# Power Control for Full-Duplex Device-to-Device Underlaid Cellular Networks: A Stackelberg Game Approach

Authors: Zhen Yang; Titi Liu; Guobin Chen
Journal: Complexity
 (2020)

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## Abstract

In spectrum sharing cognitive radio networks, unauthorized users (secondary users) are allowed to use the spectrum of authorized users (primary users) to improve spectrum utilization. Due to limited spectrum resources, how to formulate a reasonable spectrum allocation scheme is very important. As a mathematical analysis tool, game theory can solve the problem of resource allocation well. In recent years, it has been applied to the research of resource allocation in spectrum sharing networks by some literatures. In a cellular network consisting of multiple cellular users and full-duplex end-to-end communication users D2D (device-to-device), the self-interference caused by full-duplex communication and the interference caused by the D2D users to the cellular users will significantly reduce system throughput. In order to reduce the interference in the network, this paper introduces a power control algorithm based on Stackelberg game, which sets the cellular users and D2D users as the leaders and followers, respectively. The cellular users and the D2D users compete with each other to minimize the cost, and we propose new utility functions. We build an optimization problem under the outage probability constraint and power constraint and the transmission power of the users is obtained by using the Lagrangian dual decomposition method. The simulation results show that the proposed game algorithm improves network performance compared with other existing schemes.


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## Body

## 1. Introduction

With the rapid development of 5G technology, more and more mobile devices are required to access, making spectrum resources even poorer. The traditional spectrum sharing scheme is a static spectrum allocation mode. It is to allocate some fixed frequency bands to users of some specific networks. These specific users are authorized users (primary users). Users in networks that have not been allocated spectrum resources are unauthorized users (secondary users). In the past, these unauthorized users (such as Wi Fi users) could only use a small number of users.Crowded public open unauthorized spectrum, while most authorized frequency bands are sometimes idle (such as TV frequency bands). In order to improve the utilization rate of radio spectrum and make good use of radio spectrum resources reasonably, a dynamic sharing scheme must be proposed to allocate radio spectrum resources more efficiently. Therefore, cognitive radio (CR) technology is proposed in this environment. As a new concept, cognitive radio technology promotes the development of wireless communication industry, improves the utilization efficiency of spectrum resources, and provides a brand-new idea for realizing the sharing of spectrum resources.The high transmission rate of 5G cellular network can effectively promote the realization of Internet of Things (IoT) and smart city. However, the big data brought by the Internet of Things and smart cities may overload 5G cellular networks due to limited spectrum resources, and mobile communication networks gradually cannot meet people’s exponential demand for wireless mobile data traffic. D2D technology can relieve the pressure of the upcoming core network traffic and is considered as a promising way to solve this problem. D2D communication allows direct communication between devices without using base stations for relay, and the transmission distance of D2D pairs is short. Spectrum efficiency can be significantly improved through spectrum multiplexing of cellular users. Therefore, D2D communication has the advantages of reducing the network burden of cellular users, reducing the power consumption of mobile terminals, improving the robustness of network infrastructure, improving spectral efficiency and system capacity, reducing user network delay and power consumption, expanding network coverage, etc. D2D technology can be used in machine-to-machine communication, car-to-car communication, content distribution, location-aware services in smart cities, social networks, proximity online games, electronic health care, and public safety services. However, in order to realize D2D communication, several challenging problems need to be solved, such as equipment discovery, spectrum resource allocation, interference management, power control, communication security, etc.With the popularization of the fifth-generation mobile communication network and terminal equipment, the requirement for high-speed transmission of the network is becoming higher and higher. Traditional LET network technology cannot meet the requirements of high communication services, especially high-speed services. Thus, device-to-device (D2D) mode of direct communication between close-range terminals and terminals has been proposed in this context. This communication mode shares spectrum resources with cellular users in cellular networks to improve the capacity and spectrum utilization of network systems. It has become one of the most popular research technologies recently. At the same time, simultaneous full-duplex D2D communication technology can support two communication devices to transmit and receive at the same frequency band in the wireless cellular network, and further improve the spectrum utilization [1]. However, how to reduce power interference among users is a key problem, such as D2D user interference, D2D user interference to cellular users, and D2D user self-interference.


