Power Control in D2D Network Based on Game Theory

. This paper considers power control problem based on Nash equilibrium (NE) to eliminate interference in multi-cell device-to-device (D2D) network. The power control problem is modeled as a non-cooperative game model, and a user residual energy factor is introduced in the formulation. Based on the proof of the existence and uniqueness of Nash equilibrium, a distributed iterative game algorithm is proposed to realize power control. Simulation results show that the proposed algorithm can converge to Nash equilibrium quickly, and obtain a better equilibrium income by adjusting the residual energy factor.


Introduction
With the requirement for high speed and efficiency of data transmission, the limited spectrum resource brings great challenges for mobile network communication. D2D communication is a new wireless technology, where two devices can communicate with each other without exchanging information from base station, so that it cannot only reduce the burden of base station, but also improve communication quality of cellular users [1]. However, the D2D users will be suffered from interference of other users in cellular system. Therefore, the interference elimination has been investigated in recent years [2].
In [3], the authors studied that the D2D users and the cellular users use the same channel resources by multiplexed mode. Although it can improve spectrum utilization, it will introduce a new kind of interference. The authors in [4] investigated a power control method for single cellular system containing one cellular link and one D2D link. The algorithm can reduce the interference significantly between D2D users and cellular users. In [5], the authors studied the interference elimination problem of multi-cell D2D network, where the cellular users are communicated by base station schedule and the D2D link communication is guaranteed by power control.
The previous work mainly focused on the centralized power control method for D2D network, while the distributed implementation is less concerned. Therefore, we studied distributed power control method in this paper by use of game theory. We first establish a static game model in hybrid multi-cell D2D network, and then prove the existence and uniqueness of the Nash equilibrium. Finally, the distributed power control method is iteratively implemented to obtain the optimal state of Nash equilibrium. Simulation results show that the game model designed in this paper can converge quickly to Nash equilibrium, and the system can get better balance by adjusting the residual energy factor.

System Model
The system model of D2D network is shown in Figure 1. We consider a multi-cell system containing D2D links, and the adjacent cells use the same frequency band for multiplexing communication. Assume there are N cellular and D2D links in the system to use the same frequency resource. To eliminate the co-channel interference among the different cellular links and the interference between the cellular links and the D2D links, we propose a power control method based on game theory with pricing mechanism.     is the variance of the additive white Gaussian noise. Unlike [6], we introduce a user residual energy factor  in the utility function, which is defined as where j E  is the residual energy transmitted by link j , and its maximum valve is max j E . The energy factor  describes a price law. On one hand, when the supply exceeds the demand, the payment for the price of energy consumption is low and the user can consume more energy to obtain better performance. On the other hand, when the demand exceeds supply, the energy consumption should pay more prices. Therefore, adjusting  can make the performance and energy consumption in a reasonable tradeoff state. Note that  is regarded as a control factor in the simulation.
Here we formulate the problem to establish a non-cooperative power control game model with pricing mechanism, which is In this paper, we use Nash equilibrium to solve the problem. We define the policy combination   * * * * 12 , ,..., N p p p p p  as a Nash equilibrium, and establish the be- If the game players adopt the strategy combination * p , they cannot leave and change the strategy combination, so that the Nash equilibrium will be the optimal solution.

Non-cooperative Power Control Game Analysis
According to the model in eq.(2) , we prove the existence and uniqueness of Nash equilibrium in this section. After that, we use a distributed iterative power algorithm to solve the Nash equilibrium point.

Existence
Therefore, ( , ) is quasi-concave for j p of the j th player [7]. According to the game existence theorem in [8], the game G has Nash equilibrium.

Uniqueness
and we obtain It means that the optimal response strategy vector can be extended.
According to the Ref. [9], we can conclude that the non-cooperative power control game has a unique Nash equilibrium point, which can be simply expressed by

Distributed iterative game algorithm
Based on the proofs in section 3.2, we present a distributed iterative algorithm to solve the Nash equilibrium. The realization is summarized below. Step

Simulation Results
In this section, we perform the proposed non-cooperative power control game algorithm in D2D network by matlab simulation. The wireless system in simulation composes four cells 14 cc , and each cell has single cellular link and single D2D link, so that 8  N . The game player set is expressed as of the D2D link and the cellular link is also unit. The compared algorithm comes from Ref [9]. Figure 2 shows the valves of utility function varied with the iteration number. The control factor is set to be 2. Note the control factor in Ref [9] represents power coefficient. From the simulation, it can be seen that our algorithm converges to the stable state within six times, while the compared algorithm in Ref [9] needs eight times. Therefore, our method can converge more quickly than the compared one. Besides, we find that the valve of utility function of our method is bigger than the compared method, which means our method is more stable.  Figure 3 gives the bar chart of the iteration number varied with the control factor. It shows that our algorithm has smaller iteration number when the control factor is bigger than 1, which means the algorithm is more efficient.  Figure 4 gives the transmit power results for all of users in Nash equilibrium state. It observed that all of transmit power of users tend to be stable. Hence the proposed price function can make the system achieve the equilibrium stable state. From the simulations, we conclude that the power control algorithm based on pricing mechanism can effectively improve the performance of the system if pricing factor is adjusted reasonably.

Conclusion
In this paper, we investigate the power control problem in D2D network. A non-cooperative game model based on pricing mechanism is established to eliminate the interference in complex environment. We design a price function and prove its existence and uniqueness, and then propose a distributed iterative algorithm which can converge to Nash Equilibrium point. From the simulation, it can be observed that our algorithm improve the system performance compared with the conventional method.