Energy-Efficient Resource Allocation in Full-Duplex ... - IEEE Xplore

IEEE ICC 2014 - Mobile and Wireless Networking Symposium

Energy-Efficient Resource Allocation in Full-Duplex Relaying Networks ∗ Key

Gang Liu∗ , Hong Ji∗ , F. Richard Yu† , Yi Li∗ , Renchao Xie‡ Lab. of Universal Wireless Comm., Ministry of Education, Beijing University of Posts and Telecom., P.R. China † Depart. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada ‡ China Unicom, Beijing, P.R. China E-mail: {gangliu, jihong}@bupt.edu.cn; Richard [email protected]; {liyi, Renchao xie}@bupt.edu.cn;

Abstract—Recent advances of loop interference cancellation techniques enable full-duplex relaying (FDR) systems, which transmit and receive simultaneously in the same band with high spectrum efficiency. Unlike the existing works, in this paper, we study the energy efficiency aspect of resource allocation in FDR systems. We consider a OFDMA cellular network, where a shared FDR is deployed at the intersection of three sectors in a cell. Firstly, the problem of energy-efficient joint bandwidth sharing and power allocation is formulated as a three-stage Stackelberg game. Secondly, the subgame perfect equilibrium for each stage is analyzed. Then, the interplays of the three-stage game are discussed and an iterative algorithm is proposed to obtain the Stackelberg equilibrium solution. At last, simulation results are presented to show the effectiveness of the proposed game. Index Terms—Energy efficiency; OFDMA; resource allocation; shared full-duplex relay; Stackelberg game.

systems might be more crucial due to the presence of loop interference mitigation, which may cause inefficient use of the transmission power. Resource allocation plays a very important role in energy efficiency, spectrum efficiency and quality of service (QoS) provisioning in traditional wireless networks [8, 9]. Unfortunately, the existing resource allocation algorithms may not be directly applied in FDR systems because of the different manner of resource usage and the presence of loop interference in FDR systems. The above observations motivate us to investigate the problem of energy-efficient resource allocation in FDR systems. The distinct features are listed as follows: •

I. I NTRODUCTION AND M OTIVATIONS Full-duplex relaying (FDR) systems are able to improve the spectrum efficiency significantly by transmitting and receiving simultaneously in the same band [1]. Although FDR was considered impractical in the past due to inherent loop interference, recent studies [2] have shown the feasibility of FDR with the help of advanced interference cancellation techniques and transmit/receive antenna isolation. Hence, FDR has re-gained the attention from both industry and academia [3–6]. To tackle the loop interference problem in FDR systems, a broad range of mitigation schemes have been proposed, such as natural isolation, time-domain cancellation and spatial suppression [2]. It has been shown that loop interference can be mitigated sufficiently, and FDR can be a feasible alternative to traditional half-duplex relaying (HDR) in future cellular networks. To evaluate the performance of FDR systems, the end-to-end SINR and capacity [3], outage probability and diversity-multiplexing tradeoff [4] are analyzed. Besides, an optimal power allocation scheme is proposed in [5] to minimize the outage probability for FDR systems. The authors of [6] investigate the joint optimization problem of resource allocation and scheduling to maximize the weighted throughput in fullduplex MIMO-OFDMA relaying systems. Although some excellent works have been done for FDR systems, the energy efficiency aspect of FDR systems is largely ignored in the existing literature. However, energy efficiency is becoming an important design criterion in wireless communications because of the rapidly rising energy costs and increasingly rigid environmental standards [7]. Furthermore, comparing with conventional HDR systems, the energy efficiency in FDR

