Adaptive Power Control in Multi-Cell OFDM Systems - Semantic Scholar
IEICE TRANS. COMMUN., VOL.E89–B, NO.6 JUNE 2006
1951
LETTER
Adaptive Power Control in Multi-Cell OFDM Systems: A Noncooperative Game with Power Unit Based Utility∗ Lan WANG†a) and Zhisheng NIU†b) , Members
SUMMARY In this paper, we develop a new distributed adaptive power control framework for multi-cell OFDM systems based on the game theory. A specific utility function is defined considering the users’ achieved average utility per power, i.e., power unit based utility. We solve the subcarrier allocation issue naturally as well as the power control. Each user tries to maximize its utility by adjusting the transmit power on each subcarrier. A Nash equilibrium for the game is shown to exist and the numerical results show that our proposal outperforms the pure water-filling algorithm in terms of efficiency and fairness. key words: multi-cell OFDM, power control, noncooperative game, power unit based utility
1.
Introduction
With the development of the micro-cellular systems, the interference caused by reusing the same frequency in different cells can not be ignored anymore. On the other hand, in some systems where the cellular is not well organized, such as wireless LAN and ad hoc, the interference may degrade the system performance greatly. There are much literature investigating the resource allocation in single cell orthogonal frequency division multiplexing (OFDM) systems. However, not many papers study resource allocation in multi-cell OFDM systems. There is no co-channel interference in one OFDM cell because of the orthogonality of the subcarriers, while in multi-cell OFDM systems the subcarriers will be reused by users in different cells, which may cause interference to each other. Therefore, the resource allocation problem becomes more complicated since each user’s decision interacts each other. If the interference is fixed or ignored, the water-filling algorithm provides a good solution. When the subcarrier assignment is fixed, many efficient water-filling methods have been proposed to maximize the data rate under the power constraints [1], [2]. However, the water filling algorithm is not suitable for the multi-cell OFDM system because the interference level is variable. Hence, the conventional method can not solve the problem of the co-channel users’ interaction efficiently. Manuscript received August 29, 2005. Manuscript revised January 3, 2006. † The authors are with Tsinghua-Hitachi Ubiquitous IT Joint Lab, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. ∗ This paper has been partially supported by the National Nature Science Foundation of China (60272021). a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1093/ietcom/e89–b.6.1951
On the other hand, the mobile users in general do not have the knowledge of other users’ states and therefore can not cooperate each other in multi-cell systems. As a result, they can only act selfishly to maximize their own performance in a distributed way. Such fact motivates us to introduce the game theory. In [3], the authors proposed a noncooperative game theory approach to minimize the overall transmit power under each user’s power limitation and minimal rate constraints. However, in that model, the solution may not be feasible when the co-channel interference is so large that no power allocation can satisfy all the users’ rate requirements. In this paper, we propose a new framework for resource allocation in multi-cell OFDM systems. The approach proposed here relies on the power unit based utility in developing distributed subcarrier and power allocation algorithms, which is formulated as a non-cooperative game that attempts to adjust the power level of each user to maximize the user’s utility on all the subcarriers. After all users allocate the power on each subcarrier, then they allocate the bit according to the assigned power. 2.
