Performance of Energy Efficient Game Theoretical

Performance of Energy Efficient Game Theoretical-based Power Control Algorithm in a Discrete Power Environment F. de S. Chaves, R. B. Santos, F. R. P. Cavalcanti and R. A. de Oliveira Neto

Abstract— Transmitter power control is an essential tool for managing the interference in a system constituted by a set of communications nodes. In the absence of a central controller, the decentralized power control is of special interest and may be characterized as a competition among the communication nodes. This indicates the adequacy of the application of noncooperative game theory concepts to the problem. The continuous domain for transmit power is usually assumed, although, in practice, digitally power controlled systems work with discrete power levels. In this work we evaluate the performance of a distributed power control algorithm which is derived from the formulation of the power control problem as a static multi-stage nonzerosum noncooperative game. Continuous and discrete domains for transmit power are considered. Index Terms— Power control, game theory, wireless systems.

I. I NTRODUCTION Wireless communications systems may be generally characterized by multiple communication nodes and a limited amount of radio resources. Interference is one of their most important limiting factors, since it deteriorates the quality of the links [1]. Transmitter power control is an essential tool for managing the interference. The main objective of the transmitter power control is to increase the Quality of Service (QoS) or to satisfy its requirements by using minimum power resources. Distributed or decentralized power control is of special interest and importance, since it allows the use of only local information for determining a suitable transmit power [1], [2]. The decentralized power control problem may be characterized as a competition among the communication nodes. The power resources must be efficiently allocated by self-optimization, since the transmit powers of all nodes can not be jointly determined by a central entity. This indicates the adequacy of the application of noncooperative game theory concepts to the problem. Game theory is a mathematical tool for analyzing the interaction of decision makers with conflicting objectives and for the formulation of decision strategies. The potential of gametheoretical concepts begun to be explored in the distributed power control problem recently, as discussed in [3].

Utility- and pricing-based algorithms have been extensively studied in the literature [3]–[7]. The utility represents the user’s satisfaction when it uses a certain amount of the resource and the price is the cost per unit resource which the user must pay for this resource. The utility function used in [4], [5] includes details of system lower layers, as coding and modulation. Such dependence on system parameters may restrict the employment of an algorithm to some system configurations. Then, utility functions not dependent on system parameters were addressed in some works, as in [6], [7]. In this paper, we present the distributed power control algorithm derived from a static multi-stage nonzero-sum noncooperative game. We evaluate its performance in discrete domains for transmit power, taking as reference its performance in the continuous transmit power domain. The rest of the paper is organized as follows. In Section II we present the motivation of the approach taken in the paper. The mathematical development of the proposed power control algorithm is discussed in Section III. Section IV illustrates the performance of the proposed power control algorithm in the context of a CDMA system through simulation results. The conclusions of the paper are given in Section V. II. P OWER C ONTROL G AMES In this work, we approach the distributed power control problem as a set of N radio links (transmitter/receiver pairs), where each link is affected by interference caused by all other links. This is a simple and general model. Be NJ = {1, 2, ..., N } the set of radio links. Since power control actuation is discrete, Ka = {1, 2, ..., K} is defined as the set of its actuation time instants. Power used by the jth transmitter in time instant k is denoted pj (k), for all j ∈ NJ and k ∈ Ka . The path gain seen by the jth receiver in time instant k is represented by gj (k), while the correspondent interference-plus-noise power is Ij (k). Then, SINR perceived by jth receiver in time instant k, ρj (k), is given by: ρj (k) =

F. de S. Chaves is with the Signal Processing Laboratory for Communications (DSPCom), State University of Campinas, C.P. 6101, CEP: 13083-852, Campinas-SP, Brazil. email: [email protected]. R. B. Santos, F. R. P. Cavalcanti and R. A. de Oliveira Neto are with the Wireless Telecommunications Research Group (GTEL), Federal University of Cear´a, C.P. 6007, CEP: 60755-460, Fortaleza-CE, Brazil. email: {brauner, rodrigo, neto}@gtel.ufc.br. R. A. de Oliveira Neto was scholarship supported by Funcap. F. R. P. Cavalcanti was partly funded by CNPq, grant no. 304477/2002-8.

