A GAME THEORETIC APPROACH TO INTERFERENCE ...

A GAME THEORETIC APPROACH TO INTERFERENCE MANAGEMENT IN COGNITIVE NETWORKS NIE NIE∗ , CRISTINA COMANICIU∗† , AND PRATHIMA AGRAWAL‡ Abstract. In this paper, we propose a game theoretic solution for joint channel selection and power allocation in cognitive radio networks. Our proposed algorithm enforces cooperation among nodes in an effort to reduce the overall energy consumption in the network. For designing the power control, we consider both the case in which no transmission power constraints are imposed, as well as the more practical case, in which the maximum transmission power is limited. We show that an iterative algorithm for channel scheduling and power allocation can be implemented, which converges to a pure strategy Nash equilibrium solution, i.e., a deterministic choice of channels and transmission powers for all users. Our simulation results also show that, while both channel allocation and power control can independently improve the system performance, there is a significant gain for the joint algorithm. Key words. Cognitive radio, channel allocation, power control, potential game. AMS(MOS) subject classifications. 91A80, 68M10.

1. Introduction. The explosive growth of wireless services and the increased users’ population density call for intelligent ways of managing the scarce spectrum resources. With the new paradigm shift in the FCC’s spectrum management policy [2] that creates opportunities for new, more aggressive, spectrum reuse, cognitive radio technology lays the foundation for the deployment of smart flexible networks that cooperatively adapt to increase the overall network performance. The cognitive radio terminology was coined by Mitola [1], and refers to a smart radio which has the ability to sense the external environment, learn from the history, and make intelligent decisions to adjust its transmission parameters according to the current state of the environment. As the cognitive radios are essentially autonomous agents that are learning their environment and are optimizing their performance by modifying their transmission parameters, their interactions can be modeled using a game theoretic framework. In this framework, the cognitive radios are the players and their actions are the selection of new transmission parameters and new transmission frequencies, etc., which influence their own performance, as well as the performance of the neighboring players. Game theory has been extensively applied in microeconomics, and only more recently has received attention as a useful tool to design and ∗ Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030 ([email protected]). This work was supported in part by the NSF grant number: CNS-0435297. † [email protected]. ‡ Department of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849-5201 ([email protected]).

1

2

NIE NIE, CRISTINA COMANICIU AND PRATHIMA AGRAWAL

analyze distributed resource allocation algorithms (e.g., [4–8]). Some game theoretic models for cognitive radio networks were presented in [9], which has identified potential game formulations for power control, call admission control and interference avoidance in cognitive radio networks. The convergence conditions for various game models in cognitive radio networks are investigated in [10]. In our previous work [11] we have proposed a distributed channel allocation algorithm for cognitive radios using a fixed transmission power. In this paper, we extend this framework to a more realistic scenario, that also allows the radios to control their transmission powers. Here we assume that the cognitive radios are sensing the environment by sending probes and measuring the available channels, and then distributively select the best channels, then they optimize their transmission powers according to their channel selection to minimize the energy consumption. With a goal of optimizing the link throughput and establishing a fair spectrum sharing over the network, the radios measure the local interference temperature on different frequencies and adjust by maximizing the data transmission rate for a given channel quality (using adaptive modulation) and by possibly switching to a different frequency channel. To tackle the above problem, we propose a game theoretic formulation, in which the adaptive channel allocation and power control problem is modelled as a potential game similarly to the one presented in [11]. The radios are modelled as a collection of agents that distributively act to maximize their utilities in a cooperative fashion. The radios’ decisions are based on their perceived utility associated with each possible action which is related to the transmission power and to the channel selection. We study the convergence properties of the proposed adaptation algorithm and we design adaptation protocols for this algorithm. Two scenarios (power control with and without maximum transmission power limitation) are considered, and the effect of various maximum power levels on the system performance is investigated. We further study the tradeoffs related to energy consumption, throughput and fairness when power control and channel allocation are employed individually and jointly in the network. A glossary of abbreviations defined in this paper is listed as following: CA N P C Channel Allocation, No Power Control P C N CA Power Control, No Channle Allocation CP C N CA Constraint Power Control, No Channel Allocation JCAP C Joint Channel Allocation with Power Control JCACP C Joint Channel Allocation with Constraint Power Control 2. System model. The cognitive radio network we consider consists of a set of N transmitting-receiving pairs of nodes, uniformly distributed in a square region of dimension D ∗ × D∗ . We assume that the nodes are either fixed, or are moving slowly (slower than the convergence time for the proposed algorithms). Fig. 1 shows an example of a network realization,

A GAME THEORETIC APPROACH IN COGNITIVE NETWORKS

3

1000 900 800 700 600 500 400 300 200 100 0

0

200

400

600

800

1000

Fig. 1. A snapshot of the nodes’ positions and network topology.

