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wangbenchao@huawei.com. Abstract—A cooperative bandwidth and power allocation algorithm is presented for multi-relay based networks. All the.
A Cooperative Bandwidth and Power Allocation Strategy Based on Game Theory in Multi-Relay Networks Fan Jiang

Benchao Wang

School of Communications and Information Engineering Xian University of Posts and Telecommunications, Xian 710121 E-mail: [email protected] Abstract—A cooperative bandwidth and power allocation algorithm is presented for multi-relay based networks. All the relays adopt the decode-and-forward protocol and assist the transmission from the source to destination in a non-cooperative game. By using a utility based mechanism, two types of resource allocation algorithms, namely bandwidth and power allocation strategy are proposed so as to maximize the source’s utility. Specifically, we first developed the optimal bandwidth allocation schemes under equal power allocation assumption among relays. Then, we considered the optimal power and bandwidth allocation for the multi-relay cooperative transmission subject to individual power constraints on the source. Simulation studies are conducted to evaluate the effectiveness of the proposed algorithms in terms of utility. The impact of the proposed strategy on power and bandwidth allocation is also discussed. Keywords- Multi-relay network, cooperative communication, bandwidth allocation, power allocation

I.

INTRODUCTION

Cooperative relaying has been regarded as an efficient technique to realize spatial diversity in wireless network. It is based on the broadcast nature of the wireless medium and allows terminals which are in the coverage area of a source to forward an “overheard” version of the transmitted signal. In the multi-relaying system, multiple relays form a virtual antenna array to forward the information cooperatively and therefore the source can achieve a realizable diversity gain [1-2]. This helps to boost the overall system performance by means of improving the spectral efficiency, extending the coverage area, and prolonging the network lifetime. To fully realize the benefits of cooperative diversity, efficient wireless resource allocation is critical. In particular, for multi-relay system architectures in which the chief concerns are bandwidth and energy, efficient bandwidth allocation and power allocation strategy is essentially important. Recently, several researches have been reported on bandwidth and power allocation in two-hop multi-relay systems [3-5]. In [3], a joint allocation of three resources: power, subcarriers and relay nodes in multi-relay assisted systems has been studied. Given the total network power constraint, the aim of the work focus on maximizes the system transmission rate. Authors in [4] propose a random based fair allocation in multi-relay cooperative relay networks, where the selection of sub-channel is based on uniform random distribution. With the assumption

Huawei Technologies Co., Ltd. Bantian, Longgang District shenzhen 518129 [email protected]

that relays are modeled as rational agents engaging in a noncooperative game, [5] dedicated to maximize individual transmission rate according to the proposed distributed power allocation. Notice that most of the aforementioned resource allocation schemes are designed to maximize the overall system throughput, whereas the utility that the source can achieve through cooperative multi-relay transmission is often omitted. In such a situation, a game theoretic approach can be used to model the network and to guide the interactions between rational decision-makers. In [6], we propose a user cooperation stimulating strategy to solve the following two questions: ‘‘whether to cooperate’’ and ‘‘how to cooperate’’. Based on the cooperative game theory, the aim of the study focus on providing an optimal system utility and provides fairness among users. However, in the multi-relay environment, where the chief concern is to maximize the source’s utility, the conclusions provided in [6] can not be applied. In this paper we consider a DF (decode-and-forward) based cooperative two-hop multi-relay system, where we optimally allocate the two types of resources: bandwidth and power based on non-cooperative game theory. The objective is to maximize the source’s utility. By assuming each relay will contribute a certain fraction of bandwidth for cooperative data transmission; we first investigate the optimal bandwidth allocation schemes under equal power allocation among relays. Then, we generalized the optimal power and bandwidth allocation for the multi-relay cooperative transmission subject to individual power constraints on the source. The analyzing results are demonstrated by computer simulations which show the advantages of the proposed scheme compared with the conventional schemes. The rest of this paper is organized as follows. Section II presents the system model, and defines the utility functions used in this paper. Section III proposes an optimal bandwidth and power allocation scheme based on non-cooperative game theory for maximizing the source’s revenue. Section IV presents simulation results to demonstrate the effectiveness of the proposed scheme. Conclusion is provided in Section V. II.

SYSTEM MODEL

We consider a multi-relay assisted cooperative system as shown in Fig.1, where the source node S communicates with the destination node D via the help of K relay nodes. Each

This work is supported by New Century Supporting Project, Ministry of Education under project number: NCET-08-0891, Natural Science Research Project of Education Department of Shaanxi Provincial Government (project number: 11JK1009)

978-1-4244-6252-0/11/$26.00 ©2011 IEEE

relay node Rk operates in a time-division half-duplex mode using the DF protocol. To represent a user’s payoff over a set of action profiles precisely, the term ‘‘utility’’ is exploited here according to the game theory, and pricing mechanism is also introduced.