## 2. Related Work

In order to solve the resource management problem in D2D cellular network, some non-cooperative or cooperative game theory schemes are proposed in [2–5], which are used to control the power problem of DD users and cellular users in D2D cellular network, reduce the interference between users, and maximize the capacity of users in the network. In addition, the convergence of user transmit power is also analyzed in this paper. However, these works basically only consider half-duplex D2D communication mode. Therefore, the scheme of user power control and interference management in D2D full-duplex communication mode is proposed in [6–8]. Literature [6] proposes a rate maximization model to improve the total rate of full-duplex D2D communication network and minimize the interference of D2D users to cellular users. In [7, 8], a power optimization scheme is proposed to maximize the total rate of cellular subscribers and D2D pairs in the presence of two directional interference. In addition, the paper [9, 10] proposes an efficient adaptive user power control scheme, which further reduces the interference between users and improves the system performance. In document [11], in order to reduce user interference, an interference cancellation scheme of cell network selection model is designed. However, most of the existing work neglects studying the power control of multiple cellular users and multiple D2D users in full-duplex D2D communication networks. In addition, the user’s requirement for interruption probability has not been taken into account in the establishment of the model. Although reference [3] considers the interruption probability constraints of the network, it only considers the half-duplex communication mode, and its utility function does not consider the network power saving in order to maximize the network throughput. Zulhasnine et al. proposed a greedy algorithm based on the maximum throughput of the system and proposed an optimization strategy from two aspects of D2D communication users multiplexing the uplink and downlink resources of cellular users. The allocation problem of D2D to users’ cell radio resources was summarized as a mixed integer nonlinear programming problem [12]. Min et al. proposed a D2D mode selection scheme for users in downlink, which reduced the interference between communication users [13]. Yu et al. proposed a joint mode selection and resource allocation algorithm, which achieves the effects of maximum capacity and minimum communication interference of the communication system [14]. Xiao et al. studied the algorithm of D2D on the transmission power of users and cellular users and proposed a scheme to adjust the transmission power of users under different communication modes to control the interference between communication users [15]. Hassan et al. proposed a two-stage resource allocation algorithm to minimize system interference while maintaining the total rate of the target system. Firstly, a weighted two-part matching algorithm is used to minimize interference and obtain a feasible initial solution, and then local search technology is used to improve the solution [16]. Nguyen et al. and others studied from the geographical location of the cell where the users are located, established communication restricted areas according to the interference between users, and implemented different communication mode strategies in different areas, thus improving the overall communication quality of the system [17]. Ningombam and Shin proposed a radio resource allocation and power control scheme to reduce D2D interference to multiplexed uplink cellular resources [18]. A radio resource allocation scheme is proposed by using a cell sectorization method to reduce interference in D2D communication of LTE-A uplink cellular network. Jaheon et al. proposed a scheme combining resource allocation and power control. The scheme firstly coordinates the interference introduced by D2D to a certain extent by allocating reasonable multiplexing resources for D2D users and appropriately adjusts the transmission power of D2D users to ensure controllable cellular link interference [19]. Fodor et al. proposed a solution to share spectrum resources between cellular mobile devices and D2D, thus improving the spectrum efficiency and energy efficiency of traditional cellular networks [20]. Doppler et al. explained the session and management mechanism of D2D communication and how to limit the interference of D2D communication to cellular networks [21]. Kim et al. proposed a D2D link radio resource allocation strategy for interference between D2D users and cellular users [22]. Li et al. proposed a radio resource allocation scheme to minimize the interference between communication users [23, 24], aiming at the interference problem generated in the communication system where D2D communication users coexist with cellular users and D2D technology is combined with Multi-User Multiple-Input Multiple-Output (MU-MIMO) technology.Therefore, this paper proposes a power control algorithm for multiple cellular users and D2D users in full-duplex D2D communication with channel interruption constraints based on Stackelberg game. Considering the interference and self-interference in two directions among D2D users, we set cellular users and D2D users as leaders and followers, respectively. Cellular users and D2D users compete with each other to minimize their cost effectiveness (network power consumption), and propose a new utility function. Under the condition of channel interrupt constraint and power constraint, the optimization problem is constructed, and the transmission power of users is obtained by using Lagrange dual decomposition method. The simulation results show that the proposed game algorithm improves network throughput and reduces bit error rate. The second part introduces related work; the third part introduces the system model; the fourth part introduces the Stackelberg game theory method; the fifth part introduces the Stackelberg game solution; the sixth part introduces simulated experimentally.


