In this paper, the optimal power control problem in a cooperative relay network is investigated and a new power control scheme is proposed based on a non-cooperative differential game. Optimal power allocated to each node for a relay is formulated using the Nash equilibrium in this paper, considering both the throughput and energy efficiency together. It is proved that the non-cooperative differential game algorithm is applicable and the optimal power level can be achieved.
Cooperative communications have been proposed for an enhancement in the performance of wireless networks [1] and to achieve cooperative diversity for improving network coverage and capacity [2]. A relay-cooperation transmission can be seen as a kind of cooperative communication, in which a relay station helps to forward data between mobile stations and base stations. In a relay-cooperation transmission, an increase in the power level of one node always results in more interference in the other nodes, caused by the simultaneous transmissions on nearby links [3]. Meanwhile, a higher power level causes more energy to be expended. For the above reasons, it is essential to design an enabling power control approach to alleviate co-channel interference in cooperative relay networks [4].Generally, power control is considered for maximizing throughput or minimizing energy consumption [5]. The power control for relay-cooperation transmissions is challenging owing to the fact that the power control strategies of relay nodes influence each other and decisions are made dynamically under competition. Each relay node can competitively allocate the power level in a dynamic manner such that their individual target system performance can be achieved. The throughput maximization and energy consumption minimization are intertwined, and the power control scheme should therefore consider these two fundamentally conflicting objectives concurrently.A few works have been conducted on power control in cooperative relay networks [6]-[9]. In [6], sub-carrier and power allocations are considered to obtain a better BER performance. In [7], the power control is considered to increase the energy efficiency combing selective-relay cooperative communications. Game theory can be used to solve resource allocation problems of systems with conflicting components, and can be used in cooperative relay networks for settling the power control problems of relay nodes. Game theory has recently received increasing interest in the context of cooperative relay networks [3], [8], [9]. The problem of power control becomes more challenging when the relay nodes are non-cooperative, which motivates the use of game theory in research on ascertaining the importance of power control. In [3], the authors consider the multi-user power control problem in Gaussian frequency-flat interference relay channels using a game-theoretic framework. In [8], the energy efficiency of each user is maximized based on game theory and is measured in bits/joule. A game-theory-based decentralized relay selection scheme and a power allocation protocol were introduced in [9] to achieve a significant reduction in power consumption and an improvement in payoff.The novelty of our work is in considering a differential game-based power control scheme to increase network throughput and minimize energy consumption. Differential games or continuous-time infinite dynamic games are used to study a class of decision problems, under which the evolution of the state is described by a differential equation and the players act throughout a time interval. Through a comparison with existing algorithms, a different optimal objective and a new non-cooperative differential-game-based power control scheme to obtain the Nash equilibrium for power control is proposed in this paper. Our model will take the objective of minimizing the total transmission cost with a tradeoff between network throughput and energy consumption. Relay nodes act as rational, selfish players competing in a differential game to obtain the optimal transmit power and minimize their own transmission cost.The rest of this paper is organized as follows. The system model and problem statement are given in section II. Section III presents non-cooperative game solutions to the power control problem. Section IV provides the simulation results, and section V offers some concluding remarks regarding our proposal.
