A Game Approach for Cell Selection and Resource Allocation in Heterogeneous Wireless Networks Lin...

A Game Approach for Cell Selection and A Game Approach for Cell Selection and Resource Allocation in Heterogeneous Resource Allocation in Heterogeneous

Wireless NetworksWireless Networks

Lin Gao, Xinbing Wang

Institute of Wireless Communication Technology (IWCT)

Shanghai Jiao Tong University

A Game Approach for Cell Selection and Resource Allocation in Heterogeneous Wireless Networks 2

OutlineOutline IntroductionIntroduction

MotivationsMotivations Related worksRelated works ObjectivesObjectives

System Model and Problem FormulationSystem Model and Problem Formulation

Analysis of The Two-tier GameAnalysis of The Two-tier Game

Convergence Algorithm and SimulationConvergence Algorithm and Simulation

ConclusionsConclusions


MotivationMotivation

The appearance of heterogeneous OFDMA-based wireless access The appearance of heterogeneous OFDMA-based wireless access networks, such as WiMAX, LTE, Wi-Fi, etc.networks, such as WiMAX, LTE, Wi-Fi, etc.

Optimizing the cell selection and resource allocation (CS-RA) Optimizing the cell selection and resource allocation (CS-RA) processes is an important step towards maximizing the utilization of processes is an important step towards maximizing the utilization of current and future heterogeneous wireless networks.current and future heterogeneous wireless networks.

Distributed algorithm shows potential ability in CS-RA problem due to Distributed algorithm shows potential ability in CS-RA problem due to the lacking of global central node in wireless networks.the lacking of global central node in wireless networks.

An example of a An example of a cellular system with cellular system with 3 heterogeneous 3 heterogeneous base stations and 6 base stations and 6 mobile users.mobile users.


Related works Related works

The goal of cell selection (CS) procedures is to determine a base user to camp The goal of cell selection (CS) procedures is to determine a base user to camp on.on. In [5], Hanly et al. propose a cell selection algorithm to determine power allocation among In [5], Hanly et al. propose a cell selection algorithm to determine power allocation among

different users so as to satisfy per-user SINR constraints.different users so as to satisfy per-user SINR constraints. In [6], Wang et al. study an HSPA based handoff/cell-site selection technique to maximize In [6], Wang et al. study an HSPA based handoff/cell-site selection technique to maximize

the number of connected mobile stations, and propose a new scheduling algorithm to the number of connected mobile stations, and propose a new scheduling algorithm to achieve this objective.achieve this objective.

In [7], Mathar et al. provide an integrated design of optimal cell-site selection and frequency In [7], Mathar et al. provide an integrated design of optimal cell-site selection and frequency allocation, which maximizes the number of connected MSs and meanwhile maintains allocation, which maximizes the number of connected MSs and meanwhile maintains quasi-independence of radio based technology.quasi-independence of radio based technology.

In [8], Amzallag et al. formulate cell selection as an optimization problem called all-or-In [8], Amzallag et al. formulate cell selection as an optimization problem called all-or-nothing demand maximization, and propose two algorithms to achieve approximate optimal nothing demand maximization, and propose two algorithms to achieve approximate optimal solution.solution.


Related works Related works

The goal of resource allocation (RA) is to determine the radio resource The goal of resource allocation (RA) is to determine the radio resource (including time, frequency and power etc) in a particular cell to a mobile (including time, frequency and power etc) in a particular cell to a mobile user. user.

In OFDMA-based system: In OFDMA-based system: Subchannel-and-power allocation algorithms for multiuser OFDM have been Subchannel-and-power allocation algorithms for multiuser OFDM have been

investigated in [9]-[10] to maximize the overall data rate or minimize the total transmit investigated in [9]-[10] to maximize the overall data rate or minimize the total transmit power.power.

In [9], Wong et al. investigate margin-adaptive resource allocation problem and In [9], Wong et al. investigate margin-adaptive resource allocation problem and propose an iterative subcarrier and power allocation algorithm to minimize total propose an iterative subcarrier and power allocation algorithm to minimize total transmit power given fixed data rates and bit error rate (BER)..transmit power given fixed data rates and bit error rate (BER)..