Table 1 Modulation Modes in Adaptive Modulation and corresponding SIR requirement for target BER=10−3 .

Modulation Mode

SIR (dB)

1024 QAM 256 QAM 64 QAM 16 QAM QPSK BPSK

35.5 29.4 23.3 16.9 9.9 6.8

where we used dashed lines to connect the transmitting node to its intended receiving node. The nodes measure the spectrum availability and decide on the transmission channel. We assume that there are K frequency channels available for transmission, with K < N . By distributively selecting a transmitting frequency, the radios effectively construct a channel reuse distribution map with reduced co-channel interference. The transmission link quality can be characterized by a required Bit Error Rate target (BER), which is specific for the given application. An equivalent SIR target requirement can be determined, based on the modulation type selected when employing an adaptive modulation scheme (see Table 1). The Signal-to-Interference Ratio (SIR) measured at the receiver j associated with transmitter i can be expressed as: SIRij = PN

k=1,k6=i

pi Gji pk Gjk I(k, j) + σ 2

,

(2.1)

4


where pi is the transmission power at transmitter i, Gji is the link gain between transmitter i and receiver j. σ 2 denotes the received noise and it is assumed to be the same for all receiver nodes. I(i, j) is the interference function characterizing the interference created by node i to node j and is defined as   1 if transmitters i and j are transmitting over the same channel (2.2) I(i, j) =  0 otherwise. Analyzing Eq. (2.1) we see that in order to maintain a certain BER constraint the nodes can adjust at both the physical and the network layer level. At the network level, the nodes can minimize the interference by appropriately selecting the transmission channel frequency. At the physical layer, power control can reduce interference and, for a feasible system, results in all users meeting their SIR constraints. Alternatively, the target SIR requirements can be changed (reduced or increased) by using different modulation levels. As an example of adaptation at the physical layer, we have assumed that software defined radios enable the nodes to adjust their transmission rates and consequently the required SIR targets by adaptively changing the modulation scheme. Also, the nodes can adjust their transmission power level via distributed power control to ensure that all the nodes sharing the same channel meet the target SIR requirement at their intended receiver. The BER requirement selected for simulations is 10−3 , and we assume the use of an adaptive modulation scheme with six modes: BPSK, QPSK, 16 QAM, 64 QAM, 256 QAM and 1024 QAM. In Table 1 we show the modulation modes and the corresponding SIR target requirements used for our simulations [12, 13]. For our simulations we are also assuming that all users have packets to transmit at all times (worst case scenario), and that multiple users are allowed to transmit at the same time over a shared channel. We assume that users in the network are identical, which means they have an identical action set and identical utility functions associated with their possible actions. 3. Power control. For the users sharing the same frequency channel, their transmission powers affects their link quality and the interference temperature on that particular channel. The goal of power control is to adjust the transmission powers of all users to improve the link quality and to enable the group of users who are transmitting over the same channel to meet a certain target BER, which can be associated with the highest SIR target that can be met (the highest rate) using adaptive modulation. Given a target SIR γ ∗ , for a feasible system with N users, a non-negative power vector P ∗ can be obtained by: P ∗ = (I − H)−1 η,

(3.1)

5

A GAME THEORETIC APPROACH IN COGNITIVE NETWORKS G

where Hij = (hij ) is the normalized link gain matrix such that hij = γ ∗ Gij ii for i 6= j and hij = 0 for i = j, η = (ηi )i=1..N is the normalized noise vector such that ηi = γ ∗ Gσii , and σ 2 is the received noise power. We consider that there are K frequency channels, and each channel is shared by a group of users that transmit on the same frequency. Each user group determines their transmission powers via power control. Let si = 1, 2, ..., K denote the choice of transmitting channel for user i, i ∈ N , the power vector for the kth user group can be determined by: Pk∗ = (I − Hk )−1 ηk , f or k = 1, 2, ..., K,