## 3. System Model

The system model of cellular network is shown in Figure1. A single base station is located in the cell center of the network. It has K cellular users. Cellular users have the priority to use the base station to allocate the channel. M D2D user’s pair and D2D users pair are allowed to share the communication channel of cellular users, and D2D users adopt full-duplex communication mode. Suppose each D2D user has two antennas, one for transmitting signals and the other for receiving signals. Define the path loss and multipath fading channel coefficients of the cellular subscriber K and its receiver between the base stations as Lmk and hkk; for the path loss and multipath fading between the m-th D2D user pair and the k-th cellular subscriber and the base station, respectively, the channel coefficients are Lmk and hmk, respectively; the path loss and multipath fading channel coefficients between the m-th D2D pair are Lm and hmm, respectively, and for the path loss and multipath fading channel between the m-th D2D pair and the i-th user, the coefficients are Lim and him, respectively. The fading coefficient obeys the CN(0, 1) independent Gaussian distribution. Cellular user k has a transmit power of Pk. On the k-th cellular user channel, the powers of the first user (D1) and the second user (D2) of the m-th D2D pair are pmk and ampmk, respectively, where A is defined as the coefficient of the transmit power between D1 and D2. We assume that each cellular user transmits simultaneously on a single channel and that each cellular user’s channel can simultaneously transmit all D2D pairs to the user.Figure 1
Cellular network model.The signal-to-interference and noise ratio (SINR) received by the k-th cellular subscriber at the base station is defined as(1)γk=pkLkhkk2∑m=1M1+αmpmkLmkhmk2+N0,where L represents the path loss factor and L=d−3, where d represents the distance between the user’s transmitter and receiver, and N0 represents the noise power.Since the current self-interference suppression technology cannot completely eliminate self-interference [9], the power of residual self-interference is defined as pSI=ηp, where pSI is the power of residual self-interference, η0≤η≤1 is the cancellation coefficient of self-interference, and p represents the transmitting power of the transmitting end. On the k-th cellular user channel, the SINR received by D2 is defined as(2)γmk,D1⟶2=pmkLmhmm2∑i≠mM1+αipikLimhim2+ηm,2αmpmkLm,SIhm,SI2+pkLkmhkm2+N0,where Lm,SI and hm,SI represent the self-interference path loss and fading channel coefficients of the m-th D2D pair to the user, respectively. Similarly, we can get the SINR of D2 to D1 on the k-th cellular user channel as γmk,D2⟶1.When the cellular userK’s SINR value is lower than the threshold γk¯ , its communication will be interrupted, and the cellular user’s outage probability pk,outCpkP−k is expressed as [13](3)Pk,outCpk,p−k=1−exp−γ¯kN0pkLk∏m=1M1+γ¯k1+αmkpmkLmkpkLk−1,where P−k represents the power vector of the user other than the cellular user on the channel of the k-th cellular user. Since the cellular user k has the constraint ξk of the channel outage probability, pk,outC≤ξk, it can be obtained by formula (3):(4)1−ξk≤exp−γ¯kN0pkLk∏m=1M1+γ¯k1+αmkpmkLmkpkLk−1.Similarly, on the channel of the cellular userk, when the SINR value γmk of the D2D to the user is less than the target threshold γmk¯ of the SINR, the probability of the interruption of the m-th D2D pair (from D1 to D2) can be defined as(5)Pm,outD1⟶2pmk,p−mk=1−exp−γ¯mkN0+ηm,2αmpmkLm,SIhm,SI2pmkLm⋅∏i≠mM1+γ¯mk1+αipikLimpmkLm−1,where P−k represents the transmit power vector of all other users except the k-th D2D on the first channel. If the probability of break γm,outk,D2⟶1 is less than the interrupt threshold ξm, γm,outk,D2⟶1≤ξm, the constraint of the probability of interruption can be obtained:(6)1−ξm≤1−exp−γ¯mkN0pmkLm−xmD1⟶2∏i≠mM1+γ¯mk1+αipikLimpmkLm−1,where xmD1⟶2=γ¯mkηm,2αmLm,SIhm,SI2/Lm.Similarly, the probability of the first D2D pair (from D2 users to D1 users) can be defined as(7)1−ξm≤1−exp−γ¯mkN0αmpmkLm−xmD2⟶1∏i≠mM1+γ¯mk1+αipikLimαmpmkLm−1,where xmD2⟶1=γ¯mkηm,1Lm,SIhm,SI2/αmLm.