85-89748-04-9/06/$25.00 © 2006 IEEE

pj (k)gj (k) , Ij (k)

(1)

where Ij (k) is defined as:

Ii (k) =

N X

[pl (k)gl (k)] + σ 2 , l 6= i,

(2)

l=1

with σ 2 as the average AWGN power.

316

ITS2006

Different formulations for the distributed power control problem as a game are available in the literature. A game has three basic elements: a set of players, a set of possible actions for each player, and a set of objective functions mapping action profiles into real numbers. Players are the elements found in conflict. Each player has decision rights only over its own decision variables. Players are assumed to be rational, which means they decide for the strategy with the best individual game outcome. In a power control game, transmitters constitute the set of players and each player has as decision variable its own transmit power. Actions or decisions of players are confined to their strategy space, that is, the set of possible actions. The continuous set of feasible transmit power values is the strategy space of each player in a power control game. The satisfaction of each player is represented by its objective function. This is a special element in a game, since the objective function of a player must correspond to its interest with respect to the optimization process to be carried out. Objective functions can be classified as cost functions or utility functions. The concept of cost function refers to the expense or loss of the player as a result of its actions. Utility functions are referred to revenue or gain measures of players instead of their loss. Power control algorithms in the literature differentiate essentially on the objective functions. The decentralized power control game is naturally noncooperative, since players can not negotiate. Moreover, they are also nonzero-sum games, that is, the sum of objective functions of all players is not zero. It means that gains of a player do not correspond to losses of other players necessarily. Detailed discussions about game theory can be found in [8]. III. G AME -T HEORETICAL D ISTRIBUTED P OWER C ONTROL (GT-DPC) A LGORITHM We formulate the power control problem with power restrictions as a static multi-stage nonzero-sum noncooperative game, as in [9]. The multi-stage approach allows the development of algorithms able to deal with difficulties due to channel and interference time-varying nature. We denote GK = [NJ , {Pj }, {cj }] the static multi-stage nonzero-sum noncooperative power control game with K stages. The kth stage corresponds to the kth power control actuation. Transmitters constitute the set of players, with NJ = {1, 2, ..., N } as their index set. Moreover, the continuous set of power values Pj = [pj min , pj max ] is the strategy set of player j, for all j ∈ NJ . In order to simplify next mathematical developments, but without generality loss, a unique strategy space is adopted for all players: Pj = [pmin , pmax ]. We emphasize that the jth player has control only over its own power pj , which is selected such that pj ∈ Pj . The power vector p(k) = [p1 (k), ..., pn (k)] is the kth stage outcome of the game GK in terms of the selected power levels of all players. The vector consisting of the elements of p(k) other than the jth element is denoted p−j (k). The power control task can be seen as the tracking of a target SINR through power decisions. An intuitive cost function which preserves such characteristics is the squared

error between the target SINR and the actual SINR. This cost function, denoted cj , imposes a penalty to each player j which obtains SINR values far from the target SINR. Thus, the strategy of each player j in a time instant k of the game GK is defined below: 
min cj pj (k + 1), p−j (k + 1) = pj (k+1)∈Pj (3)  t 2 o ρj (k + 1) − ρj (k + 1) .