R1

m1

m1

R2

m2

1−

S

m2

K

∑ mk

k =1

mk

D

mk

Rk

mK

mK

RK

results presented in [8], a user’s utility is measured in the physical unit of bits-per-joule and is defined as

T ( P) (1) bits / joule P In this function, utility U(P) is proportional to throughput T(P) and inversely proportional to power P. The utility is then interpreted as the number of information bits received per joule of energy consumed. Consequently, the utility functions adopted by the source should not only incorporate the parameters such as throughput and power but also embody source’s preferences over the bandwidth allocation on each relay. Combined with the pricing-based algorithm described in [9-10], the utility functions in this paper for the source U s ( Pmax ) is constructed as U ( P) =

K

U s ( Pmax ) = U S ( PS ) + ∑ U Sk ( PSk )

Figure 1. Multi-relay assisted cooperative system model

As shown in Figure 1, both source and the relays assume destination as the final destination, while destination charges each transmitting node the common unit price of λ. We suppose an interference free model where user transmissions are considered as orthogonal to each other. Assume that the system is based on frequency division multiple access and source is allocated a W hertz bandwidth for transmitting its own packets. As illustrated in Figure 1, considering the channel condition differences among relays, source will divide its data into K fractions for multi-relay transmission. If the source wants its potential relay Rk to cooperatively transmit mk (m•[0, 1]) as a fraction of its own data to the destination, relay Rk must be compensated via a unit reimbursement price of μ for forwarding. The remaining 1 −

K

∑m

k

fraction of the source’s

k =1

data will be transmitted directly to destination. In this model, the cooperative diversity gain of the source heavily depends on how much fraction data is allocated to each relay for cooperative transmission. Moreover, power allocation among relays also plays an important role in optimizing the performances of the source. From the aforementioned description, it can be inferred that the choice of data fraction mk and power allocation Pk with each relay will undoubtedly affect the utility that source can achieve. In order to model the complicated interaction among each node, we will first address this issue from the aspect of utility function. Followed by the well-designed utility functions, the remaining section presents a suitable solution for the frame work described above. III.

COOPERATIVE BANDWIDTH AND POWER ALLOCATION

A. Utility Functions To appropriately denote a user’s preferences over a set of action profiles, a good representative approximation is indispensable. Here, the concept of utility function is adopted. As stated in [7], cooperative diversity is a physical layer protocol that affects physical layer variables. In particular, the two variables of interest are throughput achieved and transmission power consumed. According to the research

(2)

k =1

Where U S ( PS ) denotes the utility of the source gained under K

the conditions of direct transmission, ∑ U Sk ( PSk ) represents the k =1

utility that the source can obtain with cooperative transmission through K relays. The corresponding expression is written as ⎛ ⎞ ⎜ ⎟ k 1 ⎛ ⎞ − λ ⎟ f (γ SD ) ⎜ 1 − ∑ m k ⎟ (3) U S ( PS ) = W ⎜ K K ⎜P − ⎟ k =1 ⎝ ⎠ ⎜ max ∑ PS ⎟ ⎝ ⎠ k =1 K

K

⎡⎛ 1

USk (PSk ) = W ∑ ⎢⎜ ∑ k =1 k =1 ⎢⎝ P ⎣

k S

⎤ ⎞ − λ − μ ⎟ mk f k (γ CT )⎥ ⎠ ⎦⎥

(4)

Equation (3) accounts for the satisfaction received by the source in direct transmitting data and the associated destination charges. Where formula (4) actually represents the corresponding payoff of the source obtained through cooperative transmission along with the respective pricing rewards. More specially, in the case of cooperation, the source’s utility is the satisfaction measure achieved where cooperation subtracts the total price paid to the destination and the relay. Here, variables Pmax and PSk represent the total transmission power of the source and that allocated to each relay, respectively. Moreover, it should be followed that 1/ PSk > λ + μ , which gives the constraint of the charge provided by the destination and the relay. mk stands for the data fraction allocated by the source to each relay for cooperative transmission. f(γ) which is also called the efficiency function, denotes the probability of correct reception of a frame [8]. This is given by

(5) f (γ ) = [1 − 2BER(γ )]M With frames of M bits, BER is the bit error rate and γ denotes the received signal-to-noise ratio (SNR). The BER, for non-coherent frequency shift keyed (FSK) transmission, can be expressed as

1 γ BER(γ ) = × exp(− ); 2 2

2

γ=

h P

σ2

(6)

Where P is the transmit power. N0W is the noise power, and 2 2 h is the channel path gain. According to [11], h is calculated as h 2 = 7.75 ×10−3 / d 3.6 , d being the distance between the transmitter and the receiver in meters. Back to (3), γSD stands for the SNR of the channel from the source to the destination. Here γCT represents the effective SNR which is achieved by the source through cooperative transmission. According to DF cooperative forwarding methods, the expression of γCT is given as follows