II. System Model and Problem Statement
We consider the uplink of a relay-assisted network with K users (game players) denoted by K ≡ {1, 2,⋯,K}. Each user is assigned a given channel (assumed to be orthogonal, typically in the frequency domain). A decode-and-forward protocol is considered in this model, in which the signals have to be decoded at the relay before being forwarded. User i can serve as a relay node for other users. Let D(i) denote the set of these users. Let p_{i}(s) denote the direct transmission power of user i by time s, and p_{i,j}(s) denote the transmission power in relaying the message of user j. Let
P R_max i ( s )
denote the maximum allowable power for a relay, and we then have
p i,j ( s )= ϖ i,j P R_max i ( s )
and
∑ j∈D( i ) ϖ i,j ≤1
. The transmission energy of player i during time interval [0, T] is a predetermined positive constant.$${Q}^{iT}={\displaystyle {\int}_{0}^{T}\left({p}_{i}\left(s\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{j\in D\left(i\right)}^{}p\text{}{}_{i,j}\left(s\right)}\right)}\text{dt},\text{}\forall i\in K.$$Equation (1) can be transformed into a differential game equation with the following initial and terminal boundary conditions:$$\stackrel{\u2022}{{Q}^{i}\left(s\right)}={p}_{i}\left(s\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{j\in D\left(i\right)}^{K}p\text{}{}_{i,j}\left(s\right)},\text{}\forall s\in \left[0,T\right],\text{}i\in K,$$where an overdot denotes a derivative of a function with respect to time.Users access a wireless system through the air interface, which is a common resource; since the air interface is a shared medium, each user’s transmission is a source of interference for others [10]. The signal-to-interference ratio (SIR) is a measure of the quality of signal reception for a wireless user. Thus, there exists a minimum acceptable SIR for each user. Assuming the minimum transmission power of user i is
p i ( s ) ¯
, all users are assigned the same bandwidth W, and user i splits the m_{i} (m_{i} ∈[0,1]) fraction of its bandwidth for relaying [1], as shown in Fig. 1. The additional rate cost R_{i}(s) by time s can then be expressed as$$\begin{array}{l}{R}_{i}\left(s\right)={\alpha}_{i}[\left(1-{m}_{i}\right)W{\left(\mathrm{log}\left(1+{p}_{i}\left(s\right)\right)-\mathrm{log}\left(1+\overline{{p}_{i}\left(s\right)}\right)\right)}^{+}\\ \text{}-\mathrm{log}\left({I}_{i}\left(s\right)\right)+{m}_{i}W{\displaystyle \sum _{j\in D\left(i\right)}^{}\mathrm{log}\left(1+p\text{}{}_{i,j}\left(s\right)\right)}],\text{\hspace{0.17em}}\end{array}$$where f(z) = (z − b)^{+} = max{0, z − b}, if z − b ≤ 0 and f(z)=0, and α_{i} is a nonnegative parameter. The interference from neighboring relays is denoted by I_{i}(s) and can be expressed as
Constrained by limited resources, the decoding set D(i) is time varying. Let x_{i}(s) denote the stock, that is, the power of user i by time s, which is the sum of the transmission power and relaying power, affected by decoding set D(i). The evolution of this stock is governed by the following differential equation:$$\stackrel{\u2022}{{x}_{i}\left(s\right)}=\stackrel{\u2022}{{Q}^{i}\left(s\right)\text{\hspace{0.17em}\hspace{0.17em}}}-{g}_{i}{x}_{i}\left(s\right),\text{}\forall i\in K,\text{}s\in \left[0,T\right],$$where g_{i} is the time-varying parameter of D(i).All relay users have to decode the signals before forwarding them, and thus users have a storage cost for relaying, which is constructed based on the evolution of x_{i}(s). Let S_{i}(x_{i}(s)) denote the storage cost, and we have the following formula [11]:$${S}_{i}\left({x}_{i}\left(s\right)\right)={\beta}_{i}\left[{x}_{i}\left(s\right)-{p}_{i}\left(s\right)\right],\text{}\beta 0\text{},$$where β_{i} is a nonnegative parameter, which means the unit cost of energy consumed, and is interpreted as the number of information bits received per joule of energy consumed.As a game player, each user seeks to minimize its payments of discounted sum of the increasing transmission cost. Let C_{i}(p_{i}(s)) denote the payments of the discounted sum of the increasing transmission cost and p_{i}(s) be the control of the differential game. The optimization problem of user i can thus be expressed as$$\underset{{p}_{i}\left(s\right)}{\mathrm{min}}{C}_{i}\left({p}_{i}\left(s\right)\right)={{\displaystyle \int}}_{0}^{T}\left[\stackrel{\u2022}{{Q}^{i}\left(s\right)}+{R}_{i}\left(s\right)+{S}_{i}\text{\hspace{0.17em}}\left({x}_{i}\left(s\right)\right)\right]\text{\hspace{0.17em}}{e}^{-\lambda s}\text{dt},$$where λ is the common discount rate, which is applied to find the discount, subtracted from a future value to find the value before the game starts. The discount rate should be considered when deciding whether to spend resources on or give more resources for a transmission.We are then led to a differential game, G ≡ G (K, {p_{i}},{C_{i}}), in which the following hold:
1) The players ofGare users K ≡ {1, 2,⋯,K} .