(3.2)

where Hk = (hij ) for si = k, sj = k and i 6= j, and ηk is the normalized noise vector for si = k. The number of the elements of Pk∗ is equal to the number of the users who transmit on the same channel. For a feasible system, Pk∗ should be a non-negative vector, Pk∗ (i) > 0, i ∈ Nk , with the assumption that the transmission power can be adjusted without limitations. However, in practice, the maximum output power of a transmitter is upper-bounded. Taking this limitation into account, the transmission power vector Pk∗ can still be determined by Eq. (3.2) but with the constraint that 0 ≤ Pk∗ (i) ≤ PM AX , i ∈ Nk , f ork = 1, 2, ..., K,

(3.3)

where PM AX denotes the maximum transmitter output power depending on the physical device, and/or regulation restrictions. Consequently, the constrained transmission power P¯k∗ (i) = min{Pk∗ (i), PM AX }, i ∈ Nk . It is clear that by selecting different transmitting channels, a user will belong to different user groups and will choose its operating power level with respect to the interference environment of that particular group. When the channels are allocated adaptively, the population and the members of these user groups will change with the current channel assignment. 4. A game theoretic formulation. Game theory represents a set of mathematical tools developed for the purpose of analyzing the interactions in decision processes. In this work, we model our channel allocation problem as a normal form game (see also [11]), which can be mathematically defined as Γ = {N, {Si }i∈N , {Ui }i∈N }. N is the finite set of players (cognitive radios as decision makers). Si is the set of strategies associated with player i. In our case, the players’ strategies are the choice of a transmitting channel, si = 1, 2, ..., K. We define S = ×Si , i ∈ N as the strategy space and Ui : S → R as the set of utility functions that the players associate with their strategies. For every player i in game Γ, the utility function Ui is a function of si , the strategy selected by player i, and of the current strategy profile of its opponents: s−i . The utility function characterizes a player’s preference for a particular choice of strategy.

6


In analyzing the outcome of the game, as the players make decisions independently and are influenced by the other players’ decisions, we are interested to determine if there exist a convergence point for the adaptive channel selection algorithm, from which no player would deviate anymore, i.e., a Nash equilibrium (NE). A strategy profile for the players, S = [s1 , s2 , ..., sN ], is a NE if and only if 0

Ui (S) ≥ Ui (si , s−i ), ∀i ∈ N, s0i ∈ Si .

(4.1)

If the equilibrium strategy profile in Eq. (4.1) is deterministic, a pure strategy Nash equilibrium exists. For finite games, even if a pure strategy Nash equilibrium does not exist, a mixed strategy Nash equilibrium can be found (equilibrium is characterized by a set of probabilities assigned to the pure strategies). 4.1. Utility function. For our joint channel allocation and power control game, the utility function should characterize the preference of a user for a particular channel, given the fact the user knows that power control is employed by all the users sharing a given channel. Considering that users are willing to cooperate to achieve a fair allocation of resources, we impose that the utility function must account for both the interference perceived by the current user, as well as the interference that particular user is creating to neighboring users sharing the same channel. A possible choice for the utility function may be: Ui (si , s−i ) = −

N X

pj (sj )Gij f (sj , si )

N X

pi (si )Gji f (si , sj )

j6=i,j=1

−

(4.2) ∀i = 1, 2, ..., N.

j6=i,j=1

For the above definition, we denoted P=[p1 ,p2 ,...,pN ] as the transmission powers for the N radios and S=[s1 ,s2 ,...,sN ] as the strategy profile. The transmission powers depend on the strategy profile S (the channel allocation) as discussed in the Section 3. f (si , sj ) is an interference function defined as:   1 if sj = si , transmitter j and i choose the same strategy (same channel) f (si , sj ) =  0 otherwise

Given that the above utility function accounts for both the interference measured at the current user’s receiver, and the interference created by the user to others, the algorithm implementation for the channel selection becomes more complex, since it will require probing packets on a common access channel for measuring and estimating the interference a user will create to neighboring radios.