{

}

(7) γ CT = min γ SR , γ SD + γ RD Where γSR, γRD denote the SNR of the wireless channels from the source to the relay, the relay to the destination, respectively. Note that fk(γCT) then can be rewritten as M

⎛ ⎛ ⎧⎪ PSk hSR 2 ⎛ PSk hSD 2 PRk hRD 2 ⎞⎫⎪ ⎞ ⎞ ⎜ ⎜ + ⎟⎬ ⎟ ⎟ f k (γ CT ) = 1 − exp min ⎨− ,−⎜ 2 ⎜ 2σ 2 ⎟⎪ ⎟ ⎟ ⎜ ⎜ 2 σ 2σ 2 ⎪ ⎝ ⎠⎭ ⎠ ⎠ ⎩ ⎝ ⎝ (8) Where PRk is the transmission power of relay Rk, and the following constraints hold ⎧K k ⎪∑ PS ≤ Pmax ⎨ k =1 ⎪0 ≤ m ≤ 1 k ⎩

(9)

as well as power allocation PSk in each relay, so as to maximize the source’s utility. This is formulated as ⎧

K





k =1



k k max ⎨U S ( PS ) + ∑ U S ( PS ) ⎬ s .t .(9 )

(10)

From the conclusions given in [6], we can infer that as long as cooperative transmission can brings more benefit, the optimal data allocation is to allow the relay to cooperatively transmit all the data that originates from source to the destination. In particular, this could be explained as if cooperative transmission method is adopted, the source should let the potential relays to transmit all the data. Then, the optimization problem becomes ⎧

mk =

m k f k (γ CT ) ⎫ ⎬ PSk ⎩ k =1 ⎭ K

max ⎨∑ s .t .(9 )

(11)

As can be observed, the utility that source can obtain is proportion to mk . To obtain the optimal solution mk , PSk ,

{

}

we propose the following two algorithms. a) Equal Power Allocation based Optimal Bandwidth Allocation Assuming source will equally allocated transmitting power into each relay. According to non-cooperative game theory, the optimal data fraction allocation mk on each relay should maximize source’s utility, which also provides a Nash equilibrium point. Note that with equal power allocated on

f k (γ CT ) K

∑f

n

(12)

(γ CT )

n =1

Where PSk = PS / K . Formula (12) is interpreted as if a relay can bring larger utility for the source through cooperative transmission, this relay should be allocated more fractions of data for forwarding. By adopting the optimal value of mk , the source will achieve the maximized payoff. In other words, this condition strikes a balance among the participants, in that it is impossible to make any individual improvement, which gives the definition of Nash equilibrium point. b) Bandwidth and Power Allocation To deal with the maximization problem described in expression (11) and note that variable fk (γ CT ) is also a function of PSk , we should first obtain the optimal value of

PSk .Combined with the derivative of objective function (11), it can be formulated as

∂ (U Sk ( PSk ) ) ∂PSk

B. Cooperative Game Solutions In this subsection, with the help of non-cooperative game theory, we will discuss the optimal bandwidth allocation mk

W

each relay, to maximize the object formula described in (11), the optimal value of mk is given as

2

⎛ 1 ⎞ ⎛ 1 ⎞ = − ⎜ k ⎟ mk f k (γ CT ) + ⎜ k − λ − μ ⎟ mk Mf k (γ CT ) M −1 ⋅ P P ⎝ S ⎠ ⎝ S ⎠

2 2 2 k ⎛ ⎛ hSR 2 hSD 2 ⎞⎪⎫ Pk h Pk h ⎪⎫ ⎞⎧⎪ ⎪⎧ P h ⎟ exp ⎜ min ⎨− S SR2 , − S SD2 − R RD min ⎜ 2, ⎟ ⎬ ⎨ 2 ⎬ ⎜ ⎟ ⎜ 2σ 2σ 2 ⎭⎪ ⎟ ⎪ ⎩⎪ 2σ ⎝ 2σ 2σ ⎠⎪⎭ ⎝ ⎠⎩ (13)

Obtaining the optimal solution of PSk requires solving a mixed integer programming problem showed in (13). The power allocation results will become significantly complicated when M is large, which is a non-convex problem. On the other hand, as can be observed, the source’s utility obtained through cooperative transmission is direct proportion to f k (γ CT ) , and

inversely proportion to PSk . Meanwhile, the value of f k (γ CT ) is also affected by PSk . In order to maximize the source’s utility and according to the “water filling rule” [12], we propose the following suboptimal power allocation results: k S

P =

k γ CT K

∑γ

m CT

Pmax

(14)

m =1

This formula can be explained as: if a certain relay can provide higher effective SNR compared with other potential relays, this relay should be allocated much more power so as to maximize source’s payoff through cooperative transmission. Consequently, with reference to objective function (11), the bandwidth allocation on each relay is formulated as

mk =

f k (γ ct ) / PSk K

∑f n =1

n

(15)

(γ ct ) / P

n S

Compared with the results given in previous subsection, we can observe that the results obtained in expression (14) and (15) take the channel condition discrepancy among each relay node into consideration. This is managed as relays with better channel condition will be allocated with more bandwidth and power for cooperative transmission, which results in a competition and negotiation among relay nodes. The interaction process will finally converge at a stable point which maximizes the source’s utility, and this convergent point can be proved is the Nash equilibrium point.