In [10], Jang et al. investigate rate-adaptive problem and propose a mechanism to In [10], Jang et al. investigate rate-adaptive problem and propose a mechanism to maximize total data rate over all users subjected to power and BER constraints.maximize total data rate over all users subjected to power and BER constraints.


ObjectivesObjectives

In this paper, we formulate the CS-RA problem as a two-tier In this paper, we formulate the CS-RA problem as a two-tier game, namely inter-cell game and intra-cell game, game, namely inter-cell game and intra-cell game, respectively.respectively. In In inter-cell gameinter-cell game, mobile users select the best cell according to , mobile users select the best cell according to

optimal cell selection strategy derived from expected payoff.optimal cell selection strategy derived from expected payoff. In In intra-cell gameintra-cell game, mobile users choose the proper time-frequency , mobile users choose the proper time-frequency

resource in the serving cell to achieve maximum payoff.resource in the serving cell to achieve maximum payoff.

An illustration of the An illustration of the inter-cell game and inter-cell game and intra-cell game.intra-cell game.

Inter-cell GameInter-cell Game

Intra-cell Intra-cell GameGame



System Model and Problem FormulationSystem Model and Problem Formulation System ModelSystem Model Problem FormulationProblem Formulation





System Model System Model

A cellular system A cellular system G = (M, N)G = (M, N) with with is the set of is the set of base stations (BSs) and base stations (BSs) and is the set of mobile is the set of mobile users (MSs). Besides, users (MSs). Besides, is the set of sub- is the set of sub-channels in cell channels in cell jj. . 11

Each MS decides which cell it should camp on and which Each MS decides which cell it should camp on and which sub-channels (of the serving cell) it should occupy.sub-channels (of the serving cell) it should occupy.

11 Note : we assume that the cell number in each BS is 1 and thus the meaning of cell is Note : we assume that the cell number in each BS is 1 and thus the meaning of cell is equivalent to BS in this paper.equivalent to BS in this paper.

s2

Sub-channels

s2 s2 s2 s1 s1 s1 s1

8c

s1 s1 s4 s4

7c 6c 5c 4c 3c 2c 1c

a1 a2

s3

s2

s1s4

An example of the An example of the strategy of MS strategy of MS ss11 -- --

selecting cell selecting cell aa22 and and

sub-channels set sub-channels set {{cc1 1 ,c,c2 2 ,c,c3 3 ,c,c4 4 ,c,c7 7 ,c,c88}}..



We assume that each sub-channel in a cell can be used by We assume that each sub-channel in a cell can be used by multiple MSs multiple MSs without interferencewithout interference by means of orthogonal by means of orthogonal signals, e.g., separating the MSs in the time-dimension.signals, e.g., separating the MSs in the time-dimension.

We assume that the total available bandwidth on each We assume that the total available bandwidth on each sub-channel is sub-channel is equallyequally shared among the players selecting shared among the players selecting this sub-channel.this sub-channel.

The bandwidth of The bandwidth of cc88 is is

equally shared by equally shared by ss11 and and ss22. .

Accordingly, the power Accordingly, the power consumption by consumption by ss11 ( or ( or ss2 2 ) )

in in cc88 reduces to one half if reduces to one half if

ss11 and and ss22 use the time use the time

orthogonal signals.orthogonal signals.

s2

Sub-channels

s2 s2 s2 s1 s1 s1 s1

8c

s1 s1 s4 s4

7c 6c 5c 4c 3c 2c 1c

a1 a2

s3

s2

s1s4



s2

Sub-channels

s2 s2 s2 s1 s1 s1 s1

8c

s1 s1 s4 s4

7c 6c 5c 4c 3c 2c 1c

a1 a2

s3

s2

s1s4

Game ModelGame Model PlayersPlayers : the MSs set : the MSs set U U = {= {1,2,…,N1,2,…,N }.}.

StrategiesStrategies (or actions) of player (or actions) of player u u : (i) selecting a cell, i.e., , (ii) : (i) selecting a cell, i.e., , (ii) selecting a set of sub-channels in the serving cell, i.e., selecting a set of sub-channels in the serving cell, i.e., . .

PayoffPayoff function: defined in latter slide. function: defined in latter slide.