## 4. Stackelberg Game Theory

### 4.1. Traditional Noncooperative Game Model

In cognitive radio network system, it is considered that secondary users are access users who will not cooperate with each other; then there must be conflicts between secondary users, which plays an extremely important role in analyzing complex and competitive cognitive radio networks with noncooperative game theory. LetΘ=κ,PkUkp be NPG model and κ=1,2,3,…,K be the set of perceived users. Pk is the power set of secondary user k; its revenue function is Uk∗. In order to solve the problem that the game model cannot obtain the global optimal transmission power solution, a linear cost function is introduced into the utility function of the secondary user, and the benefit function expression of NPGP model [25] is defined as follows:(8)NPGP:maxpk∈PkUkpk,P−k=LRMpk1−e−γk/2M−c1pk,k∈κ.In the formula,M is the total number of bits per frame; L is that number of valid information bit (L & lt; M); R is transmission rate; c1 is a penalty factor; P is secondary user transmit power vector: p=p1,…,pk, P−k is the power of other users besides perceiving user k.Utility functionUk represents the final income of users after the game in bits/joules. However, the above penalty factor is a constant, and due to different interferences caused by users, it is unfair to set it in this way. Through the introduction of the above-mentioned related game schemes, the improved penalty function [26] is expressed as follows:(9)NPGP−MSFLA:Ukpk,P−k=LRMpk1−e−γk/2M−c1epk−c3γk−γkmin.In the formula,c2 and c3 are constant cost factors.Letf1γk be an effective function when data transmission is successful. Generally, it is modulated by frequency shift keying. The expression is defined as follows:(10)f1γk=1−e−γk/2M.In the game model of [27], a new benefit function is designed, in which the penalty function is a nonlinear function set according to the interference of users and the signal-interference-noise ratio, and the utility function is defined.(11)NPGP−ESIA:Ukpk,P−k=LRMpk1−e−γk1+eγk−γkmin−αeβγk/γkmin−1pkpth.In the formula,α is that trim factor of the penalty function and is constant; β is that trim factor of the penalty function and is constant; pth is threshold of average power.