7

4.2. Potential game formulation and equilibrium convergence. In the previous section we have defined an utility function and discussed the physical meaning to which the formulation gears. Now, we further study the mathematical properties impacted on this function in order to have good convergence properties for the adaptation algorithm. It has been shown that certain classes of games converge to a Nash equilibrium when a best or better response adaptive strategy is employed, see Refs. [14–18]. In what follows, we show that for the utility function defined in Eq. (4.2), an exact potential game can be formulated that ensures that a pure strategy Nash equilibrium solution to be reached for the joint power control and channel selection algorithm. Characteristic for a potential game is the existence of a potential function that exactly reflects any unilateral change in the utility function of any player. The potential function models the information associated with the improvement paths of a game instead of the exact utility of the game [15]. An exact potential function is defined as a function Fp : S → R, if for all i, and si , s0i ∈ Si , with the property that Ui (si , s−i ) − Ui (s0i , s−i ) = Fp (si , s−i ) − Fp (s0i , s−i ).

(4.3)

If a potential function can be defined for a game, the game is an exact potential game. In an exact potential game, for a change in actions of a single player, the change in the potential function is equal to the value of the improvement deviation. Any potential game in which players take actions sequentially converges to a pure strategy Nash equilibrium that maximizes the potential function. For our previously formulated joint power control and channel allocation game with utility function U , we can define an exact potential function to be N N X X Fp (S) = Fp (si , s−i ) = −α pj (sj )Gij f (sj , si ) i=1

j6=i,j=1

−(1 − α)

N X


j6=i,j=1

∀i = 1, 2, ..., N,

(4.4)

0 < α < 1.

The function in Eq. (4.4) essentially reflects the network utility. It can be seen thus that the potential game property Eq. (4.3) ensures that an increase in individual users’ utilities contributes to the increase of the overall network utility. Without loss of generality, in this work we set α = 0.5. See Appendix A, for a proof that the function defined in Eq. (4.4) is an exact potential function.

8


We note that the above property holds only if users take actions sequentially, following a best response strategy. Consequently, a certain coordination among users should be implemented in the distributed algorithm. The easiest way to implement a certain level of coordination is to allow users to play the game only when they win a Bernouli trial with probability p = N1 , where N is the number of users currently in the system. Another observation is that the evaluation of the utility function in Eq. (4.2) implies that an estimate of the vector powers should be computed for each considered configuration of users on channels, in order to determine the interference power. For this purpose, knowledge on channel gains for all users should be available, and can also be obtained by channel probing, measurements, and exchanging control messages in the network. 5. Performance evaluation. In this section, we present some numerical results to illustrate the performance of the proposed adaptive channel allocation and power control algorithm for both non-constraint power control and constraint power control scenarios. For simulation purposes, we consider a fixed wireless ad hoc network (as described in the system model section) with N = 20 and D = 1000 (20 transmitters and their receivers are randomly distributed over a 1000m × 1000m square area). The joint adaptation algorithm is illustrated for a network of 20 transmitting radios, sharing K = 4 available channels. A random channel assignment is selected as the initial assignment. For a fair performance comparison, all the simulations start from the same initial channel allocation. The BER requirement is 10−3 and the noise power σ 2 is set to be 10−13 . For the numerical results, a path loss coefficient of n = 2 was selected. From the simulation results, we can see that the proposed adaptive channel allocation and power control game preserves the convergence property of the cooperative spectrum sharing algorithm proposed in [11] for fixed power transmission. Both cases with non-constraint and with constraint power control converge to a pure strategy Nash equilibrium (a determined channel assignment), but reaching different equilibrium points. As an example, Fig. 2 illustrates the convergence property for the joint power control and channel allocation algorithm, when no constraints on the maximum transmission power are imposed. As performance measures for the proposed algorithm we consider the achieved SIRs and throughputs (adaptive modulation is used to ensure a certain BER target, as previously explained in Section 2). We consider the average performance per user as well as the variability in the achieved performance (fairness), measured in terms of variance. We also consider the total transmission power as a measure related to the energy efficiency. We also study the improvement in performance that can be achieved by employing a joint optimization over channels and powers (JCAPC), compared to the case for which either only power control is used to improve the performance, given a random allocation of the channels (Power Control,


9

The strategies taken by the nodes 4

S: Strategies

3

2

1

0

100

200

300

400

500

600

700

800

900

T: Number of Trials

Fig. 2. Potential game: convergence of users’ strategies.