IV.

gradually approaches the X-axis, or more specifically, when d3=0, since R3’s channel condition is better compared with other two relays, more data originated from source will be allocated to it. This could be explained as with the adaptive data allocation scheme, the data allocation process will finally converge at the point when the source receives maximized diversity gain by cooperative transmission. This point is called the Nash equilibrium point.

SIMULATION RESULTS

In this section we present some simulation results to evaluate the performance of the proposed resource allocation algorithms. The simulation scenario we adopted is illustrated in Fig.2, which comprise of three relay nodes. The destination is located in the origin and the source is situated 800 meters far from the destination in the X-axis, the coordinates of the source being given as (800,0). R1 and R2 move along the X-axis toward the source and the destination, respectively, while R3 moves along the line parallel to Y-axis toward the destination. Then coordinates are (d1, 0), (d2, 0) and (400, d3), accordingly. The value of d1 varies from 400 meters to 1200 meters, and that of d2 just oppositely. The coordinate of d3 varies from 400 to -400. Other parameters used in the simulation include M=80, W=106 Hz, and N0W =5×10−15W. The transmitting power is assumed to be 0.1W for each relay nodes and 0.3W for the source node. The unit price of λ and μ charged by the destination and the relay is assumed to be 0.1 respectively.

Figure 3. Data proportion with equal power allocation

Figure 4. Source’s utility under four resource allocation scheme

Figure 4 displays source’s utility under four different allocation schemes. As can be observed, with the proposed optimal power and data allocation scheme, the payoff source can achieved through cooperative transmission is much greater compared with other three allocation scheme. This is the original framework we set forth, and one which underlies how the proposed scheme can enhance system performance. Note the curve of the proposed scheme overlap with the curve of Opt. power algorithm at certain part. Combine with the following figures, this phenomenon is explained as: at some region where power allocation effectively dominates the performance of the scheme, optimal bandwidth allocation could only bring little enhancement.

Figure 2. Simulation Model

Without loss of generality, we will compare the proposed allocation scheme with other three allocation method: ① equal bandwidth and power allocation algorithm, represented as “Equal”; ②equal power allocation with optimal bandwidth allocation algorithm, represented with “Opt power”; ③equal bandwidth allocation with optimal power allocation algorithm, denoted as “Opt data”; ④the proposed power and bandwidth allocation scheme, expressed as “Proposed”.

Figure 3 shows the data allocation with each relay node under equal power allocation assumption. In accordance with the expression given in (12), it can be observed that initially for the relay node with better channel condition, such as R1 and R3, more bandwidth will be allocated for cooperative transmission. Moreover, as all the relays are moving according to the track set by the experiment, data fraction allocation on each relay also varies accordingly. It also worth noting that as R3

Figure 5. Proposed power allocation

Figure 6 . Proposed data allocation

Figure 5 and 6 illustrate the changing value of power as well as bandwidth with each relay. When each relay starts to move towards the set direction, its channel conditions also varies. Consequently, by adopting the proposed scheme, the source will adjust power and bandwidth allocation on each relay accordingly, so as to maximize its payoff through cooperative transmission. Moreover, as can be observed, the variation of power allocation is in accordance with the

adaptation of the data allocation. This is in accordance with our previous analysis. More specifically, in the simulation scenario, when R1 and R3 are 800 meters away from the destination, which overlap with the source. At the exact time, R3 lies just in the middle between the source and the destination. Compared with other relay nodes, this location is translated as R3 can provide much more cooperative diversity gain. According to the proposed scheme, more data and power will be allocated to it subsequently, which leads to the maximized system performance. V.

CONCLUSION

This paper presents cooperative bandwidth and power allocation strategy in the context of a multi-relay network. The relays are modeled as rational agents engaging in a noncooperative game. Using a utility based mechanism, two types of resource allocation algorithms, namely bandwidth and power allocation strategy are proposed so as to maximize the source’s utility. First, we developed the optimal bandwidth allocation schemes under equal power allocation assumption among relays. Then, we considered the optimal power and bandwidth allocation for the multi-relay cooperative transmission subject to individual power constraints on the source. Finally, simulation results demonstrate the benefit of the proposed strategy compared with conventional schemes in terms of utility. REFERENCES [1]

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