### 4.2. Stackelberg Game Model

Stackelberg’s game model is actually an orderly game model; that is, some actors can act first and are called leaders. However, the users who act later make corresponding action strategies according to the users who act earlier, which are called followers. This is the Stackelberg game. At the top of the game, leaders will choose the best action to maximize their benefits when they know the relevant information of followers. At the bottom of the game, followers make the best strategy through noncooperative game under the condition of understanding the relevant behaviors of leaders. The Nash equilibrium solution of Stackelberg game is called Stackelberg equilibrium (SE). The game leader will formulate his own optimal response function considering the follower’s response information, which is defined as follows:(12)SF∗≤argUSL.Leaders get their best action strategySF∗ when maximizing their own benefit function. At this time, followers choose the best action strategy SF∗ according to the leader’s response SL∗ to maximize their own interests.(13)SL∗≤argUSL.After many games, Stackelberg equilibrium solutionsSFSE,SLSE can be obtained. Both parties change their strategies according to the information of other participants to further improve their income. Stackelberg equilibrium solution: the behavioral strategy set SFSE,SLSE is the strategic equilibrium solution of Stackelberg’s game. For any leader ∀i∈K,(14)UiS1,…,SiSE,…,SK,…,SLSE≥UiS1,…,Si,…,SK,…,SLSE,∀i.For any one of the followers∀j∈N,(15)UiS1,…,SjSE,…,SK,…,SFSE≥UiS1,…,Sj,…,SK,…,SFSE,∀j.The power control model can adopt Stackelberg game algorithm to control the transmission power of cognitive users according to the different order of primary and secondary users’ strategies. In addition, since most literatures do not consider the outage probability constraints of primary users and secondary users into the system model, taking the constraints into account will improve the practicability of the algorithm, and Lagrange optimization method and KTT scheme are used to obtain the optimal transmission power of each user on the corresponding channel.The power control strategy between cellular users and D2D to user can use the Stackelberg game model to solve power control problems. Specifically, cellular users are leaders, first taking action, setting the market price to sell the spectrum resources they use; D2D pairs are followers, observing the leader’s actions and taking appropriate actions, by paying the main user’s corresponding fees to share the licensed band of cellular users. Both cellular users and D2D pairs are selfish, and they are designed to minimize their own costs under outage constraints, thereby building a corresponding benefit function.We first analyze the leader subgame of cellular user power control issues. Cellular users aim to minimize their cost by providing interference prices to D2D pairs. The cost utility of the leader (cellular users) can be defined as their own transmit power minus the gain from the D2D pair by sharing resources with all D2D pairs. The utility cost function for cellular users is expressed as(16)Qkpk,p−k,ck=pk−ck∑m=1M1+αmpmkLmkhmk2,where ck represents the interference price of the cellular user k.The optimization issues for cellular users are as follows:(17)minQkpk,p−k,cks.t.4.For a follower’s subgame, the cost function for each D2D pair is the total power allocated on the channel of all cellular users plus the cost paid for using the cellular user’s channel. Therefore, the cost function of the first D2D to the user can be expressed as(18)Umpm,p−m,c=∑k=1K1+αmpmk+∑k=1Kck1+αmpmkLmkhmk2,where Pm represents the power vector of the m-th D2D pair to users on all k-cellular user channels, P−m=P1,P2,…,Pm−1,…,Pm+1,…,PM represents the transmit power matrix of all other D2D pairs, and c=c1,c2,…,ck represents the price vector that the m-th D2D user pays to all cellular users.Therefore, the first D2D pair is intended to solve the problem of multi-channel power control. The specific problems are(19)minUmpm,p−m,cs.t.67.Since constraints (4), (6), and (7) are nonconvex, the games (17) and (19) under constraints are nonconvex. Therefore, we can optimize problems (17) and (19) by using the logarithm of the variable: p˜k=logpk, p˜mk=logpmk. Therefore, the optimization problems in (17) and (19) are translated into(20)minQkep˜k,ep˜−k,ck=ep˜k−ck∑m=1M1+αmep˜mkLmkhmk2,∀ks.t.γ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mkLmkep˜kLk≤log11−ξk,∀k,(21)minUmep˜m,ep˜−m,c=∑k=1K1+αmep˜mk+∑k=1Kck1+αmep˜mkLmkhmk2,∀ms.t.γ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm≤log11−ξm,∀kγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm≤log11−ξm,∀k.In the system herein, the cellular user’s channel is fixed. For example, each cellular user occupies one subchannel, and then the D2D user selects whether to share the cellular user’s channel for communication according to the cellular user channel condition. Therefore, this paper introduces the Stackelberg game theory to control the user power and manage the D2D user access cellular user channel; that is, the cellular user performs its own transmit power according to its own utility function (equation (16)), while the D2D user selectively accesses the cellular user’s channel and controls its own transmit power according to its utility function (equation (18)).The solution to the Stackelberg game is to analyze the existence and uniqueness of the Nash equilibrium. The tree constraint functions of (20) and (21) have P˜m,outcp˜k,P˜−k, P˜m,outD1⟶2p˜mk,P˜−mk, and P˜m,outD2⟶1p˜mk,P˜−mk. The performance problem of Nash Equilibrium can be expressed as follows.Theorem 1.
The subgame of D2D to the user (follower) is the existence of Nash equilibrium and is unique.Proof.
For each D2D pair,∀m∈M, the interrupt constraint functions P˜m,outD1⟶2p˜mk,P˜−mk and P˜m,outD2⟶1p˜mk,P˜−mk are strictly convex functions and are monotonically decreasing in p˜mk. The power vectors Pk=pmk;∀m∈M in the Hessian matrices P˜m,outD1⟶2 and P˜m,outD2⟶1 are both non-negative. Therefore, problem (21) is obviously an optimization problem of strictly convex constraints, NE always exists, and there is a unique optimal solution.Theorem 2.
The subgame of a cellular user (leader) is the existence of Nash equilibrium and is unique.Proof.
For each cellular user, the∀k∈K, the k-th cellular user’s interrupt constraint function is P˜m,outcp˜k,P˜−k; the interrupt constraint function is a strict convex function and is about the p˜mk monotonically decreasing function. Therefore, problem (20) is an optimization problem of strict convex constraints, and there is a unique optimal solution.
The Nash equilibrium of the Stackelberg game consists of the Nash equilibrium of the follower subgame and the Nash equilibrium of the leader subgame and analyzes the existence and uniqueness of the Nash equilibrium of the Stackelberg game. The Nash equilibrium of the leader’s subgame exists and is unique. For a given power allocation scheme, the Nash equilibrium of the follower subgame exists and is unique.