No Channel Allocation (PC NCA)), or only channel adaption is employed for a fixed transmission power level (Channel Allocation, No Power Control (CA NPC)). We first illustrate the results for the case of non-constraint power control. We assume that all the users have the same initial transmission power. We examine different initial power levels with a range covering 1.5 × 10−5 , 1.5 × 10−6 , 1.5 × 10−7 , 1.5 × 10−8 , 1.5 × 10−9 , 1.5 × 10−10 W . We find that the performance of the algorithm with non-constraint power control is independent on the initial power level. As an example, in Fig. 3, we show the histograms of the users’ achieved SIRs for a) the initial randomly channel assignment with fixed transmission power; b) PC NCA ; c) CA NPC ; d) JCAPC, with an initial transmission power of 1.5 × 10−7 W . It can be seen that, for PC NCA, power control reduces the interference temperature by adjusting the transmission power of neighboring nodes, and consequently improves the SIRs of the users who suffer from poor link conditions, such that no user has an SIR below 7 dB. Furthermore, we see that in CA NPC, adaptive channel allocation can provide further improvement by creating a better frequency reuse allocation even without power control: no user will have an SIR below 10 dB. However, when channel allocation and power control are employed jointly, the advantage is obvious in that the distribution of the users’ achieved SIRs is concentrated around a mean value (with very low variance), with almost all the users maintaining an achieved SIR around 20 dB, which demonstrates a fair and efficient spectrum sharing.

10


a) initial SIRs

b) PC_NCA

6

10 8

4

6 4

2

2 0 −20

0

20

40

60

0 −20

0

c) CA_NPC 20

6

15

4

10

2

5

0

20

40

60

40

60

d) JCAPC

8

0 −20

20

40

60

0 −20

0

20

Fig. 3. Histogram of users’ achieved SIRs. a) Initial State b) PC NCA c) CA NPC d) JCAPC, initial transmission power = 1.5 × 10−7 W .

The performance in terms of the normalized achievable throughput at each receiver is similar, as illustrated in Fig. 4. By exploiting both of the channel allocation and power control techniques in the proposed cooperative spectrum sharing algorithm, a more fair allocation of the throughput is achieved throughout the network. The improvement for the users that initially had a low performance can be noticed, at the expense of a slight penalty in performance for the users with initially high throughput. We further investigated the performance of the proposed algorithm under the constraint of maximum data transmission power (JCACPC). In the simulation, all the users operate at the maximum data transmission power initially. The initial power level range used in the study of JCAPC is utilized to examine the effects of various maximum data transmission power limitations on the performance of JCACPC. In Fig. 5, we illustrate the histogram of achieved SIRs with the maximum data transmission power set to be 1.5 × 10−7 W . It is shown that, CPC NCA with a limitation of transmission power has a poorer performance in terms of achieved SIR without the help of frequency reusing planning. This happens because the benefits of the power control are limited by the maximum transmission power constraint.


a) initial Throughputs

11

b) PC_NCA

8

15

6

10

4 5

2 0 −5

0

5

10

15

0 −5

0

c) CA_NPC

5

10

15

10

15

d) JCAPC

10

20

8

15

6 10 4 5

2 0 −5

0

5

10

15

0 −5

0

5

Fig. 4. Histogram of users’ normalized Throughputs. a) Initial State b) PC NCA c) CA NPC d) JCAPC, initial transmission power = 1.5 × 10−7 W .

a) initial SIRs

b) CPC_NCA

6

6

4

4

2

2

0 −20

0

20

40

60

0 −20

0

c) CA_NPC 20

6

15

4

10

2

5

0

20

40

60

40

60

d) JCACPC

8

0 −20

20

40

60

0 −20

0

20

Fig. 5. Histogram of users’ achieved SIRs. a) Initial State b) CPC NCA c) CA NPC d) JCACPC, maximum transmission power = 1.5 × 10−7 W .

12


a) Pmax = 1.5e−8 W 20 10 0 −5

0

5

10

15

20

25

30

35

40

45

30

35

40

45

30

35

40

45

b) Pmax = 1.5e−9 W 10 5 0 −5

0

5

10

15

20

25

c) Pmax = 1.5e−10 W 10 5 0 −5

0

5

10

15

20

25

Fig. 6. Histogram of users’ achieved SIRs of JCACPC algorithm with various maximum data transmission power.