As can be seen, this is a general formulation, since target SINR ρtj (k + 1) is time-varying for j ∈ NJ . In the case of best effort services, which admit flexible resource allocations and experiment better performances with higher SINR levels, it would be adequate to have a power control algorithm which works with flexible target SINR. The target SINR choice is an important aspect, since it can bring the system to a more energy efficient operation point and further maintain the system stable. Therefore, a choice rule for the target SINR must be considered. A. Mapping function of transmit power into target SINR An adequate criterion for determining the target SINR must consider the required transmit power level. Players in favorable situation use few power resources. Then, a mapping function of transmit power into target SINR may determine high target SINRs for such players, without significative performance degradation of other players. When transmit power levels of players initially in favorable situation become insufficient to reach the determined SINR values, such transmit powers will be increased with the action of a power control algorithm and the overall interference level will increase consequently. In this case, in order to avoid performance degradation of other players, a mapping function responsible for defining the target SINR must determine lower levels of SINR to be targeted. A choice rule for the target SINR with these characteristics makes possible a stable and efficient operation point. An infinite number of functions may assume the role of the mapping function of transmit power into target SINR. We adopt the function shown in Fig. 1. This mapping function is suitable for high loaded systems, where only players in very good situations are allowed to target SINR levels significatively higher than the minimum one, as shown in Fig. 1(a). Fig. 1(b) illustrates the same function in logarithmic scale, where it is a linear function and can be defined as follows: ρtj (k + 1)dB = α − βpj (k + 1)dBm ,

(4)

with α and β as constant defined through extreme  parameters   dB dB t dBm t points pdBm , ρ and p , ρ . Furthermore, max max min min β is non-negative. Thus, their expressions are:   dB dB t t ρ − ρ dB min , (5) α = ρtmin +  max pdBm min 1 − pdBm

317

max

β=

dB ρtmax pdBm max

dB − ρtmin . − pdBm min

(6)

establishes a set of conditions to the existence of a NE solution: Theorem 1: For each j ∈ NJ , let Pj be a closed, bounded and convex subset of a finite-dimensional euclidian space, and the cost functional cj : P1 × · · · × PN −→ R be jointly continuous in all its arguments and strictly convex in pj for every pl ∈ Pl , l 6= j. Then, the associated nonzero-sum game admits a Nash equilibrium solution. In GK , the strategy space Pj = [pmin , pmax ] is a closed, bounded and convex subset of the euclidian space R, for all j ∈ NJ . Then, the verification of cost function continuity with respect to all its arguments is needed. Furthermore, cj must be strictly convex in pj for every pl ∈ Pl , l 6= j. In the following expressions, the time instant indicator (k + 1) will be omitted. In order to express the cost function cj in terms of transmit powers of all players, one must use the linear version of the mapping function expressed in logarithmic scale in (4). Moreover, SINR and game definition equations, respectively (1) and (3) must be used. So, cost function cj may be expressed in terms of transmit powers of all players:

Target SINR

ρtmax

ρtmin pmin

pmax Transmit power [mW]

(a) Linear scale.

dB

Target SINR [dB]

ρtmax

ρtmin

dB

(8)

 2   g  gj j α/10 ; ξ = 2 10 ; δ= . Ij Ij

(9)

with: pdBm min

Transmit power [dBm]

pdBm max

θ = 10

(b) Logarithmic scale. Fig. 1.

cj = θp−2β − ξp−β+1 + δp2j , j j

Mapping function of transmit power into target SINR.

It is important to observe that the choice of the target SINR level for time instant (k +1) is function of transmit power also in time instant (k + 1). Then, target SINR and transmit power are coupled and jointly determined. Furthermore, the cost function cj can provide a safe SINR evolution for all players, since the target SINR extreme values (ρtmax and ρtmin ) can be defined higher than a threshold SINR which corresponds to the minimum acceptable quality. B. Nash equilibrium point The transmit power that optimizes individual cost function depends on the transmit powers of all other transmitters. Therefore, it is necessary to determine a set of powers where each player is satisfied with the cost that it has to pay, given the power selections of other players. Such an operating point is called equilibrium point. The Nash equilibrium solution (or Nash equilibrium point) is one of the most celebrated equilibrium solutions [8]. Nash equilibrium (NE) concepts offer predictable and stable outcomes of a game where multiple agents with conflicting interests compete through self-optimization and reach a point where no player wishes to deviate from. Formally, a power vector p∗ (k) = [p∗1 (k), ..., p∗N (k)] is a NE point of game GK if, for each j ∈ NJ and k ∈ Ka , it holds:

cj p∗j (k), p∗−j (k) ≤ cj pj (k), p∗−j (k) . (7) The following theorem, whose proof can be found in [8],