## 4.1. Traditional Noncooperative Game Model

In cognitive radio network system, it is considered that secondary users are access users who will not cooperate with each other; then there must be conflicts between secondary users, which plays an extremely important role in analyzing complex and competitive cognitive radio networks with noncooperative game theory. LetΘ=κ,PkUkp be NPG model and κ=1,2,3,…,K be the set of perceived users. Pk is the power set of secondary user k; its revenue function is Uk∗. In order to solve the problem that the game model cannot obtain the global optimal transmission power solution, a linear cost function is introduced into the utility function of the secondary user, and the benefit function expression of NPGP model [25] is defined as follows:(8)NPGP:maxpk∈PkUkpk,P−k=LRMpk1−e−γk/2M−c1pk,k∈κ.In the formula,M is the total number of bits per frame; L is that number of valid information bit (L & lt; M); R is transmission rate; c1 is a penalty factor; P is secondary user transmit power vector: p=p1,…,pk, P−k is the power of other users besides perceiving user k.Utility functionUk represents the final income of users after the game in bits/joules. However, the above penalty factor is a constant, and due to different interferences caused by users, it is unfair to set it in this way. Through the introduction of the above-mentioned related game schemes, the improved penalty function [26] is expressed as follows:(9)NPGP−MSFLA:Ukpk,P−k=LRMpk1−e−γk/2M−c1epk−c3γk−γkmin.In the formula,c2 and c3 are constant cost factors.Letf1γk be an effective function when data transmission is successful. Generally, it is modulated by frequency shift keying. The expression is defined as follows:(10)f1γk=1−e−γk/2M.In the game model of [27], a new benefit function is designed, in which the penalty function is a nonlinear function set according to the interference of users and the signal-interference-noise ratio, and the utility function is defined.(11)NPGP−ESIA:Ukpk,P−k=LRMpk1−e−γk1+eγk−γkmin−αeβγk/γkmin−1pkpth.In the formula,α is that trim factor of the penalty function and is constant; β is that trim factor of the penalty function and is constant; pth is threshold of average power.