The proposed JCACPC algorithm maintains its advantages and demonstrates a performance close to that of JCAPC in Fig. 3. However, its performance is dependent on the upper bound of the transmission power. In Fig. 6, we illustrate the evolution of the distribution of users’ achieved SIRs as the upper bound of transmission power drops from 1.5 × 10−7 W to 1.5 × 10−10W . It can be found that the JCACPC algorithm degenerates into CA NPC algorithm gradually as the maximum transmission power decreases. When the upper bound drops to 1.5 × 10−10W or below, JCACPC performs as the same as CA NPC does, which means constraint power control can not provide constructive contributions to the algorithm anymore. A similar trend can be observed for the throughput in Fig. 7 and Fig. 8. The reason behind this fact is that when the power upper bound is relatively high, the joint channel allocation and power control algorithm has more freedom to adjust the users’ power, and benefit from the optimal power allocation. The transmission power distribution at equilibrium point in this case is shown in Fig. 9. As the upper bound drops, some of the users may have to transmit at the maximum power and still not meet the target yet, as shown in Fig. 10. But, to some extent, the adaptive channel allocation can compensate part of the loss of performance. However, when the transmission power constraint becomes more and more strict, more and

A GAME THEORETIC APPROACH IN COGNITIVE NETWORKS a) initial Throughputs

b) CPC_NCA

8

8

6

6

4

4

2

2

0 −5

0

5

13

10

0 −5

15

0

c) CA_NPC

5

10

15

10

15

d) JCACPC

10

20

8

15

6 10 4 5

2 0 −5

0

5

10

0 −5

15

0

5

Fig. 7. Histogram of users’ normalized Throughputs. a) Initial State b) CPC NCA c) CA NPC d) JCACPC, maximum transmission power = 1.5 × 10−7 W .

a) Pmax = 1.5e−8 W 20 10 0 −4

−2

0

2

4

6

8

10

12

8

10

12

8

10

12

b) Pmax = 1.5e−9 W 10 5 0 −4

−2

0

2

4

6

c) Pmax = 1.5e−10 W 10 5 0 −4

−2

0

2

4

6

Fig. 8. Histogram of users’ normalized Throughputs of JCACPC algorithm with various maximum data transmission power.

14


−7

1.8

x 10

1.6

Maximum Power

1.4 1.2 1 0.8 0.6 0.4 0.2 0

Fig. 9. Transmission power distribution at equilibrium point of JCACPC, maximum transmission power = 1.5 × 10−7 W .

−8

5

x 10

4.5 4 3.5 3 2.5 2 Maximum Power 1.5 1 0.5 0



15

−10

x 10

2.5

2 Maximum Power 1.5

1

0.5

0


more users are unable to meet the target, and consequently the performance degrades. Eventually, when the upper bound drops below a certain value, 1.5×10−10W in this example, all the users are forced to transmit at an equal maximum power (see Fig. 11 ), and JCACPC yields the same performance as that of CA NPC. In Fig. 12, we compare the energy consumption in terms of total transmission power for all the five scenarios: CA NPC, PC NCA, CPC NCA, JCAPC and JCACPC. We find that the energy consumption of CA NPC increases linearly with the initial fixed transmission power. CPC NCA’s total transmission power demonstrates the same trend. PC NCA and JCAPC show a constant energy consumptions with different initial power levels, but PC NCA may converge to a point with much higher energy consumption due to the lack of adaption in the channel selection, which may require some users to use high powers to overcome the interference on their current channel. In Fig. 13 we summarize the performance comparisons among all five cases in terms of average throughput per user and variance of the throughput per user. The variance measure quantifies the fairness, with the fairest scheme having the lowest variance. 6. Conclusion. In this work, we have proposed a game theoretic solution for joint channel selection and power allocation in a cognitive ad hoc

16


Total Transmission Power vs. Initial(MAX) Transmission Power

2

10

0

Total Transmission Power

10

Without PC PC without MAX Power PC with MAX Power Only PC without MAX Power Only PC with MAX Power

−2

10

−4

10

−6

10

−8

10

−10

10

−10

10

−9

10

−8

10

−7

−6

10

10

Initial(MAX) Transmission Power

Fig. 12. Total transmission power vs. Maximum transmission power (constraint power control) or Initial transmission power (other cases).

7 Average Throughput per User Variance of the Throughput per User

6

5

4

3

2

1

0

PC_NCA

CPC_NCA

CA_NPC

JCAPC

JCACPC

Fig. 13. Average-Throughput and Variance of the Throughput per user for all the cases.