System Model
We consider a multi-cell OFDM system where L subcarriers in each cell are reused among multiple cells. Since the same frequency bands are reused by multiple cells, users assigned with the same frequency interfere with each other. Assume the number of cells which reuse the same frequency is K and within a cell, users transmit with all the subcarriers and the multiple access scheme among them is either TDMA or FDMA. The ith user’s signal-to-interference ratio (SIR) at subcarrier l (l = 1, 2, · · · , L) can be expressed as [3] γil = ji
hlii pli hlji plj + N0
, (i = 1, 2, · · · , K),
(1)
where pli is the transmit power of the ith user at the lth subcarrier and hlji denotes channel gain from the jth user to the base station of the ith user at the lth subcarrier. N0 is the background noise for all the users and subcarriers. Given γil , when continuous rate adaptation is used, the achievable data rate per Hz of user i at the lth subcarrier can be expressed as [4] cli = log2 (1 + βγil ),
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
(2)
IEICE TRANS. COMMUN., VOL.E89–B, NO.6 JUNE 2006
1952
where β = −1.5/ ln(5BER) is called SIR gap, which indicates the gap of SIR that is needed to reach a certain capacity between practical implementations and informationtheoretical results. The total data rate of the ith user is then L given by cΣi = cli . l=1
Here, we use the power unit based utility to describe the degree of satisfaction according to current QoS provided by networks, which is related to the total utility that a user can achieve in the lifetime of its battery and can be expressed as L l Ui ci Ui (cΣi ) l l=1 Fi (p) = = , pi ≥ 0, (3) L pΣi l pi l=1
where p denote the power allocation vector of all users at all subcarriers and Ui (cΣi ) is the obtained satisfaction when the total data rate at all subcarriers of user i is cΣi . The form of Ui (·) should be properly selected in order to reflect the nature of satisfaction level. Since the multimedia service is widely used, we choose the utility function for multimedia service as an example. The form of utility function for multimedia applications is [5] Ui (x) =
Uimax . 1 + exp[σi (µi − x)]
(4)
In this function, µi is called shift parameter representing the required average data rate of the application; σi is called slope parameter, which is positive and determines whether the application has a hard QoS requirement or a soft QoS requirement; Uimax is the maximum of achievable utility. Figure 1 shows the total utility of two subcarriers belonged to one user. In this figure, the level of interference is kept constant, and the user’s power unit based utility is plotted versus the transmit power. The peak value can be achieved when the user reaches the equilibrium.
3.
Distributed Noncooperative Power Control
The power unit based utility level for the ith user is Fi (p). We use an alternative notation Fi (pi , p−i ) where p−i = (p1 , · · · , pi−1 , pi+1 , · · · , pk ). This notation emphasizes that the ith user has control over its own power, pi only. It is important to note that even though the problem is formulated as a power control problem, it covers the subcarrier allocation problem as well. When the assigned power on the subcarrier is zero, the subcarrier is not used and vice versa. Therefore, the subcarrier allocation problem can be solved naturally. Formally, the noncooperative power control game (NPG) is expressed as (NPG) max Fi (pi , p−i ), (i = 1, 2, . . . , K).
(5)
In the NPG, each user optimizes its own power unit based utility depending on the transmit power of all the other users in the system. It is necessary to characterize a set of power where given the power levels of other users, no user can improve its utility level by making individual changes in its power. Such an operating point is called a Nash equilibrium. Now we analyze the equilibrium of the game. Mathematically, we can represent the necessary condition for equilibrium as ∂Fi = 0, (l = 1, 2, . . . , L). (6) ∂pli By mathematical manipulation, it reduces to the following equation Ui (cΣi )
βγil pΣi 1 = Ui (cΣi ), (i = 1, 2, . . . , K), (7) ln 2 1 + βγil pli
where Ui (cΣi ) = dU i (cΣi )/dcΣi . Equation (7) is the equilibrium equation of the game model (5). The solution of the equation, represented by p∗ , is the Nash equilibrium of the noncooperative game. We can prove the existence of the equilibrium by using Debreu’s theorem since the utility function is quasi-concave [7]. Now we investigate the property of the equilibrium. Theorem 1: When the NPG achieves Nash equilibrium, the optimal power allocation, pl∗ i , satisfies 1 Iil (p) = Ωi , (i = 1, 2, . . . , K), (8) β hlii l l∗ where Iil (p) = h ji p j + N0 denote the interference repl∗ i +
ji
Fig. 1 Power unit based utility function of a multimedia user for fixed interference.
ceived by user i at the lth subcarrier including the background noise. The proof of the above theorem can be deduced from the equilibrium Eq. (7) easily. Ωi is a constant for user i. The theorem can be regarded as a kind of water-filling algorithm for one user’s all subcarriers, where Ωi can be regarded as the water level. However, the water level Ωi differs for different users. Therefore, the game-based multi-level waterfilling in (8) can be regarded as an extension of the pure water-filling.