2α/10

It is clear that θ, ξ and δ are continuous. Then, analyzing (8) we guarantee that the cost function cj is continuous with respect to all its arguments if pj 6= 0, j ∈ NJ . Then, as negative power values are unfeasible, by defining pmin > 0 the continuity of the cost function is assured. Conditions for cost function strict convexity are met in the following. The necessary optimality condition for a differentiable function is that its first-order derivative be equal to zero. After some algebraic manipulation and through the change of variable: −(β+1)

pj

= F,

(10)

one can write the partial derivative of the cost function cj with respect to pj as a quadratic function of the variable F . Solving the equation in F and returning to variable pj by using (10), we obtain the NE point in logarithmic scale (with the time instant indicator):  1  α + Ij (k + 1)dBm − gj (k + 1)dB . β+1 (11) The sufficient optimality condition for a two-time differentiable function is that its second-order derivative be different from zero. The second-order partial derivative of cj with respect to pj , after using (10) once more, can also be written as a quadratic function of the variable F , whose minimum value ǫ is given by:   [ξβ(1 − β)]2 − 4 4β 2 θ + 2βθ 2δ (12) ǫ=− . 4 [4β 2 θ + 2βθ] pj (k + 1)dBm =

Making ǫ > 0 we guarantee that the second-order derivative of the cost function cj is strictly positive, and obtain the following inequality:

318

β 3 < 2β 2 + 7β + 4 =⇒ β < 4,

(13)

approximation may converge to the NE solution in an iterative process, as demonstrated to the approximation:

which corresponds to the practical constraints below: 0 ≤ ρtmax

dB

− ρtmin

dB

  dBm < 4 pdBm max − pmin .

Ij (k + 1) Ij (k) ≈ . gj (k + 1) gj (k) (14)

Then, GK admits a NE solution, given by (11). Restrictions for the existence of the equilibrium are pmin > 0 and (14). C. Stability analysis of the Nash equilibrium point An important characteristic of an equilibrium solution is the stability. A NE solution is stable with respect to a determined deviation of a player, i.e., a choice different from the NE solution, if an iterative process converges to the originally satisfactory solution (NE point). The following definition can be used to carry out a stability analysis [8]. Definition 1: A Nash equilibrium solution u∗j , j ∈ M is stable with respect to the deviation scheme Υ if it may be obtained as: u∗j = lim uj (k),

(15)

k uj (k + 1) = arg min Jj (uj (k + 1), uΥ −j ),

(16)

k→∞

uj ∈Uj

k where u is the decision variable, uΥ −j is the deviation of players except player j in the time instant k, and J is the objective function. The NE point given by (11) can be written in linear scale:

pj (k + 1) = 10

α/[10(β+1)]



 1 Ij (k + 1) ( β+1 ) . gj (k + 1)

(17)

After using the expression of interference-plus-noise power (2) in equation above and from Definition 1, the NE point may be written as follows: p∗j = lim 10α/[10(β+1)] k→∞

"P

N l=1

[pl (k)gl ] + σ 2 gj

1 #( β+1 )

, (18)

where l 6= j and gj is the channel gain of player j, j ∈ NJ . A special case of deviation corresponds to the situation where players adjust their actions in response to the more k recent actions of the other players, that is, uΥ −j = u−j (k). In Υk GK , such situation is represented by p−j = p−j (k), which corresponds to: pj (k + 1) = arg min cj (pj (k + 1), p−j (k)) .