17


network. Based on channel probing and measurements, the users extract information to independently assess their channel preferences and compute their optimal transmission power. We prove that the distributed algorithm can be modeled as an exact potential game, which is guaranteed to converge to a pure strategy Nash equilibrium, i.e., to a deterministic selection of channels and powers for all users. Our simulation results quantify the performance gain obtained by the proposed joint algorithm compared to simply employing power control for a fixed channel allocation, or to adaptively choosing the channels but with a fixed transmission power level.

APPENDIX A. Proof of the exact potential function. Suppose there is a potential function of game Γ is defined in Eq. (4.4) as:   N N N X X X −α pi (si )Gji f (si , sj ) pj (sj )Gij f (sj , si )−(1−α) Fp (S) = i=1

j6=i,j=1

j6=i,j=1

where 0 < α < 1. Then for all i ∈ {1, 2, ..., N }, Fp (si , s−i ) N X



i=1

−α

= −α

N X

=

N X

pj (sj )Gij f (sj , si ) −(1−α)

j6=i,j=1

+



−α

k6=i,k=1

= −α

pj (sj )Gij f (sj , si ) − (1 − α)

N X

pj (sj )Gkj f (sj , sk )−(1−α)

+

k6=i,k=1


N X



pk (sk )Gjk f (sk , sj )

j6=k,j=1

pj (sj )Gij f (sj , si ) − (1 − α)

j6=ij=1

N X

N X

j6=i,j=1

j6=k,j=1

N X

pi (si )Gji f (si , sj )

j6=i,j=1

j6=i,j=1

N X



N X

N X


j6=ij=1



−αpi (si )Gki f (si , sk ) − α

N X

j6=k,j6=ij=1

pj (sj )Gkj f (sj , sk )

18




N X

−(1−α)pk (sk )Gik f (sk , si )−(1−α)


j6=k,j6=i,j=1

N X

= −α

pj (sj )Gij f (sj , si ) − (1 − α)

j6=i,j=1

+

N X

k6=i,k=1

+

−α

N X




N X


j6=k,j6=i,j=1

N X

pj (sj )Gij f (sj , si ) − (1 − α)

N X

pi (si )Gki f (si , sk )(1 − α)

j6=i,j=1

−α

(−(1−α)pk (sk )Gik f (sk , si ))

j6=k,j6=i,j=1

−(1 − α)

= −α

N X

k6=i,k=1



k6=i,k=1


j6=i,j=1

(−αpi (si )Gki f (si , sk )) +

N X

N X


j6=i,j=1

k6=i,k=1

+

N X

N X

pk (sk )Gik f (sk , si )

k6=i,k=1

N X



k6=i,k=1

−α

−(1 − α)

N X


j6=k,j6=i,j=1

N X

j6=k,j6=i,j=1



pk (sk )Gjk f (sk , sj ) .

Let Q(s−i ) =

N X

k6=i,k=1



−α

N X

j6=k,j6=i,j=1


A GAME THEORETIC APPROACH IN COGNITIVE NETWORKS N X

−(1 − α)



j6=k,j6=i,j=1

pk (sk )Gjk f (sk , sj ) .

Then, Fp (si , s−i ) = −α

N X


j6=i,j=1 N X

−(1 − α)


j6=i,j=1

−α

N X

pi (si )Gki f (si , sk )

k6=i,k=1

−(1 − α)

N X

pk (sk )Gik f (sk , si ) + Q(s−i ).

k6=i,k=1

Substitute k with j, = −(α + (1 − α))

N X


N X

pi (si )Gji f (si , sj ) + Q(s−i ).

j6=i,j=1

−(α + (1 − α))

j6=i,j=1

If user i changes its strategy from si to s0i , we can get: Fp (s0i , s−i ) = −α

N X

pj (sj )Gij f (sj , s0i )

j6=i,j=1 N X

−(1 − α)

pi (s0i )Gji f (s0i , sj )

j6=i,j=1

−α

N X

pi (s0i )Gki f (s0i , sk )

k6=i,k=1

−(1 − α)

N X

pk (sk )Gik f (sk , s0i ) + Q(s−i )

k6=i,k=1

= −(α + (1 − α))

N X


N X

pi (s0i )Gji f (s0i , sj ) + Q(s−i ).

j6=i,j=1

−(α + (1 − α))

j6=i,j=1

19

20


Here Q(s−i ) is not affected by the strategy changing of user i. Hence, Fp (s0i , s−i ) − Fp (si , s−i ) = −(α + (1 − α))

N X


N X


j6=i,j=1

−(α + (1 − α))

j6=i,j=1



− −(α + (1 − α)) −(α + (1 − α))

N X

N X

j6=i,j=1

=−

N X

− −




pj (sj )Gij f (sj , s0i ) −

j6=i,j=1




j6=i,j=1

N X


j6=i,j=1

N X

pj (sj )Gij f (sj , si ) −

j6=i,j=1

N X

j6=i,j=1



pi (si )Gji f (si , sj ) .