978-1-4799-2003-7/14/$31.00 ©2014 IEEE

•

•

Different from the existing works, the energy efficiency aspect of resource allocation in FDR systems is studied in this paper. The problem of joint power and subcarrier allocation is formulated to maximize the energy efficiency. A widely used metric bits/Joule [8, 9] is adopted to measure the performance of energy efficiency. We consider a shared FDR deployment scenario (shown in Fig. 1(a)), where a FDR relay is deployed at the intersection of three sectors in the cell. Comparing with separate FDR deployment (shown in Fig. 1(b)), shared FDR deployment can improve the overall network performance by maintaining connections to multiple base stations and serving multiple users at the same time [10, 11]. Consequently, the considered system becomes a multipoint-to-multipoint FDR system, where the multi-access interference, multiuser interference and loop interference co-exist at the same time. To the best of our knowledge, how to deal with these interferences together has not been well studied before. In this work, we propose a simple and practical transmission policy to tackle the involved interferences. The problem of energy-efficient resource allocation is modeled as a three-stage Stackelberg game. We analyze the subgame perfect equilibrium for each stage using a backward induction method and develop an iterative algorithm to obtain the Stackelberg equilibrium. Meanwhile, the existence and uniqueness of the equilibrium solution is proved by theoretical analysis and simulations.

The rest of this paper is organized as follows. In Section II and III, the system model and problem formulation are presented, respectively. The proposed game is analyzed in Section IV. Simulation results are presented in Section V. Finally, we conclude this study in Section VI with future work.

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II. S YSTEM D ESCRIPTION We consider a downlink cellular network with shared relay (shown in Fig. 1(a)), where each cell is divided into M = 3 sectors. Each sector is equipped with one single-antenna base station (BS). The shared relay station (RS) with M antennas is deployed at the center of the cell. Different from [10], decode and forward full-duplex relaying scheme [6] is adopted at RS and hence the RS is able to transmit and receive data simultaneously in the same band, which leads to different manner of resource allocation. Each user has only one antenna. Onehop users are directly served by the BS in their corresponding sector and two-hop users are served by the relay. Since the relay has M antennas, it can support at most M two-hop users simultaneously in the same band through spatial division. A fixed bandwidth B is shared by the whole cell and the dynamic resource partition among different sectors is coordinated by a radio resource manager. To facilitate the bandwidth allocation process, we divide the two-hop users with similar channel condition and rate requirement into several groups employing user paring algorithm in [12], and each group has at most M users. Hence, the two-hop users in the same group can share the same band. Assume that K1 one-hop users and K2 groups of two-hop users are scheduled and served in a certain time slot. Then, the total spectrum resource will be dynamically partitioned into K (K ≥ K1 + K2 ) bands with different bandwidth bj (each band comprises of one or multiple successive subcarriers) and adaptively allocated to users. Quasistatic flat-fading is considered and OFDMA multiple access scheme is used. A. Transmission Strategies for One-hop Users For one-hop user i(i = 1, 2, ..., K1 ), let gij be the channel gain in band j of user i from its corresponding BS, and pij is the power assigned to user i in band j. If band j is assigned to user i, then the band allocation indicator aij = 1; Otherwise, aij = 0. The resource allocation policy for one-hop user i will be determined by maximizing the energy efficiency, which can be expressed as follows K gij pij j=1 aij bj log2 (1 + δ 2 ) ηi = bits/Joule (1) K pc + j=1 pij where pc denotes the circuit power (including the energy consumption of mixers, filter, and digital-to-analog converters), which is independent of the data transmission power. Without loss of generality, we assume the noise variance δ 2 is the same for all the users. Note that the energy efficiency metric bits/Joule in (1) is a widely adopted in the existing literature on energyefficient communications [7, 13]. B. Transmission Strategies for Two-hop Users For two-hop users, the downlink transmission can be split into two parts: feeder link (from the BSs to the RS) and access link (from the RS to two-hop users). The direct links between BSs and two-hop users are ignored because of large path-loss. The signal model for feeder link is actually a multi-access channel. The received signal at the FDR involves both loop interference and multi-access interference. In order to mitigate the

Fig. 1. Two full-duplex relaying deployment scenarios: (a) Shared full-duplex relaying; (b) Separate full-duplex relaying.