LETTER
1953
4.
Numerical Results
To show the performance of the NPG, we set up a sevencell simulation scenario. Around a central cell, we consider six hexagonal cells with the distance of 2 km, which are far away from the central cell because the frequency is not reused in neighbor cells in practice. The base station is located at the center of each cell and one co-channel mobile per cell is generated as a uniform distribution within the corresponding cell for each simulation instance, such as Fig. 2. The propagation model takes into consideration of path loss and frequency selective fading. The path loss is obtained using the simple model gi = 0.097/di4 where di is the distance from the ith user to the corresponding base station. The five-path Rayleigh model is used to the frequency selective fading with an exponential power profile. We consider the OFDM system with 8 subcarriers in total. The background noise N0 is 2 × 10−15 Watts. In such a scenario, the NPG converges to the Nash equilibrium after 2-3 iterations. Figure 3 shows the path gain in all the subcarriers of each user, which is combined effect of path loss and frequency selective fading. Generally speaking, the users with small path gain have to use larger transmit power to achieve the target SIR. Figures 4–6 compare the performance of the proposed NPG with the pure water-filling algorithm. The classic water-filling algorithm must have the total power constrains, but in the NPG, the power allocation is adaptive and we can control the total power in a reasonable range. Therefore, to ensure a fair comparison, the total transmit power limitation to each user is assumed to be 0.5 Watts, 1.0 Watts and 1.5 Watts respectively in the water-filling algorithm. Figure 4 shows that, although the proposed NPG has no power limitation, all the power in the Nash equilibrium is reasonable from the view of practice. From Fig. 4 and Fig. 5, we observe that although the total data rate of water-filling is higher than our proposal when the path gain is large, it is at the cost of much higher power consume, such as users
Fig. 2
The locations of terminals and base stations.
1-4. When the transmit power is almost the same, the NPG can achieve higher data rate than the water-filling, such as user7. On the other hand, it is important to note that in our NPG, the total data rates of all uses are almost the same while they are very different in the water-filling. That is to say, even though there are no data rate requirements, all of the users achieve almost the same data rate when the system is in equilibrium. Therefore, the proposal can guarantee the fairness among users automatically. It owes the characteristic of the proposed power unit based utility function. Figure 6 is the comparison of power unit based utility in the NPG and in the water-filling. It shows that the achieved utility gets larger as path gain increases. Therefore, we can conclude that the better channel condition, the less transmit power and the larger utility. We also observe that the power unit based utility of the NPG is much higher than that of the water-filling for all the users since the water-filling algorithm ignores the co-channel interference, which makes
Fig. 3
The path gain of different subcarriers for different users.
Fig. 4 Comparison of the total transmit power in the NPG and in the water-filling.
IEICE TRANS. COMMUN., VOL.E89–B, NO.6 JUNE 2006
1954
sidering the users’ achieved average utility per power, i.e., power unit based utility. The problem is formulated as a distributed noncooperative power control game (NPG). All users choose appropriate strategy to maximize their own utilities in a distributed fashion. The NPG converges to the Nash equilibrium after 2-3 iterations. After all users allocate the power on each subcarrier,they then allocate the bit according to the assigned power. The simulation results show that the proposal achieves much better utility than the pure water-filling algorithm since the water-filling makes some subcarriers over-crowed. On the other hand, the characteristic of the proposed power unit based utility guarantees the fairness among users in equilibrium. Therefore, all the users achieve almost the same total data rate in equilibrium. Fig. 5 filling.