(19)

lim pj (k + 1) = lim pj (k) = p∗j , ∀j ∈ NJ ,

(20)

pj ∈Pj

Thus: k→∞

k→∞

assuring the stability of the NE point of GK with respect to the situation where the players adjust their transmit power in response to the status given in the previous power control actuation. In practice, NE solution given by (11) is not realizable, since values of channel gain and interference-plus-noise power at time instant (k + 1) are not available at the time instant k. However, from the stability analysis, we conclude that an

(21)

Then, we can finally present the GT-DPC algorithm:  1  α + pj (k)dBm − ρj (k)dB (22) β+1 with α and β defined in (5) and (6). dB It is important to observe in (5) and (6) that if ρtmax dB t t and ρmin tend to a unique value ρ dB , parameters α and β tend, respectively, to ρt dB and 0. It means that if the flexibility on the SINR to be targeted decreases, the GT-DPC algorithm tends to the classical DPC algorithm [2]. pj (k + 1)dBm =

IV. S IMULATION R ESULTS In this work, the distributed power control problem is approached in a general manner. In order to illustrate the performance of such algorithms by computational simulations, we consider a CDMA cellular system in the uplink direction. In this case, the intracell interference corresponds to a significative fraction of total interference. Then, we use a single-cell cellular model. A snapshot simulation model is adopted where mobile stations are uniformly distributed over the cell area. In each snapshot, up to 600 iterations of the power control algorithm are performed, in intervals of 0.666 ms. Other simulation parameters are set as follows. The cell radius is set to 1 km and omnidirectional antennas are considered. A generic macrocell path loss model for suburban areas is assumed, where P L(d) = 129.4 + 35.2 log10 (d). The distance d is expressed in kilometers and represents the distance between mobiles and base stations. Shadowing is modeled as a zero-mean lognormal random variable with standard deviation of 7 dB. Fast fading follows the Jake’s model [10] with carrier frequency assumed 2 GHz. According to previous discussions, the proposed GT-DPC algorithm is naturally a power control algorithm for best effort services, since it allows a flexible QoS. Then, the following best effort service classes are considered: • Service Class 1: Throughput between 64 and 256 kbps. • Service Class 2: Throughput between 64 and 512 kbps. Throughput requirements can be translated into radio link quality requirements. The Shannon’s capacity rule provides a theoretical limit to this mapping [11]. Using the 1 Shannon’s basic formula one can express the radio link quality, given by the bit energy per interference spectral density ratio (Eb /N0 ), as a function of the throughput Rb in bits/s and of the bandwidth W in Hz [12]: Eb /N0 =

2Rb /W − 1 . Rb /W

(23)

1 The use of the Shannon’s formula may be optimistic, but it is in accordance with recent breakthrough research in coding theory such as turbo codes. Moreover, a slight change in this mapping relationship does not change the conclusions of our work.

319

Nmax = 1 +

power to demonstrate the independence of NE solution on the initial power values. It is observed that GT-DPC and GT-DPC(Discrete) converge in few iterations. As expected, the GT-DPC(Up/Down) presents a slower convergence, due to its limitation with respect to the power adaptation step. One can also note that the algorithms converge to different values of Eb /N0 . The GT-DPC algorithm converge to the Eb /N0 which corresponds to the NE solution. However, their other versions achieve approximate values. Then, the GT-DPC algorithm converge to an appropriate solution even with discrete power sets.

1 0.9 0.8

Throughput [kbps]

In all simulations, we consider a constant bandwidth (64 kHz). Moreover, the noise power is set to −103 dBm and the maximum transmit power is limited to 21 dBm with a power range of 70 dB. The considered processing gain of the simulated CDMA system is 21 dB. We are interested in evaluating the performance of GTDPC algorithm in discrete transmit power domains, towards practical implementations. Then, three different versions of the algorithm are considered: • GT-DPC: The presented algorithm. It is executed over a continuous transmit power set. • GT-DPC(Discrete): GT-DPC algorithm executed over a discrete transmit power set with resolution of 1 dB. At each power control actuation, the transmit power given by the GT-DPC is approximated to the nearer value in the discrete power set. • GT-DPC(Up/Down): GT-DPC algorithm executed with restrictions on the step of the transmit power adaptation. At each power control actuation, the transmit power is increased or reduced in 1 dB. The conceptual maximum load Nmax for a CDMA system is determined by the pole capacity equation [13]. If used for a 256 kbps QoS guaranteed service class in the considered system:

0.7 0.6 0.5 0.4 0.3 0.2

GT−DPC GT−DPC(Discrete) GT−DPC(Up/Down)

0.1

PG ≈ 34, (Eb /N0 )t

(24)

0 64

80

100

120

140

160

180

200

220

240

256

Number of users

where P G is the processing gain and (Eb /N0 )t corresponds to the throughput of 256 kbps. The maximum load for the 512 kbps QoS guaranteed service class is 4 users. The maximum number of simulated users is 25.