2) The strategy of playeriispi(s), which denotes the allocated power, and the game’s strategy profile isP(s) = (p1(s),p2(s),...,pk(s)).
3) The players’ utilities are the objective of the minimum problem in (6).
Generally, the payments (utilities) of defined game G are not linear, which is different from the original normal game form.
III. Non-cooperative Optimal Power Control
In this section, we consider the model of a finite horizon, a series of dynamic optimization programs, and one feedback Nash equilibrium problem. The dynamic optimization program technique was developed by Bellman [12] and is given in Theorem 1.Theorem 1: Bellman’s dynamic programming. A set of controls, u^{＊} (s) = ϕ^{＊} (s, x), constitutes an optimal solution to control (6) if there exist continuously differentiable functions, V(s, x), defined [t_{0} ,T]× R^{m} → R and satisfying the following Bellman equation,$$\begin{array}{l}-{V}_{s}(s,x)=\underset{u}{\mathrm{min}}\left\{g[s,x,u]+{V}_{x}(s,x)f[s,x,u]\right\}\\ \text{}=\left\{g[s,x,{\varphi}^{\ast}(s,x)]+{V}_{x}(s,x)f[s,x,{\varphi}^{\ast}(s,x)]\right\},\end{array}$$$$V\left(T,x\right)=q\left(x\right).$$In this case, time is presented by s , the state is denoted by x, and V^{i} (s, x ) is the utility function (cost) of player i in time interval [t_{0}, T] For the optimization problem of (6), value function V^{i} (s, x) can be represented as follows:$${V}^{i}\left(s,x\right)={\displaystyle {{\displaystyle \int}}_{0}^{T}\left[\stackrel{\u2022}{{Q}^{i}\left(s,x\left(s\right)\right)}+{R}_{i}\left(s,x\left(s\right)\right)+{S}_{i}\left(x\left(s\right)\right)\right]}{e}^{-\lambda s}\text{dt}.$$Performing the indicated maximization for the optimization problem indicated above, we can obtain the Nash equilibrium for users, that is,$$\begin{array}{c}{p}_{i}\left(s\right)={\phi}_{i}(s)=\frac{{\alpha}_{i}\left(1-{m}_{i}\right){\mathrm{log}}_{2}\left(e\right)}{{\beta}_{i}-{V}_{x}^{i}\left(s,x\right){e}^{\lambda s}-1}-1,\\ \forall i\in K,\text{}s\in \left[0,T\right].\end{array}$$Proposition 1. System (7), (8) provides the solution.$${V}^{i}\left(s,x\right)=\left[{A}_{i}\left(s\right)x+{B}_{i}(s)\right]{e}^{-\lambda s},\text{}\forall i\in K$$with {A_{1} (s), A_{2}(s), A_{3} (s),…, A_{K} (s)} satisfies the following equations:$${A}_{i}\left(s\right)=\left(\theta -\frac{{\beta}_{i}}{\lambda +{g}_{i}}\right){e}^{\left(\lambda +{g}_{i}\right)\left(t-T\right)}+\frac{{\beta}_{i}}{\lambda +{g}_{i}},$$$${A}_{i}\left(T\right)=\theta ,$$and {B_{1} (s), B_{2}(s), B_{3} (s),…, B_{K} (s)} is given by$${B}_{i}\left(s\right)={e}^{\lambda s}\left[{\displaystyle {{\displaystyle \int}}_{0}^{s}{f}_{i}\left(t\right){e}^{-\lambda t}\text{dt}}+{B}_{i}^{0}\right].