Comparison of the total data rate in the NPG and in the water-
Acknowledgment The authors would like to express their sincere thanks to Hitachi R&D Headquarter for the continuous supports, and special thanks to Dr. Xiang Duan and other colleagues in Networking Theory Lab for their valuable comments and suggestions. References
Fig. 6 filling.
Comparison of the obtained utility in the NPG and in the water-
some subcarriers over-crowed. 5.
Conclusions
In this paper, we have presented a noncooperative game for power control and subcarrier assignment in multi-cell OFDM systems. A specific utility function is defined con-
[1] G. Munz, S. Pfletschinger, and J. Speidel, “An efficient waterfilling algorithm for multiple access OFDM,” Proc. IEEE Global Telecommunications Conf. (GLOBECOM 2002), vol.1, pp.681–685, Nov. 2002. [2] W. Yu, W. Rhee, S. Boyd, and J.M. Cioffi, “Iterative water-filling for Gaussian vector multiple-access channels,” IEEE Trans. Inf. Theory, vol.50, no.1, pp.145–152, Jan. 2004. [3] Z. Han, Z. Ji, and K.J.R. Liu, “Power minimization for multi-cell OFDM networks using distributed non-cooperative game approach,” Proc. IEEE Global Telecommunications Conf. (GLOBECOM 2004), vol.6, pp.3742–3747, Nov. 2004. [4] X. Qiu and K. Chawla, “On the performance of adaptive modulation in cellular systems,” IEEE Trans. Commun., vol.47, no.6, pp.884–895, June 1999. [5] M. Xiao, N.B. Shroff, and E.K.P. Chong, “Utility-based power control in cellular wireless systems,” Proc. IEEE Twentieth Annual Joint Conference of Computer and Communications Societies (INFOCOM 2001), vol.1, pp.412–421, April 2001. [6] R. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Sel. Areas Commun., vol.13, no.7, pp.1341–1347, Sept. 1995. [7] W.Y. Zhang, Game Theory and Information Economics, in Chinese, Shanghai People’s Publishing House, Shanghai, China, 1996.
1951
LETTER
Adaptive Power Control in Multi-Cell OFDM Systems: A Noncooperative Game with Power Unit Based Utility∗ Lan WANG†a) and Zhisheng NIU†b) , Members
SUMMARY In this paper, we develop a new distributed adaptive power control framework for multi-cell OFDM systems based on the game theory. A specific utility function is defined considering the users’ achieved average utility per power, i.e., power unit based utility. We solve the subcarrier allocation issue naturally as well as the power control. Each user tries to maximize its utility by adjusting the transmit power on each subcarrier. A Nash equilibrium for the game is shown to exist and the numerical results show that our proposal outperforms the pure water-filling algorithm in terms of efficiency and fairness. key words: multi-cell OFDM, power control, noncooperative game, power unit based utility
1.