1 1 1

2 2 1

3

4 4 1

1 1 ,

2 1 ,

3 2

4 1 ,

: , = 1,1,1,1,0,0,1,1 ;

: , = 0,0,0,0,1,1,1,1 ;

: ;

: , = 0,0,1,1,0,0,0,0 ;

s s a

s s a

s

s s a

s x a y

s x a y

s x a

s x a y

1 2, , , ,= , ,...,

iu i u c u c u cy Y Y Y

ux


Problem Formulation Problem Formulation

Key NotationsKey Notations



The The exclusive-payoffexclusive-payoff of player of player uu in sub-channel in sub-channel cc of cell of cell ii is defined is defined as following:as following:

As As TTcc players occupying sub-channel players occupying sub-channel cc, the bandwidth of , the bandwidth of cc is is

equallyequally shared by shared by TTcc players. Accordingly, the power consumption players. Accordingly, the power consumption

of player of player uu reduce to reduce to PPu,cu,c//TTcc . Hence, the . Hence, the achievedachieved payoffpayoff of player of player uu

in sub-channel in sub-channel cc of cell of cell ii is :is :

,u c

c

U

T


The The overall-payoffoverall-payoff of player of player uu in cell in cell i i can be written as:can be written as:

wherewhere


, ,1 means player occupying and 0 otherwise.u c u cY u c Y



, ,

,

,

: Indication of the cell player selected.

: State of player in sub-channel , 1 means player

occupying and 0 otherwise.

: The power of player in sub-channel

u c k u c

u c

u c

i u

Y u c Y u

c Y

P u

.c

The The optimization problemoptimization problem for each player for each player uu ::

wherewhere




Analysis of The Two-tier GameAnalysis of The Two-tier Game Optimal Power Allocation Optimal Power Allocation Analysis of Intra-Cell GameAnalysis of Intra-Cell Game Analysis of Inter-Cell GameAnalysis of Inter-Cell Game





We decouple the optimization problem as three sub-We decouple the optimization problem as three sub-problems: power optimization problem, intra-cell game and problems: power optimization problem, intra-cell game and inter-cell game.inter-cell game.

Select cell 1

Player u

The intra-cell game in 1

… …

Find the optimal sub-channel state vector in cell 1

Select cell 2

Find the optimal sub-channel state vector in cell 2

Select cell M

Find the optimal sub-channel state vector in cell M

The inter-cell game

Calculate the optimal power allocation vector Pu,1 Pu,2 …

in all cells

The intra-cell game in 2 The intra-cell game in M

Optimal power allocation

Payoff: Uu,1 Payoff: Uu,2 Payoff: Uu,M

… …

… …


Optimal Power AllocationOptimal Power Allocation

Optimal Power AllocationOptimal Power Allocation


Optimal Power Allocation Optimal Power Allocation

s1a3

a2

a1

Channels Gain Channels Gain Channels Gain

Sub-channels Sub-channels Sub-channels

a1 a2 a3

Recall the equation of achieved payoff of player Recall the equation of achieved payoff of player uu in sub- in sub-channel channel cc ::

The information The information player player uu needed for needed for the calculation of the calculation of optimal power optimal power allocation. allocation.



Lemma 2Lemma 2 : The optimal power allocation for player : The optimal power allocation for player uu in sub- in sub-channel channel cc is: is:

Lagrange EquationLagrange Equation



The illustration of optimal power allocation and the optimal The illustration of optimal power allocation and the optimal absolute payoff in sub-channels. Note that the optimal absolute payoff in sub-channels. Note that the optimal power is power is independentindependent to to TTcc from Lemma 2. from Lemma 2.

c1 channels

Channels Gain

c2 c3 c4 c5 c6 c7 c8 c9

c1 channels

Optimal Power

c2 c3 c4 c5 c6 c7 c8 c9

Pt

0 c1 channels

Optimal Absolute Payoff

c2 c3 c4 c5 c6 c7 c8 c9

*,u cU

*,u cP

,u ch



c1 channels


c2 c3 c4 c5 c6 c7 c8 c9

*,u cU

c1 channels

Player Number

c2 c3 c4 c5 c6 c7 c8 c9

cT

1

23

c1 channels


c2 c3 c4 c5 c6 c7 c8 c9

Optimal Achieved Payoff*,u cU

The illustration of the optimal achieved payoff in sub-The illustration of the optimal achieved payoff in sub-channels. channels.