$$The proof of Proposition 1 is given in the Appendix. In (14), f_{i}(t) is suppressed by {α_{i}}, {m_{i}}, and {β_{i}}, which can be found in the Appendix.Based on the above proposition, (6) can be solved through a non-cooperative differential game, as follows:$$\begin{array}{l}\overline{\underset{\_}{\text{}Procedure.\text{Optimalpower}{p}_{i}\left(s\right)\text{assignedtouser}i\text{}}}\text{}\\ \text{}1)\text{Chooseobjectiveuser}i,\text{andstartwithatimeinterval,}\\ \text{[}{t}_{0}\text{,}T\text{],andinitialparametersforoptimalpowercontrol:}\\ \text{2)Constructtheminimumpaymentsproblemofuser}i:\\ \text{3)CalculatetheNashequilibriumoftheoptimization}\\ \text{problemfromequation(13).Controlthepowerlevelof}\\ \underset{\_}{\text{node}i\text{basedontheNashequilibrium.}}\end{array}$$Proposition 2. The Nash equilibrium obtained from (13) varies with a change in parameter A_{i}(s) and has the same changing trend.The result in Proposition 2 is straightforward, based on Proposition 1.
IV. Performance Evaluation
As A_{i} (s) has an obvious impact on the Nash equilibrium, we first consider the variation of A_{i} (s) with time and λ in this simulation, which is shown in Fig. 2. It is easy to find that A_{i} (s) is proportional to the time vector.In Fig. 2(a), A_{i} (s) increases with time and varies from 0 to 5. As can be seen from (10), Nash equilibrium φ_{i} (s) is a function of A_{i} (s), the variation of which will significantly reflect on φ_{i} (s), which means the optimal power assigned to each user will be modified by A_{i} (s). On the other hand, A_{i} (s) is the coefficient of x in (11). The system solution V^{i} (t,x) in (11) will also be influenced by A_{i} (s). It can be seen from Fig. 2(b) that A_{i} (s) is inversely proportional to discount rate λ.
Variation of A_{i}(s) with (a) time and (b) discount rate.
We can easy find the optimal power assigned to users through the differential power control game algorithm, which is shown in Fig. 3(a). It can be seen from Fig. 3(a) that the Nash equilibrium obtained has the same variation trend as A_{i} (s), which is consistent with Proposition 2. Furthermore, we can find that the optimal power is proportional to the discount rate (shown in Fig. 3(b)), which means we can assign more power for a transmission with a higher discount rate.
Variation of p_{i}(s) with (a) time and (b) discount rate.
V. Conclusion
In this paper, a non-cooperative differential-game based power-control approach for a cooperative relay network was proposed, which is distributed in each node and acts independently to achieve coordinate power control of the network resources. It was proved that a differential game can be used in cooperative relay networks for power control problems. Minimum payments of each node can be achieved, and the optimal power level assigned to relay nodes can be obtained.