loop interference, a time-domain loop interference cancellation technology in [2] is employed at the relay receiver. Moreover, to eliminate the multi-access interference, we can apply the multiuser detection technologies. However, for the consideration of complexity, we adopt a MMSE receiver [6] to separate signals from different BSs, which can also further suppress the residual loop interference. To facilitate the transmission in feeder link, base station coordination [14] can be applied to make distributed BSs act as a single M -antenna transmitter. In this way, the feeder link could be regarded as a virtual point-to-point MIMO channel, and classical MIMO technologies [15] can be applied to parallelize the feeder link. Assume two-hop users in group i are served in band j. Let Hij ∈ CM ×M be the channel matrix from the virtual BS antenna array to the RS in band j, and its singular value M ×M decomposition is Hij = Uij Σij VH is a ij , where Σij ∈ C m diagonal matrix consists of M singular values {σij } of Hij . According to [15], the virtual point-to-point MIMO channel from BSs to the RS can be decomposed as M independent parallel subchannels. Similar to (1), the energy efficiency of the feeder link for two-hop users in group i can be denoted as K ζi =

j=1

aij

M

pc +

m=1 bj log2 (1 + K M m j=1 m=1 pij

m m σij pij δr2 )

(2)

where aij is the band allocation indicator, pm ij is the transmit power assigned to subchannel m and δr2 the equivalent variance of the total of noise and residual loop interference In fact, the access link is a typical broadcast channel. To orthogonalize the data stream for different two-hop users in the same group, we adopt the well-known zero-forcing beamforming [16] in the access link. In this way, the multiuser interference between two-hop users in a same group can be totally eliminated. Let Gij ∈ CM ×M be the equivalent channel matrix from the RS to the two-hop users in group i in band j. Then, according to [16], the transmission rate of access link for two-hop users in group mi mcan be expressed K M γij qij m as Ria = j=1 aij m=1 bj log2 (1 + δ 2 ), where γij = 1 m and qij are the effective channel gain and −1 ] [(Gij GH m,m ij ) transmit power for two-hop user m in group i in band j. [A]m,m

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denotes the mth element in the main diagonal of matrix A. Similarly, the energy efficiency of the access link for two-hop users in group i is expressed as K ςi =

j=1

aij

M

pc +

m=1 bj log2 (1 + K M m j=1 m=1 qij

m m γij qij δ2 )

(3)

III. E NERGY- EFFICIENT R ESOURCE A LLOCATION P ROBLEM F ORMULATION In the resource allocation process, the radio resource manager first determines the price for each band according to the demand of the BSs. Secondly, the BSs compete for bands from the radio resource manager and then allocate them to one-hop users and the shared relay. At last, the shared RS decides the appropriate power level for two-hop users. Based on the observations above, the energy-efficient bandwidth sharing and power allocation problem can be formulated as a three-stage Stackelberg game model (as shown in Fig. 2). In our proposed game, the up-stage acts as leader, which makes decision first. The down-stage is the follower and moves subsequently on the basis of the leader’s strategy. In stage 1, the radio resource manager serves as leader and offers the price cj for each band bj according to the band demand of the BSs. In stage 2, the BSs, as the follower of stage 1, decide how many and which bands to buy from the radio resource manager based on the price of each band. And then, the BSs, as the leader of stage 2, sell the bands and power to one-hop users and the feeder link of two-hop users to gain their profit. In stage 3, the RS, as the follower of stage 2, performs resource allocation for the access link of two-hop users to gain its revenue. Since the decisions of leaders will affect the strategy of the followers, to formulate this three-stage Stackelberg game model, it is wise to adopt the backward induction method as follows. In stage 3, the RS performs optimal power allocation to maximize the energy efficiency of each access link for the twohop users organized in groups. For two-hop users in group i, the energy efficiency of their access link is given in (3). Then, we define the utility of the RS on the access link for the two-hop users in group i as URS,i = αi ςi − βi ςi

(4)