(a) Service Class 1 1 0.9

15

Throughput [kbps]

0.8

10

Eb /N0 [dB]

5

0

0.7 0.6 0.5 0.4 0.3 0.2

−5 GT−DPC; max. initial power GT−DPC; min. initial power GT−DPC(Discrete); max. initial power GT−DPC(Discrete); min. initial power GT−DPC(Up/Down); max. initial power GT−DPC(Up/Down); min. initial power

−10

−15 5

10

15

20

25

30

35

40

45

50

GT−DPC GT−DPC(Discrete)

0.1 0 64

GT−DPC(Up/Down) 100

150

200

250

300

350

400

450

512

Number of users

(b) Service Class 2

55

Algorithm iteration

Fig. 3.

Fig. 2. Convergence curves of Eb /N0 obtained by GT-DPC algorithm with extreme initial power values.

In order to evaluate convergence aspects of GT-DPC algorithm, we consider experiments where users are static. Fig. 2 illustrates a sample of the Eb /N0 evolution achieved by one user in a typical snapshot where ten best effort users are placed in the cell. Similar curves were obtained for the other users. The extreme transmit power values are used as initial transmit

CDF curve of achieved throughput.

In order to obtain performance results statistically more representative, simulations of 10,000 snapshots are performed. The capacity of the algorithms to provide throughput in the specified range is verified with the Cumulative Distribution Function (CDF) curve of the throughput achieved by the users. Fig. 3 shows the throughput CDF curves for system loads of 25 users. We consider that all users move at 10 km/h, which correspond to a Doppler spread of 18.5 Hz.

320

0.008

Average fraction of time in which the Eb /N0 is 1 dB below the minimum Eb /N0

Service Classes 1 and 2 are considered separately in Figs. 3(a) and 3(b), respectively. In both cases, the use of GT-DPC algorithm makes the users experiment throughput levels coherent with the specified range, i.e., between 64 and 256 kbps for Service Class 1 and between 64 and 512 kbps for Service Class 2. It is important to observe that GTDPC(Discrete) and GT-DPC(Up/Down) produce almost the same throughput distribution of the GT-DPC algorithm. Although the cumulative distribution of throughput is almost the same, it is important to observe the energy efficiency of the algorithms. The energy efficiency is defined as the quantity of bits transmitted with a unit of energy. Fig. 4 shows the average energy efficiency per user for different system loads. Average energy efficiency per user [Mbits/J]

700

0.007

0.006

0.005

0.004

0.003

0.002

0.001

5

15

25

Number of users 600

Fig. 5. Average fraction of time in which the Eb /N0 is 1 dB below the (Eb /N0 )t

500

400

300

In practice, power controlled systems work with discrete power sets. Then, we evaluate the GT-DPC algorithm performance in discrete transmit power domains through extensive computer simulations in the context of a CDMA cellular system. The algorithm was shown to be suitable for elastic services, since it keeps the achieved Eb /N0 values inside the specified range. Further results indicate low performance degradation of GT-DPC algorithm in discrete power domains when compared to the same algorithm in the continuous power domain.

GT−DPC; Service Class 1 GT−DPC; Service Class 2 GT−DPC(Discrete); Service Class 1 GT−DPC(Discrete); Service Class 2 GT−DPC(Up/Down); Service Class 1 GT−DPC(Up/Down); Service Class 2

200

100

5

15

25

Number of users

Fig. 4.