*,*

,

u c

u cc

UU

T


Intra-Cell GameIntra-Cell Game

Intra-Cell GameIntra-Cell Game


Analysis of Intra-Cell Game Analysis of Intra-Cell Game

The optimal sub-channels state vector for player The optimal sub-channels state vector for player uu, i.e., , i.e., , can be derived as following : , can be derived as following :*

,yu i

An example of the An example of the optimal sub-channels optimal sub-channels state vector for player u state vector for player u with with kku,i u,i =2.=2.

c1 channelsc2 c3 c4 c5 c6 c7 c8 c9

*,u cU

Player u

*,y (0,0,0,0,0,0,1,0,1)u i



Lemma 3Lemma 3 ::22 The best response function for player The best response function for player uu is: is:

wherewhere

,

1 2

: the required subchannel number of player in cell .

{ , ,..., }: permutation of sub-channels according to the

ascending order of .

: the number of players

i

u c

uc

uc

k k u i

T

T

excluding player in sub-channel .u c22 Note : we assume that the average channel gains of different sub-channels are Note : we assume that the average channel gains of different sub-channels are approximately the same. We use this assumption to facilitate the description of Nash approximately the same. We use this assumption to facilitate the description of Nash equilibrium state.equilibrium state.



The meaning of Lemma 3 is thatThe meaning of Lemma 3 is that each player will choose each player will choose

the sub-channels with the sub-channels with leastleast other players occupied.other players occupied.

An example of the An example of the optimal sub-channels optimal sub-channels state vector for player state vector for player uu with with kku,i u,i =2.=2.


1

23

ucT

Number of Players except u

Player u

*,( ) (0,0,0,0,1,0,1,0,0)u iy


Theorem 1Theorem 1 : A strategy profile: A strategy profile is a is a Nash equilibrium of the intra-cell game (in cell Nash equilibrium of the intra-cell game (in cell ii) iff the ) iff the following conditions hold:following conditions hold:


* * * *1, 2, ,{ , ,..., }i i i Ni iy y yY

An example of the Nash An example of the Nash Equilibrium in a cell with Equilibrium in a cell with 9 sub-channels.9 sub-channels.c1 channels

c2 c3 c4 c5 c6 c7 c8 c9

1

23

cT Number of Players

Nash Equilibrium



Property Property : all Nash equilibria sub-channel allocations : all Nash equilibria sub-channel allocations achieve achieve load balancingload balancing over the sub-channels in a cell. over the sub-channels in a cell.

The average of the maximal achieved payoff of player The average of the maximal achieved payoff of player uu in in cell cell ii : :


Inter-Cell GameInter-Cell Game

Inter-Cell GameInter-Cell Game


Analysis of Inter-Cell Game Analysis of Inter-Cell Game

The optimal cell for player The optimal cell for player uu, i.e., , can be derived as , i.e., , can be derived as following :following :

*i

*

,

* *, , ,* arg max ,

u i

u i u i u ii

U

i U P y



An example of the optimal cell for player An example of the optimal cell for player ii..


*,1uU

Player u

Optimal payoff in cell 1


Player u

Optimal payoff in cell 2

Cell Index (i)1 2

*,u iU

Player u



can be obtained only when the intra-cell game in cellcan be obtained only when the intra-cell game in cell ii achieves Nash achieves Nash equilibria, which implies that the player equilibria, which implies that the player uu has connected with cell has connected with cell ii. However, the . However, the serving cell serving cell ii is is indirectlyindirectly determined by . This leads to a determined by . This leads to a non-causalnon-causal problem. problem.

Thus we introduce the Thus we introduce the mixed strategymixed strategy of player of player uu as follows: as follows:

wherewhere

*,u iU

*,u iU

1: the probability of player selecting cell , 1.

Mi iu ui

p u i p



1 2 3 4 5 Cells Index (i)

*,u iU

player u

1up

2up

3up

4up

5up

Lemma 4Lemma 4 : The mixed-strategy matrix is a : The mixed-strategy matrix is a mixed-strategy Nash equilibrium if for each player mixed-strategy Nash equilibrium if for each player uu, the , the following conditions hold:following conditions hold:

wherewhere

An example of the An example of the mixed strategy for mixed strategy for player u.player u.