From Eq. (4.2), Ui (s0i , s−i ) − Ui (si , s−i ) =−

N X


N X


j6=ij=1

−

j6=ij=1



− −

N X

pj (sj )Gij f (sj , si ) −

j6=i,j=1

N X

j6=i,j=1




∀i = 1, 2, ..., N,

Ui (s0i , s−i ) − Ui (si , s−i ) = Fp (s0i , s−i ) − Fp (si , s−i )∀i = 1, 2, ..., N. (A.1) So, we prove that Fp (S) defined in Eq. (4.4) is an exact potential function of game Γ.


21

REFERENCES [1] J. Mitola III, Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio, Doctor of Technology Dissertation, Royal Institute of Technology (KTH), Sweden, May 2000. [2] Facilitating Opportunities for Flexible, Efficient, and Reliable Spectrum Use Employing Cognitive Radio Technologies, FCC Report and Order, FCC-0557A1, March 11, 2005. [3] J. Mitola III, Cognitive Radio for Flexible Mobile Multimedia Communications, IEEE 1999 Mobile Multimedia Conference (MoMuC), November, 1999. [4] David J. Goodman and Narayan B. Mandayam, Network Assisted Power Control for Wireless Data, Mobile Networks and Applications, 6(5): 409–415, 2001 [5] S. Ginde, J. Neel, and R. Buehrer, Game Theoretic Analysis of Joint Link Adaptation and Distributed Power Control in GPRS, Vehicular Technology Conference, Orlando October 2003. [6] D. Krishnaswamy, Game-theoretic formulations for network-assisted resource management in wireless networks, Proceeding of IEEE Vehicular Technology Conference, Vancouver, September 2002. [7] Shin Horng Wong and Ian J. Wassell, Application of Game Theory for Distributed Dynamic Channel Allocation, IEEE 55th Vehicular Technology Conference, Spring 2002, Birmingham, AL, pp. 404–408, May 2002. [8] R. Menon, A. MacKenzie, R. Buehrer, and J. Reed, Game Theory and Interference Avoidance in Decentralized Networks, SDR Forum Technical Conference, November 15–18, 2004. [9] J. Neel, J.H. Reed, and R.P. Gilles, The Role of Game Theory in the Analysis of Software Radio Networks, SDR Forum Technical Conference, November 2002. [10] J. Neel, J.H. Reed, and R.P. Gilles, Convergence of Cognitive Radio Networks, Wireless Communications and Networking Conference, 2004. [11] N. Nie and C. Comaniciu, Adaptive Channel Allocation Spectrum Etiquette for Cognitive Radio Networks, IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN) 2005, Nov. 2005. [12] S.T. Chung and A.J. Goldsmith, Degree of Freedom in Adaptive Modulation: A Unified View, IEEE Transactions on Communications, 49(9), September 2001. [13] J.G. Proakis, Digital Communications, The McGraw-Hill Companies, Inc. 2001. [14] R. Rosenthal, A class of games possessing pure-strategy Nash equilibria, International Journal of Game Theory, 2: 65–67, 1973. [15] D. Monderer and L. Shapley, Potential Games, Games and Economic Behavior 14: 124–143, 1996. [16] G. Arslan and J. Shamma, Distributed convergence to Nash equilibria with local utility measurements, In 43rd IEEE Conference on Decision and Control, pp. 1538–1543, 2004. [17] Jeff S. Shamma and Grdal Arslan, Dynamic Fictitious Play, Dynamic Gradient Play, and Distributed Convergence to Nash Equilibria, IEEE Transactions on Automatic Control, 50(3), March, 2005. [18] Samuel Ieong, Robert McGrew, Eugene Nudelman, Yoav Shoham, and Qixiang Sun, Fast and Compact: A Simple Class of Congestion Games, In Proceedings of American Association for Artifical Intelligence (AAAI), 2005.