Fig. 2. Three-stage Stackelberg game modeling ({aij }: the set of band allocation indicator for the ith user; Φ2 : the set of index for two-hop users; )

of the BSs is defined as a quadratic utility function as follows UBSs =

K1 i=1

εi ηi +

K 1 +K2

(βi ςi + ρi ζi ) −

i=K1 +1

⎛ ⎞ K 1 ⎝ 2 − b + 2ν bi bj ⎠ 2 j=1 j

K

c j bj

j=1

(5)

i=j

where εi and ρi represent the revenue obtained from onehop user i and the feeder link of two-hop users in group i, respectively, cj is the price of band bj offered by the radio K1 +K2 βi ςi is the cost paid by the RS to resource manager, i=K 1 +1 the BSs, and ν ∈ (−1, 1) is the band substitutability parameter. Note that the motivations to choose quadratic utility function in (5) are mainly as follows: 1) it is concave with respect to the band demand bj , and this makes it easy to obtain the maximum revenue of the BSs, 2) the linear band demand function can be readily obtained by differentiating the utility function (5). In stage 1, the goal of radio resource manager is to maximize its profit by offering the price cj according to the band demand bj of the BSs. Hence, the total revenue of the radio resource manager can be denoted by the following utility function URRM,j =

K

(cj bj − wj bj )

(6)

j=1

where wj is the cost for band bj at the radio resource manager. Usually, we have cj ≥ wj ; Otherwise, the resource manager will not sell the band bj to the BSs. IV. A NALYSIS OF THE P ROPOSED S TACKELBERG G AME

where αi and βi can be interpreted as the revenue and cost to transmit one bit of data, respectively. Moreover, βi is charged by the BSs due to the assigned bands. Then, the utility function is the profit of the RS gained from the data transmission using one Joule of energy for the access link of two-hop users in group i. Note that we assume αi > βi . Otherwise, the RS will not provide service for two-hop users in group i. In stage 2, the responsibility of BSs is to buy bands from the radio resource manager and allocate the bands and power to one-hop users and the feeder link of two-hop users. The aim of the BSs is to maximize the energy efficiency expressed in (1) and (2) to gain their profit. Inspired by [8, 17], we assume the band demand bj satisfies linear structure. Then, the utility

In this part, the proposed game is analyzed in a backward induction manner. After each stage is solved in Subsection IV-A, IV-B and IV-C, the interplays between them and the Stackelberg equilibrium are discussed in Subsection IV-D. A. Stage 3: Energy-Efficient Power Allocation for RS By observing (1), (2) and (3), we realize that the energy efficiency expressions of one-hop and two-hop users take the similar form. Therefore, in this subsection, we will only give the power allocation policy for the access link of two-hop users. As for one-hop users and the feeder links of two-hop users, the power allocation policies can be obtained in a similar way. After investigating the property of the utility function of the RS, we

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have the following theorem and the proof is omitted due to space limitation. Theorem 1. For the access link of two-hop users in group i, the utility function URS,i in (4) is strictly quasiconcave with respect to the power allocation vector qi = 1 2 M 1 2 M T [qi1 , qi1 , ..., qi1 , ...qiK , qiK , ..., qiK ] . Furthermore, there exits a unique optimal power allocation vector q∗i , which satisfies

1) When Ria (pc + Pt (qi )) − Ria

(0)

|qi =qi

> 0, URS,i is first

m strictlyincreasing and then strictly decreasing in qij and ∂U hence ∂qRS,i = 0. m ∗ ij |qi =qi < 0, URS,i is 2) When Ria (pc + Pt (qi )) − Ria (0) |qi =qi

m stricltly decreasing in qij and hence q∗i = 0. a K M (0) ∂R m = where Ria = ∂qmi , Pt (qi ) = j=1 m=1 qij and qi ij

m−1 m+1 1 2 M 1 2 M T [qi1 , qi1 , ..., qi1 , ..., qij , 0, qij , ...qiK , qiK , ..., qiK ] .