For simplicity, we consider an example with 2 BSs and 2 MSs. We assume that for For simplicity, we consider an example with 2 BSs and 2 MSs. We assume that for all players. The payoff of two players : all players. The payoff of two players :

wherewhere


strategy of player 2

stra

tegy

of

1 ,u i ik

*,, = : The maximum exclusive-payoff of player in cell u iu i iF U u i

MS2

BS2BS1

MS1

F2,2

F1,1

F2,1

F1,2



We denote the mixed strategy of players by The expected We denote the mixed strategy of players by The expected payoff of player payoff of player 11 can be written as: can be written as:

The optimal The optimal pp11, , pp22 and the mixed-strategy Nash equilibria are shown as and the mixed-strategy Nash equilibria are shown as

following :following :



The optimal The optimal pp11, , pp2 2 , , pp33 and the mixed-strategy Nash and the mixed-strategy Nash

equilibria for the cellular system with 2 BSs and 3 MSs are equilibria for the cellular system with 2 BSs and 3 MSs are shown as following:shown as following:





Convergence Algorithm and SimulationConvergence Algorithm and Simulation Convergence algorithmConvergence algorithm Simulation Results Simulation Results



Convergence algorithm Convergence algorithm

To apply the algorithm in practical system, the following To apply the algorithm in practical system, the following three essential assumptions are necessary.three essential assumptions are necessary. First, we assume that each MS has ability to initiate inter-frequency First, we assume that each MS has ability to initiate inter-frequency

measurement, from which each MS can measure the average sub-measurement, from which each MS can measure the average sub-channels gain.channels gain.

Second, we assume that each cell periodically broadcasts the Second, we assume that each cell periodically broadcasts the number of MSs connecting with this cell.number of MSs connecting with this cell.

Third, we assume that each cell counts the load on each sub-Third, we assume that each cell counts the load on each sub-channel and multicast to all MSs connecting with him.channel and multicast to all MSs connecting with him.

CS-Algorithm CS-Algorithm : converge to the inter-cell game Nash : converge to the inter-cell game Nash equilibrium.equilibrium.

RA-Algorithm RA-Algorithm : converge to the intra-cell game Nash : converge to the intra-cell game Nash equilibrium.equilibrium.


CS-AlgorithmCS-Algorithm The basic idea of the CS-algorithm :The basic idea of the CS-algorithm :

Problem 1Problem 1: mixed strategy of one player can : mixed strategy of one player can notnot be observed by other players, which makes the calculation of expected be observed by other players, which makes the calculation of expected payoff impractical.payoff impractical.

Problem 2Problem 2: the mixed strategy : the mixed strategy zzuu will will degeneratedegenerate to the pure strategy due to the to the pure strategy due to the non-smoothnon-smooth characteristic of best response characteristic of best response

functions.functions.




Solution of problem 1Solution of problem 1 Lemma 5Lemma 5 : The expected payoff of player : The expected payoff of player uu is equivalent to is equivalent to Qu Qu defining as defining as

follows:follows:

wherewhere

,

*,

: The probability of players (excluding player itself) in cell .

( 1) : The average of the maximal achieved payoff of player

in cell if there exists

i r

u i

r u i

U r u

i

E

other players in cell .r i


Solution of problem 2Solution of problem 2 To overcome the degeneration problem, we introduce the To overcome the degeneration problem, we introduce the

smoothedsmoothed best response functions : best response functions :


1 2( , ,..., )Mu u u uz p p p

The illustration of The illustration of standard best response standard best response and smoothed best and smoothed best response functions.response functions.1

2p

1

0

1p Standard Best Response

Smoothed Best Response with small

Smoothed Best Response with large

2p



The detail pseudo-code ofThe detail pseudo-code of CS-Algorithm CS-Algorithm



RA-AlgorithmRA-Algorithm

The basic idea of the RA-algorithm : The basic idea of the RA-algorithm : Greedy occupationGreedy occupation

Problem 1Problem 1: the unstable sub-channel allocations caused by simultaneously moving of : the unstable sub-channel allocations caused by simultaneously moving of different players,different players,

Solution of Problem 1Solution of Problem 1: the technique of : the technique of backoffbackoff mechanism. mechanism.