Introduction
With the development of the micro-cellular systems, the interference caused by reusing the same frequency in different cells can not be ignored anymore. On the other hand, in some systems where the cellular is not well organized, such as wireless LAN and ad hoc, the interference may degrade the system performance greatly. There are much literature investigating the resource allocation in single cell orthogonal frequency division multiplexing (OFDM) systems. However, not many papers study resource allocation in multi-cell OFDM systems. There is no co-channel interference in one OFDM cell because of the orthogonality of the subcarriers, while in multi-cell OFDM systems the subcarriers will be reused by users in different cells, which may cause interference to each other. Therefore, the resource allocation problem becomes more complicated since each user’s decision interacts each other. If the interference is fixed or ignored, the water-filling algorithm provides a good solution. When the subcarrier assignment is fixed, many efficient water-filling methods have been proposed to maximize the data rate under the power constraints [1], [2]. However, the water filling algorithm is not suitable for the multi-cell OFDM system because the interference level is variable. Hence, the conventional method can not solve the problem of the co-channel users’ interaction efficiently. Manuscript received August 29, 2005. Manuscript revised January 3, 2006. † The authors are with Tsinghua-Hitachi Ubiquitous IT Joint Lab, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. ∗ This paper has been partially supported by the National Nature Science Foundation of China (60272021). a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1093/ietcom/e89–b.6.1951
On the other hand, the mobile users in general do not have the knowledge of other users’ states and therefore can not cooperate each other in multi-cell systems. As a result, they can only act selfishly to maximize their own performance in a distributed way. Such fact motivates us to introduce the game theory. In [3], the authors proposed a noncooperative game theory approach to minimize the overall transmit power under each user’s power limitation and minimal rate constraints. However, in that model, the solution may not be feasible when the co-channel interference is so large that no power allocation can satisfy all the users’ rate requirements. In this paper, we propose a new framework for resource allocation in multi-cell OFDM systems. The approach proposed here relies on the power unit based utility in developing distributed subcarrier and power allocation algorithms, which is formulated as a non-cooperative game that attempts to adjust the power level of each user to maximize the user’s utility on all the subcarriers. After all users allocate the power on each subcarrier, then they allocate the bit according to the assigned power. 2.
System Model
We consider a multi-cell OFDM system where L subcarriers in each cell are reused among multiple cells. Since the same frequency bands are reused by multiple cells, users assigned with the same frequency interfere with each other. Assume the number of cells which reuse the same frequency is K and within a cell, users transmit with all the subcarriers and the multiple access scheme among them is either TDMA or FDMA. The ith user’s signal-to-interference ratio (SIR) at subcarrier l (l = 1, 2, · · · , L) can be expressed as [3] γil = ji
hlii pli hlji plj + N0
, (i = 1, 2, · · · , K),
(1)
where pli is the transmit power of the ith user at the lth subcarrier and hlji denotes channel gain from the jth user to the base station of the ith user at the lth subcarrier. N0 is the background noise for all the users and subcarriers. Given γil , when continuous rate adaptation is used, the achievable data rate per Hz of user i at the lth subcarrier can be expressed as [4] cli = log2 (1 + βγil ),
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
(2)
IEICE TRANS. COMMUN., VOL.E89–B, NO.6 JUNE 2006
1952
where β = −1.5/ ln(5BER) is called SIR gap, which indicates the gap of SIR that is needed to reach a certain capacity between practical implementations and informationtheoretical results. The total data rate of the ith user is then L given by cΣi = cli . l=1
Here, we use the power unit based utility to describe the degree of satisfaction according to current QoS provided by networks, which is related to the total utility that a user can achieve in the lifetime of its battery and can be expressed as L l Ui ci Ui (cΣi ) l l=1 Fi (p) = = , pi ≥ 0, (3) L pΣi l pi l=1
where p denote the power allocation vector of all users at all subcarriers and Ui (cΣi ) is the obtained satisfaction when the total data rate at all subcarriers of user i is cΣi . The form of Ui (·) should be properly selected in order to reflect the nature of satisfaction level. Since the multimedia service is widely used, we choose the utility function for multimedia service as an example. The form of utility function for multimedia applications is [5] Ui (x) =
Uimax . 1 + exp[σi (µi − x)]
(4)
In this function, µi is called shift parameter representing the required average data rate of the application; σi is called slope parameter, which is positive and determines whether the application has a hard QoS requirement or a soft QoS requirement; Uimax is the maximum of achievable utility. Figure 1 shows the total utility of two subcarriers belonged to one user. In this figure, the level of interference is kept constant, and the user’s power unit based utility is plotted versus the transmit power. The peak value can be achieved when the user reaches the equilibrium.
3.