According to Theorem 1, the optimal power allocation vector of access link of two-hop users in group i can be obtained by solving nonlinear equation. However, this suffers from high computational complexity. Fortunately, since the utility function (4) is strictly quasi-concave, the binary search assisted ascent (BSAA) algorithm [13] can be adopted to realize the energyefficient power allocation. B. Stage 2: Optimal Bandwidth Sharing for Users After the energy-efficient power allocation policy of RS is determined, the BSs will decide the optimal size of bands to buy from the radio resource manager and allocate the bands to different users in this part. Since the utility function of BSs in (5) is concave with respect to the band demand bj , the optimal band demand b∗j , which maximizes the utility, can be obtained by solving dUdbBSs = 0, i.e., j K1

aij εi ηij +

K 1 +K2

aij (βi ςij + ρi ζij ) − cj − bj − ν

i=K1 +1

i=1

M

m=1

p

log2 (1+ ijδ2 ij ) , pc + K j=1 pij log2 (1+

bi = 0

i=j g

where ηij =

m pm σij ij δ2

)

ςij =

M

pc +

m qm γij ij ) δ2 m q m=1 ij

log2 (1+ K M

m=1

j=1

and

ζij = p +K M rpm . After several manipulations, the c j=1 m=1 ij optimal band demand b∗j is given as follows K ν(Kκ − i=1 ci ) κ − cj ∗ − (7) bj = 1−ν (1 − ν)[1 + ν(K − 1)] K 1 K1 +K2 where κ = i=1 aij εi ηij + i=K a (βi ςij + ρi ζij ). Actu1 +1 ij ally, b∗j is the best reaction function [18] for the bandwidth price cj . As we assumed in (5), it is a linear function with respect to cj . In addition, we can also notice that b∗j is a function of the band allocation indicators aij and the power allocation policy in Subsection IV-A. Different band allocation policy leads to different band demand b∗j and different revenue UBSs of the BSs. Therefore, how to allocate the bands to different users is an important problem. For the purpose of maximizing the profit of BSs, we develop a revenue based band allocation Algorithm 1 to complete the band allocation. In each iteration of Algorithm 1, only one band will be assigned. εi ηi or βi ςi +ρi ζi is the revenue

Algorithm 1 Revenue based band allocation algorithm 1: Initialize the set of bands Ω = {1, 2, ..., K}, the set of users U = {1, 2, ..., K1 + K2 }, the set of rate requirement R = {R1 , R2 , ..., RK1 +K2 } and the band allocation indicator aij = 0 for ∀i ∈ U, ∀j ∈ Ω. Let j = 1. 2: for Ω = Φ do 3: Find l∗ = arg max∀i∈U {εi ηi , βi ςi + ρi ζi } then let al∗ j = 1 if the minimum rate requirement Rl∗ is satisfied, then let U = U \ {l∗ }. 4: j = j + 1 and Ω = Ω \ {j}. 5: end for 6: Output the band allocation indicators {aij }.

of the BSs, which can be obtained by assigning a band to onehop user i or two-hop user group i. Therefore, the intuitive explanation of algorithm 1 is that the BSs will allocate the band to the one with higher revenue to maximize their profit. C. Stage 1: Bandwidth Pricing In stage 1, to maximize its profit, the radio resource manager needs to determine the optimal price according to the band demand of BSs. After substituting the optimal band demand (7) into the utility function in (6), we have

K K ν(Kκ − i=1 ci ) κ − cj URRM = − (cj − wj ) 1−ν (1 − ν)[1 + ν(K − 1)] j=1 (8) In order to obtain the optimal price vector c, we investigate the property of the utility function URRM . Then, we have Property 1. The utility function in (8) is jointly concave with respect to the price vector c, when the band substitutability parameter ν satisfies one of the following conditions: 1 2 1) ν ∈ (−1, 3−K ) ( 1−K , 1), when K = 1, 3; 2) ν ∈ (−1, 1), when K = 1; 3) ν ∈ (− 12 , 1), when K = 3. In practical multi-user and multi-carrier systems, the number of bands K is always large enough (e.g., K ≥ 10). Therefore, according to Property 1, the utility function is usually concave for most ν. Hence, the optimal price c∗j can be obtained by RRM solving ∂U∂c = 0. Then, we have j 2ν −2cj + κ + wj + 1−ν

K − ν i=1 wj − νKκ =0 (1 − ν)[1 + ν(K − 1)]

K

i=1 cj

After several manipulations, we get a closed-form solution c∗j =

κ + wj 2

(9)

where κ is same as that in (7). We can realize that the optimal price is jointly determined by the power allocation policy in Stage 3, the band allocation policy in Stage 2 and the cost of each band at the radio resource manager.