GT−DPC; Service Class 1 GT−DPC; Service Class 2 GT−DPC(Discrete); Service Class 1 GT−DPC(Discrete); Service Class 2 GT−DPC(Up/Down); Service Class 1 GT−DPC(Up/Down); Service Class 2

Average energy efficiency per user.

In terms of energy efficiency, the GT-DPC and the GTDPC(Dicrete) algorithms have similar performances. The GTDPC(Up/Down) presents an efficiency loss of approximately 5% for Service Class 1 and 8% for Service Class 2 in all considered system loads. Finally, we verify the capacity of the algorithms to maintain the achieved throughput above a threshold value. In practical systems the minimum Eb /N0 represents the Eb /N0 required to maintain the minimum quality of the service. Therefore, a Eb /N0 margin below the target is considered in which signal quality is assumed acceptable. Then, the average fraction of time in which the achieved Eb /N0 is 1 dB below the minimum Eb /N0 is calculated for all algorithms and shown in Fig. 5. This figure brings results for both service classes. As expected, high system loads produce more difficulties on the satisfaction of minimum requirements as a minimum throughput. This is observed for all considered algorithms. However, one can see that GT-DPC(Up/Down) presents slightly better results than GT-DPC algorithm, while GT-DPC(Discrete) slightly worse ones. Once more, the discrete versions of GT-DPC have performance similar to the original algorithm. V. C ONCLUSIONS In this paper we present a distributed power control algorithm with power restrictions originated from a gametheoretical framework, the GT-DPC algorithm. This algorithm is developed over a continuous transmit power domain.

R EFERENCES [1] J. Zander, “Distributed Cochannel Interference Control in Cellular Radio Systems,” IEEE Trans. Veh. Technol., vol. 41, no. 3, pp. 305–311, Aug. 1992. [2] G. J. Foschini and Z. Miljanic, “A Simple Distributed Autonomous Power Control Algorithm and its Convergence,” IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 641–646, Nov. 1993. [3] A. B. MacKenzie and S. B. Wicker, “Game Theory and the Design of Self-Configuring, Adaptive Wireless Networks,” IEEE Commun. Magazine, vol. 39, no. 11, pp. 126–131, Nov. 2001. [4] C. U. Saraydar, N. B. Mandayam and D. J. Goodman, “Efficient Power Control via Pricing in Wireless Data Networks,” IEEE Trans. Commun., vol. 50, no. 2, pp. 291–303, Feb. 2002. [5] D. Goodman and N. Mandayam, “Power Control for Wireless Data,” IEEE Pers. Commun., vol. 7, no. 2, pp. 48–54, Apr. 2000. [6] S. Gunturi and F. Paganini, “Game Theoretic Approach to Power Control in Cellular CDMA,” in Proc. IEEE Veh. Technol. Conf. (VTC), vol. 4, pp. 2362–2366, Oct. 2003. [7] C. W. Sung and W. S. Wong, “A Noncooperative Power Control Game for Multirate CDMA Data Networks,” IEEE Trans. Wireless Commun., vol. 2, no. 1, pp. 186–194, Jan. 2003. [8] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd ed. SIAM, 1998. [9] F. de S. Chaves, R. A. de Oliveira Neto and F. R. P. Cavalcanti, “A Game-Theoretic Approach to the Distributed Power Control Problem in Wireless Systems and the Application of a Simple Prediction Method,” in Proc. Brazilian Symp. Telecommun., vol. 1, pp. 1–6, Sep. 2005. [10] W. C. Jakes, Microwave Mobile Communications, 2nd ed. Wiley, 1974. [11] C. E. Shannon, “The Zero Error Capacity of a Noisy Channel,” IRE Trans. Information Theory, vol. 2, pp. S8–S19, Sep. 1956. [12] J. G. Proakis, Digital Communications, 4th ed. McGrall-Hill, 2001. [13] J. S. Lee and L. E. Miller, CDMA Systems Engineering Handbook, Artech House Publishers, 1997.

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