Distributed Noncooperative Power Control
The power unit based utility level for the ith user is Fi (p). We use an alternative notation Fi (pi , p−i ) where p−i = (p1 , · · · , pi−1 , pi+1 , · · · , pk ). This notation emphasizes that the ith user has control over its own power, pi only. It is important to note that even though the problem is formulated as a power control problem, it covers the subcarrier allocation problem as well. When the assigned power on the subcarrier is zero, the subcarrier is not used and vice versa. Therefore, the subcarrier allocation problem can be solved naturally. Formally, the noncooperative power control game (NPG) is expressed as (NPG) max Fi (pi , p−i ), (i = 1, 2, . . . , K).
(5)
In the NPG, each user optimizes its own power unit based utility depending on the transmit power of all the other users in the system. It is necessary to characterize a set of power where given the power levels of other users, no user can improve its utility level by making individual changes in its power. Such an operating point is called a Nash equilibrium. Now we analyze the equilibrium of the game. Mathematically, we can represent the necessary condition for equilibrium as ∂Fi = 0, (l = 1, 2, . . . , L). (6) ∂pli By mathematical manipulation, it reduces to the following equation Ui (cΣi )
βγil pΣi 1 = Ui (cΣi ), (i = 1, 2, . . . , K), (7) ln 2 1 + βγil pli
where Ui (cΣi ) = dU i (cΣi )/dcΣi . Equation (7) is the equilibrium equation of the game model (5). The solution of the equation, represented by p∗ , is the Nash equilibrium of the noncooperative game. We can prove the existence of the equilibrium by using Debreu’s theorem since the utility function is quasi-concave [7]. Now we investigate the property of the equilibrium. Theorem 1: When the NPG achieves Nash equilibrium, the optimal power allocation, pl∗ i , satisfies 1 Iil (p) = Ωi , (i = 1, 2, . . . , K), (8) β hlii l l∗ where Iil (p) = h ji p j + N0 denote the interference repl∗ i +
ji
Fig. 1 Power unit based utility function of a multimedia user for fixed interference.
ceived by user i at the lth subcarrier including the background noise. The proof of the above theorem can be deduced from the equilibrium Eq. (7) easily. Ωi is a constant for user i. The theorem can be regarded as a kind of water-filling algorithm for one user’s all subcarriers, where Ωi can be regarded as the water level. However, the water level Ωi differs for different users. Therefore, the game-based multi-level waterfilling in (8) can be regarded as an extension of the pure water-filling.
LETTER
1953
4.
Numerical Results
To show the performance of the NPG, we set up a sevencell simulation scenario. Around a central cell, we consider six hexagonal cells with the distance of 2 km, which are far away from the central cell because the frequency is not reused in neighbor cells in practice. The base station is located at the center of each cell and one co-channel mobile per cell is generated as a uniform distribution within the corresponding cell for each simulation instance, such as Fig. 2. The propagation model takes into consideration of path loss and frequency selective fading. The path loss is obtained using the simple model gi = 0.097/di4 where di is the distance from the ith user to the corresponding base station. The five-path Rayleigh model is used to the frequency selective fading with an exponential power profile. We consider the OFDM system with 8 subcarriers in total. The background noise N0 is 2 × 10−15 Watts. In such a scenario, the NPG converges to the Nash equilibrium after 2-3 iterations. Figure 3 shows the path gain in all the subcarriers of each user, which is combined effect of path loss and frequency selective fading. Generally speaking, the users with small path gain have to use larger transmit power to achieve the target SIR. Figures 4–6 compare the performance of the proposed NPG with the pure water-filling algorithm. The classic water-filling algorithm must have the total power constrains, but in the NPG, the power allocation is adaptive and we can control the total power in a reasonable range. Therefore, to ensure a fair comparison, the total transmit power limitation to each user is assumed to be 0.5 Watts, 1.0 Watts and 1.5 Watts respectively in the water-filling algorithm. Figure 4 shows that, although the proposed NPG has no power limitation, all the power in the Nash equilibrium is reasonable from the view of practice. From Fig. 4 and Fig. 5, we observe that although the total data rate of water-filling is higher than our proposal when the path gain is large, it is at the cost of much higher power consume, such as users
Fig. 2
The locations of terminals and base stations.