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4:

cj

[n]

−cj 1

[n]

cj 1

> }

Output the energy-efficient power allocation strategy and band allocation scheme.

D. Interplays and The Stackelberg Equilibrium Up to now, each stage of the three-stage Stackelberg game has been solved. However, it is easy to realize that: 1) the energyefficient power allocation policy in stage 3 depends on the band allocation policy in stage 2, 2) the optimal band demand and band allocation policy in stage 2 are closely related to the power allocation policy in stage 3 and the optimal price in stage 1, 3) the optimal price for each band in stage 1 are jointly determined by the power allocation policy in stage 3 and the band allocation policy in stage 2. Hence, to get the Stackelberg equilibrium [18], we must take these interplays into account. The existence and uniqueness of the Stackelberg equilibrium is guaranteed by the following proposition.

V. S IMULATION R ESULTS AND D ISCUSSIONS Simulation results will be given in this part. The total bandwidth is 25.6MHz, which comprises of 256 subcarriers of 10kHz. Each band consists of several continuous subcarriers. The 3GPP path loss model is used to realize different channels

310

c1

Subgame perfect 300 equilibrium for stage 1

c

c3

290

c 280

2

8

4

10 12 14 The price of each band

The utility of base stations UBSs

320

240

220

200

16

2

b

3 4

0

5

10 15 Band demand

20

(d) The three−stage Stackelberg game 25

15 14.5 14

13

b

b

(c) The subgame of stage 3

13.5

b1

Subgame perfect equilibrium for stage 2

180

160

15.5

q1

Subgame perfect equilibrium for stage 3

q

2

c1 b

20

1

c2 15

b2 c

10

3

12.5

0 0.2 0.4 0.6 0.8 1 Allocated power on each subchannel(W)

3

b3 c

5

q

90 The average energy efficiency (kbits/Joule)

(b) The subgame of stage 2

330

Fig. 3.

Proposition 1. The optimal energy-efficient power allocation policy in Subsection IV-A, the optimal band demand b∗j in (7) and the optimal band price c∗j in (9) are the subgame perfect equilibrium in each stage, respectively. Based on Proposition 1, we know that every subgame perfect equilibrium is a Nash equilibrium [18]. Therefore, there exists a Stackelberg equilibrium for the three-stage Stackelberg game. Because each subgame perfect equilibrium solved above is unique, the three-stage Stackelberg equilibrium is also unique. Hence, the existence and uniqueness of the Stackelberg equilibrium are guaranteed. Based on the analysis above, we can develop the following iterative algorithm to obtain the Stackelberg equilibrium solution. In Algorithm 2, the BSs need to decide the optimal band demand and assign the bands to different users in each iteration. Also, the BSs and RS will perform the optimal power allocation. The algorithm will stop when the prices converge. Therefore, Algorithm 2 will converge to the Stackelberg equilibrium. This is because the optimal band price is a best reaction function of the optimal power allocation policy and the band allocation policy. Hence, when the prices converge, the power allocation policy and the band allocation policy will also achieve their optimum. Therefore, according to Proposition 1, the convergence of Algorithm 2 is guaranteed.

(a) The subgame of stage 1 340

Price or band demand

[n+1]

while{

Energy efficiency of the access link(kbits/Joule) The utility of radio resource manager URRM

Algorithm 2 Energy-efficient resource allocation algorithm based on Stackelberg game 1: Give a default band price cj at the radio resource manager; do 2: Perform the optimal power allocation using BSAA algorithm at BSs and RS; Allocate the bands to different users using Algorithm 1; Decide the optimal band demand based on (7) at BSs; [n+1] 3: Update the price cj using (9);

0

4

b4 0

5 10 Iteration times

15

The convergence of the three-stage Stackelberg game.