Appendix
Proof of Proposition 1.Assume that V^{i} (s, x) is a polynomial that can be expressed as follows:$${V}^{i}\left(s,x\right)=\left[{A}_{i}\left(s\right)x+{B}_{i}(s)\right]{e}^{-\lambda s}.$$Using (15), system (7), (8) can be expressed as$$\begin{array}{l}\left[\lambda {A}_{i}\left(s\right)-{A}_{i}^{\prime}\left(s\right)\right]x+\left[\lambda {B}_{i}\left(s\right)-{B}_{i}^{\prime}\left(s\right)\right]=\\ \stackrel{\u2022}{{Q}^{i}\left(s\right)}+{R}_{i}\left(s\right)+{\beta}_{i}\left[x-{p}_{i}\right]+{A}_{i}\left(s\right)\left(\stackrel{\u2022}{{Q}^{i}\left(s\right)}-{g}_{i}x\right),\end{array}$$$${A}_{i}\left(T\right)x+{B}_{i}(T)=\theta {\left({x}_{i}\left(T\right)-\overline{{x}_{i}}\right)}^{+}.$$We then have$${A}_{i}^{\prime}\left(s\right)=\left(\lambda +{g}_{i}\right){A}_{i}\left(s\right)-{\beta}_{i},$$$${A}_{i}\left(T\right)=\theta ,$$$${B}_{i}^{\prime}\left(s\right)=\lambda {B}_{i}\left(s\right)-\left(1+{A}_{i}\left(s\right)\right)\stackrel{\u2022}{{Q}^{i}\left(s\right)}+{R}_{i}\left(s\right)-{\beta}_{i}{p}_{i},$$and$${B}_{i}(T)=\text{\u2212}\theta \overline{{x}_{i}}.$$Plugging equations into (20), we have$$\begin{array}{l}{B}_{i}^{\prime}\left(s\right)=\lambda {B}_{i}\left(s\right)-\left(1+{A}_{i}\left(s\right)\right)(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}-1\\ +{\displaystyle \sum _{j\in D\left(i\right)}^{K}{\varpi}_{i,j}{P}_{R\_\mathrm{max}}^{i}\left(s\right)})+{\alpha}_{i}[\left(1-{m}_{i}\right)W(\mathrm{log}\left(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}\right)\\ {-\mathrm{log}\left(1+\overline{{p}_{i}\left(s\right)}\right)\underset{\begin{array}{l}\\ \end{array}}{\overset{}{\text{}}})}^{+}+{m}_{i}W{\displaystyle \sum _{j\in D\left(i\right)}^{}\mathrm{log}\left(1+{\varpi}_{i,j}{P}_{R\_\mathrm{max}}^{i}\left(s\right)\right)}\\ -\mathrm{log}\left({I}_{i}\left(s\right)\right)\underset{}{\overset{}{\begin{array}{l}\\ \text{}\end{array}}}]-{\beta}_{i}\left(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}-1\right)=\lambda {B}_{i}\left(s\right)+{f}_{i}\left(s\right),\end{array}$$where
α i ¯
and
β i ¯
, for ∀i ∈ K , are constraints involving the model parameters, {α_{1}, α_{2}, α_{3}, …, α_{K}}, {m_{1}, m_{2}, m_{3} ,…, m_{K} }, {β_{1}, β_{2}, β_{3},…, β_{K}} , and$$\begin{array}{l}{f}_{i}\left(s\right)=-\left(1+{A}_{i}\left(s\right)\right)(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}-1+\\ {\displaystyle \sum _{j\in D\left(i\right)}^{K}{\varpi}_{i,j}{P}_{R\_\mathrm{max}}^{i}\left(s\right)})-{\beta}_{i}\left(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}-1\right)\\ +{\alpha}_{i}[\left(1-{m}_{i}\right)W{\left(\mathrm{log}\left(\frac{\overline{{\alpha}_{i}}}{\overline{{\beta}_{i}}-{A}_{i}\left(s\right)}\right)-\mathrm{log}\left(1+\overline{{p}_{i}\left(s\right)}\right)\right)}^{+}\\ -\mathrm{log}\left({I}_{i}\left(s\right)\right)+{m}_{i}W{\displaystyle \sum _{j\in D\left(i\right)}^{}\mathrm{log}\left(1+{\varpi}_{i,j}{P}_{R\_\mathrm{max}}^{i}\left(s\right)\right)}].\end{array}$$Solving (18) and (19), we have$${A}_{i}\left(s\right)=\left(\theta -\frac{{\beta}_{i}}{\lambda +{g}_{i}}\right){e}^{\left(\lambda +{g}_{i}\right)\left(t-T\right)}+\frac{{\beta}_{i}}{\lambda +{g}_{i}}.$$B_{i} (s) can be solved as$${B}_{i}\left(s\right)={e}^{\lambda s}\left[{\displaystyle {{\displaystyle \int}}_{0}^{s}{f}_{i}\left(t\right){e}^{-\lambda t}\text{dt}}+{B}_{i}^{0}\right]$$with$${B}_{i}^{0}=\text{\u2212}\theta {\overline{x}}_{i}-{\displaystyle {{\displaystyle \int}}_{0}^{T}{F}_{i}\left(t\right){e}^{-\lambda t}\text{dt}}.$$Hence, Proposition 1 follows. ⧠
This paper is supported by the National High Technology Research and Development Program of China (863 Program) (No. 2012AA121604).