The detail pseudo-code ofThe detail pseudo-code of RA-Algorithm RA-Algorithm


Simulation results Simulation results

The cell selection interval: The cell selection interval: The minimal scheduling interval:The minimal scheduling interval: The number of cells: The number of cells: The number of MSs: The number of MSs: The number of sub-channels in each cell The number of sub-channels in each cell ii:: Channel model: free-space propagation Channel model: free-space propagation



Convergence of RA-algorithm (in a given cell)Convergence of RA-algorithm (in a given cell)

Variance Ratio always Variance Ratio always equals to 1 for any Nash equals to 1 for any Nash EquilibriumEquilibrium



Expectation of maximum acquired-subchannel-number of Expectation of maximum acquired-subchannel-number of players 1 to 15players 1 to 15

Red: estimation results Red: estimation results according to Eq. (27)according to Eq. (27)

Blue: simulation resultsBlue: simulation results



CLPDF learned by player 1CLPDF learned by player 1

Dash: analytical resultsDash: analytical results

Bar: learning resultsBar: learning results



Convergence of CS-algorithm Convergence of CS-algorithm



Impacts of Price and DCR on Mixed-strategy Nash Impacts of Price and DCR on Mixed-strategy Nash equilibrium in the inter-cell gameequilibrium in the inter-cell game

Fig. 2: Increasing price of BS2 Fig. 2: Increasing price of BS2 Reduce the load of BS 2Reduce the load of BS 2

Fig. 3: Increasing bandwidth of BS3 Fig. 3: Increasing bandwidth of BS3 Increase the load of BS 3 Increase the load of BS 3






ConclusionsConclusions ConclusionsConclusions


ConclusionsConclusions In this paper, We propose a distributed cell selection and resource In this paper, We propose a distributed cell selection and resource

allocation mechanism, in which the CS-RA processes are performed allocation mechanism, in which the CS-RA processes are performed by the mobile user independently.by the mobile user independently.

We formulate the problem as a two-tier game named as inter-cell We formulate the problem as a two-tier game named as inter-cell game and intra-cell game, respectively. We study the two-tier game game and intra-cell game, respectively. We study the two-tier game in details and analyze the existence and property of the Nash in details and analyze the existence and property of the Nash equilibria of the proposed games.equilibria of the proposed games.

We analyze the structure of Nash equilibria and find some We analyze the structure of Nash equilibria and find some interesting properties: interesting properties: load balance, load regulating, connecting load balance, load regulating, connecting directingdirecting etc. etc.

Thank you !Thank you !


ReferenceReference [5] S. V. Hanly, “An algorithm for combined cell-site selection and power control to maximize [5] S. V. Hanly, “An algorithm for combined cell-site selection and power control to maximize

cellular spread spectrum capacity,” IEEE J. Select. Areas Commun., vol. 13, no. 7, pp. 1332-1340, cellular spread spectrum capacity,” IEEE J. Select. Areas Commun., vol. 13, no. 7, pp. 1332-1340, 1995.1995.

[6] A. Sang and X. Wang, “Coordinated load balancing, handoff/cell-site selection, and scheduling [6] A. Sang and X. Wang, “Coordinated load balancing, handoff/cell-site selection, and scheduling in multi-cell packet data systems,” in Proc. ACM MOBICOM ’04, pp. 302-314, 2004.in multi-cell packet data systems,” in Proc. ACM MOBICOM ’04, pp. 302-314, 2004.

[7] R. Mathar and M. Schmeink, “Integrated optimzal cell site selection and frequency allocation for [7] R. Mathar and M. Schmeink, “Integrated optimzal cell site selection and frequency allocation for cellular radio networks,” Telecommunication Systems, vol. 21, pp. 339-347, 2002.cellular radio networks,” Telecommunication Systems, vol. 21, pp. 339-347, 2002.

[8] D. Amzallag, Reuven Bar-Yehuda, Danny Raz and Gabriel Scalosub, “Cell Selection in 4G [8] D. Amzallag, Reuven Bar-Yehuda, Danny Raz and Gabriel Scalosub, “Cell Selection in 4G Cellular Networks,” In Proc. IEEE INFOCOM ’08, pp. 700-708, April 2008.Cellular Networks,” In Proc. IEEE INFOCOM ’08, pp. 700-708, April 2008.

[9] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive [9] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747-1758, Oct. subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747-1758, Oct. 1999.1999.

[10] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. [10] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171-178, 2003..Areas Commun., vol. 21, no. 2, pp. 171-178, 2003..

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