The proposed game WFPA with P =20dBm

80

total

APA with P

=20dBm

total

70

WFPA with P

60

APA with P

=25dBm

total

=25dBm

total

WFPA with P

=30dBm

total

50

APA with P

=30dBm

total

40 30 20 10 0

0

5

Fig. 4.

10 The signal to noise ratio(dB)

15

20

The comparison of energy efficiency.

[6]. The small scale fading from the RS to two-hop users and that from the BS to one-hop users are modeled as independent and identically distributed Rayleigh random variables. Since a strong line of sight is always existed between the BS and the RS in practice, the channel from BS to RS is modeled as Rician random variables with Rician factor κ = 6dB. The circuit power is pc = 100mW and the band substitutability parameter is ν = 0.2 in all the results. The convergence of the proposed game is first verified by Fig. 3. For ease of illustration, we consider a simple scenario with three one-hop users and one group of two-hop users. Each one-hop user or the group of two-hop users can only be allocated one band. Fig. 3(a), (b) and (c) show that there exists an unique subgame perfect equilibrium for each stage, which guarantees the existence and uniqueness of the proposed three-stage Stackelberg game. Fig. 3(d) illustrates the global convergence of the proposed game. We can find that the price

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1.4

1.2 The total transmission rate (Mbits/s)

tradeoff between energy efficiency and spectrum efficiency will be considered.

The proposed game WFPA with Ptotal=20dBm APA with P

ACKNOWLEDGMENT

=20dBm

total

WFPA with P

=25dBm

This work is jointly supported by the National 863 Project of China (No. 2012AA01A508), the National Natural Science Foundation (No. 61271182) and the National Natural Science Foundation for Young Scholar (No. 61302080).

total

1

APA with P

=25dBm

total

WFPA with P

=30dBm

total

0.8

APA with P

=30dBm

total

0.6

R EFERENCES

0.4

0.2

0

0

5

Fig. 5.

10 The signal to noise ratio(dB)

15

20

The comparison of transmission rate.

and the band demand converge soon at the same time and that higher demand will lead to higher price, which coincides with the linear structure assumption of the price and demand. In Fig. 4 and Fig. 5, we consider another common scenario with multiple one-hop users and multiple groups of two-hop users. We compare the performance of the proposed game with two classical algorithms: water-filling power allocation scheme (WFPA) and average power allocation scheme (APA). Both WFPA and APA scheme adopt the subcarrier allocation algorithm in [6] to maximize the total transmission rate. From Fig. 4, we can observe that the proposed game outperforms WFPA scheme and APA scheme in terms of energy efficiency. However, as shown in Fig. 5, the high performance in energy efficiency of the proposed game is obtained at the expense of total transmission rate. In addition, by increasing the total transmission power from 20dBm to 30dBm, the total transmission rate of WFPA and APA scheme increases, however, the energy efficiency of WFPA and APA schemes first increases and then decreases. The above observations reveal that their exists a tradeoff between energy efficiency and spectrum efficiency, as in other wireless networks [7]. VI. C ONCLUSIONS AND F UTURE W ORK We investigated the problem of energy-efficient joint bandwidth sharing and power allocation in OFDMA cellular network with shared FDR. Firstly, the problem of energy-efficient resource allocation was formulated as a three-stage Stackelberg game. In stage 1, the radio resource manager offers the price of each band to the BSs. In stage 2, the BSs buy bands from radio resource manager and allocate the bands to different users. In stage 3, the RS performs the energy-efficient power allocation for two-hop users. Then, we analyzed the subgame perfect equilibrium for each stage using backward induction method and developed an iterative algorithm to obtain the Stackelberg equilibrium. Besides, the existence and uniqueness of the Stackelberg equilibrium were proved through theoretical analysis and verified by simulations. Simulation results demonstrated that the excellent performance in terms of energy efficiency. The

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