary user in is N (N ≤ NR ).
As secondary user i ∈ S0 , 1 ≤ i ≤ NR and S0 is a possible
relay set, so secondary user NR satisfies the selection criteria
(16):
NR − 1
1
>
wk Gk,P GP,k Rk
wNR GNR ,P GP,NR RNR
(30)
And wj Gj,P GP,j Rj ≥ wi Gi,P GP,i Ri , so the following
inequation can be obtained:
′
NR − N
≥
wNR GNR ,P GP,NR RNR
∑
k∈S0 &&k∈S
/ ′
1
wk Gk,P GP,k Rk
(31)
∑
k∈S ′
α2
2
k∈S0
∑
Ci Ei Fi
(Di Pi + Ei )2
Since α, Pi , M, Ci , Di , Ei , Fi > 0, ∂∂αU2P < 0. So Primary
user’s utility funtion UP is concave in α.
∑
′
then, from (30)and (31), for ∀j ∈ S , we can have:
′
1
N −1
−
wk Gk,P GP,k Rk
wj Gj,P GP,j Rj
′
1
N −1
−
wk Gk,P GP,k Rk
wNR GNR ,P GP,NR RNR
∑
+
′
k∈S0 &&k∈S
/ ′
=
∑
k∈S0
NR − N
1
−
wk Gk,P GP,k Rk
wNR GNR ,P GP,NR RNR
1
NR − 1
−
wk Gk,P GP,k Rk
wNR GNR ,P GP,NR RNR
>0
(32)
i.e.,
∑
k∈S ′
′
′
1
N −1
>
wk Gk,P GP,k Rk
wj Gj,P GP,j Rj
So S is also a possible relay set.
(33)