1-4. When the transmit power is almost the same, the NPG can achieve higher data rate than the water-filling, such as user7. On the other hand, it is important to note that in our NPG, the total data rates of all uses are almost the same while they are very different in the water-filling. That is to say, even though there are no data rate requirements, all of the users achieve almost the same data rate when the system is in equilibrium. Therefore, the proposal can guarantee the fairness among users automatically. It owes the characteristic of the proposed power unit based utility function. Figure 6 is the comparison of power unit based utility in the NPG and in the water-filling. It shows that the achieved utility gets larger as path gain increases. Therefore, we can conclude that the better channel condition, the less transmit power and the larger utility. We also observe that the power unit based utility of the NPG is much higher than that of the water-filling for all the users since the water-filling algorithm ignores the co-channel interference, which makes
Fig. 3
The path gain of different subcarriers for different users.
Fig. 4 Comparison of the total transmit power in the NPG and in the water-filling.
IEICE TRANS. COMMUN., VOL.E89–B, NO.6 JUNE 2006
1954
sidering the users’ achieved average utility per power, i.e., power unit based utility. The problem is formulated as a distributed noncooperative power control game (NPG). All users choose appropriate strategy to maximize their own utilities in a distributed fashion. The NPG converges to the Nash equilibrium after 2-3 iterations. After all users allocate the power on each subcarrier,they then allocate the bit according to the assigned power. The simulation results show that the proposal achieves much better utility than the pure water-filling algorithm since the water-filling makes some subcarriers over-crowed. On the other hand, the characteristic of the proposed power unit based utility guarantees the fairness among users in equilibrium. Therefore, all the users achieve almost the same total data rate in equilibrium. Fig. 5 filling.
Comparison of the total data rate in the NPG and in the water-
Acknowledgment The authors would like to express their sincere thanks to Hitachi R&D Headquarter for the continuous supports, and special thanks to Dr. Xiang Duan and other colleagues in Networking Theory Lab for their valuable comments and suggestions. References
Fig. 6 filling.
Comparison of the obtained utility in the NPG and in the water-
some subcarriers over-crowed. 5.
Conclusions
In this paper, we have presented a noncooperative game for power control and subcarrier assignment in multi-cell OFDM systems. A specific utility function is defined con-
[1] G. Munz, S. Pfletschinger, and J. Speidel, “An efficient waterfilling algorithm for multiple access OFDM,” Proc. IEEE Global Telecommunications Conf. (GLOBECOM 2002), vol.1, pp.681–685, Nov. 2002. [2] W. Yu, W. Rhee, S. Boyd, and J.M. Cioffi, “Iterative water-filling for Gaussian vector multiple-access channels,” IEEE Trans. Inf. Theory, vol.50, no.1, pp.145–152, Jan. 2004. [3] Z. Han, Z. Ji, and K.J.R. Liu, “Power minimization for multi-cell OFDM networks using distributed non-cooperative game approach,” Proc. IEEE Global Telecommunications Conf. (GLOBECOM 2004), vol.6, pp.3742–3747, Nov. 2004. [4] X. Qiu and K. Chawla, “On the performance of adaptive modulation in cellular systems,” IEEE Trans. Commun., vol.47, no.6, pp.884–895, June 1999. [5] M. Xiao, N.B. Shroff, and E.K.P. Chong, “Utility-based power control in cellular wireless systems,” Proc. IEEE Twentieth Annual Joint Conference of Computer and Communications Societies (INFOCOM 2001), vol.1, pp.412–421, April 2001. [6] R. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Sel. Areas Commun., vol.13, no.7, pp.1341–1347, Sept. 1995. [7] W.Y. Zhang, Game Theory and Information Economics, in Chinese, Shanghai People’s Publishing House, Shanghai, China, 1996.