BIO
alex_xuht@hotmail.comHaitao Xu received his BS degree from Zhong Shan University, China, in 2007, and his MS degree at the University of Bristol in 2009. He is currently studying for a PhD at the University of Science and Technology Beijing, China. His research interests include mathematical models, management algorithms, self-organized networks, game theory and next-generation networks.
xwzhouli@sina.comXianwei Zhou received his BS degree from the Department of Mathematics at Southwest Teachers’ University in 1986 and his MS degree from the Department of System Science and Mathematics from Zhengzhou University in 1992, his PhD from the Department of Transportation Engineering of Southwest Jiao-tong University, China in 1999. He was engaged in post doctor study at School of Electronic and Information Engineering of Beijing Jiao-tong University, China, from 1999 to 2000. He is currently a professor at the Department of Communication Engineering, School of Computer & Communication Engineering, University of Science and Technology Beijing, China. His research interests include communication network, security next-generation networks, mobile IPv6, scheduling theory and game theory.
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@article{ HJTODO_2014_v36n2_280}
,title={Optimal Power Control in Cooperative Relay Networks Based on a Differential Game}
,volume={2}
, url={http://dx.doi.org/10.4218/etrij.14.0113.0434}, DOI={10.4218/etrij.14.0113.0434}
, number= {2}
, journal={ETRI Journal}
, publisher={Electronics and Telecommunications Research Institute}
, author={Xu, Haitao
and
Zhou, Xianwei}
, year={2014}
, month={Apr}
TY - JOUR
T2 - ETRI Journal
AU - Xu, Haitao
AU - Zhou, Xianwei
SN - 1225-6463
TI - Optimal Power Control in Cooperative Relay Networks Based on a Differential Game
VL - 36
PB - Electronics and Telecommunications Research Institute
DO - 10.4218/etrij.14.0113.0434
PY - 2014
UR - http://dx.doi.org/10.4218/etrij.14.0113.0434
ER -
Xu, H.
,
&
Zhou, X.
( 2014).
Optimal Power Control in Cooperative Relay Networks Based on a Differential Game.
ETRI Journal,
36
(2)
Electronics and Telecommunications Research Institute.
doi:10.4218/etrij.14.0113.0434
Xu, H
,
&
Zhou, X
2014,
Optimal Power Control in Cooperative Relay Networks Based on a Differential Game,
ETRI Journal,
vol. 2,
no. 2,
Retrieved from http://dx.doi.org/10.4218/etrij.14.0113.0434
[1]
H Xu
,
and
X Zhou
,
“Optimal Power Control in Cooperative Relay Networks Based on a Differential Game”,
ETRI Journal,
vol. 2,
no. 2,
Apr
2014.
Xu, Haitao
and
,
Zhou, Xianwei
and
,
“Optimal Power Control in Cooperative Relay Networks Based on a Differential Game”
ETRI Journal,
2.
2
2014:
Xu, H
,
Zhou, X
Optimal Power Control in Cooperative Relay Networks Based on a Differential Game.
ETRI Journal
[Internet].
2014.
Apr ;
2
(2)
Available from http://dx.doi.org/10.4218/etrij.14.0113.0434
Xu, Haitao
,
and
Zhou, Xianwei
,
“Optimal Power Control in Cooperative Relay Networks Based on a Differential Game.”
ETRI Journal
2
no.2
()
Apr,
2014):
http://dx.doi.org/10.4218/etrij.14.0113.0434