## 4.2. Stackelberg Game Model

Stackelberg’s game model is actually an orderly game model; that is, some actors can act first and are called leaders. However, the users who act later make corresponding action strategies according to the users who act earlier, which are called followers. This is the Stackelberg game. At the top of the game, leaders will choose the best action to maximize their benefits when they know the relevant information of followers. At the bottom of the game, followers make the best strategy through noncooperative game under the condition of understanding the relevant behaviors of leaders. The Nash equilibrium solution of Stackelberg game is called Stackelberg equilibrium (SE). The game leader will formulate his own optimal response function considering the follower’s response information, which is defined as follows:(12)SF∗≤argUSL.Leaders get their best action strategySF∗ when maximizing their own benefit function. At this time, followers choose the best action strategy SF∗ according to the leader’s response SL∗ to maximize their own interests.(13)SL∗≤argUSL.After many games, Stackelberg equilibrium solutionsSFSE,SLSE can be obtained. Both parties change their strategies according to the information of other participants to further improve their income. Stackelberg equilibrium solution: the behavioral strategy set SFSE,SLSE is the strategic equilibrium solution of Stackelberg’s game. For any leader ∀i∈K,(14)UiS1,…,SiSE,…,SK,…,SLSE≥UiS1,…,Si,…,SK,…,SLSE,∀i.For any one of the followers∀j∈N,(15)UiS1,…,SjSE,…,SK,…,SFSE≥UiS1,…,Sj,…,SK,…,SFSE,∀j.The power control model can adopt Stackelberg game algorithm to control the transmission power of cognitive users according to the different order of primary and secondary users’ strategies. In addition, since most literatures do not consider the outage probability constraints of primary users and secondary users into the system model, taking the constraints into account will improve the practicability of the algorithm, and Lagrange optimization method and KTT scheme are used to obtain the optimal transmission power of each user on the corresponding channel.The power control strategy between cellular users and D2D to user can use the Stackelberg game model to solve power control problems. Specifically, cellular users are leaders, first taking action, setting the market price to sell the spectrum resources they use; D2D pairs are followers, observing the leader’s actions and taking appropriate actions, by paying the main user’s corresponding fees to share the licensed band of cellular users. Both cellular users and D2D pairs are selfish, and they are designed to minimize their own costs under outage constraints, thereby building a corresponding benefit function.We first analyze the leader subgame of cellular user power control issues. Cellular users aim to minimize their cost by providing interference prices to D2D pairs. The cost utility of the leader (cellular users) can be defined as their own transmit power minus the gain from the D2D pair by sharing resources with all D2D pairs. The utility cost function for cellular users is expressed as(16)Qkpk,p−k,ck=pk−ck∑m=1M1+αmpmkLmkhmk2,where ck represents the interference price of the cellular user k.The optimization issues for cellular users are as follows:(17)minQkpk,p−k,cks.t.4.For a follower’s subgame, the cost function for each D2D pair is the total power allocated on the channel of all cellular users plus the cost paid for using the cellular user’s channel. Therefore, the cost function of the first D2D to the user can be expressed as(18)Umpm,p−m,c=∑k=1K1+αmpmk+∑k=1Kck1+αmpmkLmkhmk2,where Pm represents the power vector of the m-th D2D pair to users on all k-cellular user channels, P−m=P1,P2,…,Pm−1,…,Pm+1,…,PM represents the transmit power matrix of all other D2D pairs, and c=c1,c2,…,ck represents the price vector that the m-th D2D user pays to all cellular users.Therefore, the first D2D pair is intended to solve the problem of multi-channel power control. The specific problems are(19)minUmpm,p−m,cs.t.67.Since constraints (4), (6), and (7) are nonconvex, the games (17) and (19) under constraints are nonconvex. Therefore, we can optimize problems (17) and (19) by using the logarithm of the variable: p˜k=logpk, p˜mk=logpmk. Therefore, the optimization problems in (17) and (19) are translated into(20)minQkep˜k,ep˜−k,ck=ep˜k−ck∑m=1M1+αmep˜mkLmkhmk2,∀ks.t.γ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mkLmkep˜kLk≤log11−ξk,∀k,(21)minUmep˜m,ep˜−m,c=∑k=1K1+αmep˜mk+∑k=1Kck1+αmep˜mkLmkhmk2,∀ms.t.γ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm≤log11−ξm,∀kγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm≤log11−ξm,∀k.In the system herein, the cellular user’s channel is fixed. For example, each cellular user occupies one subchannel, and then the D2D user selects whether to share the cellular user’s channel for communication according to the cellular user channel condition. Therefore, this paper introduces the Stackelberg game theory to control the user power and manage the D2D user access cellular user channel; that is, the cellular user performs its own transmit power according to its own utility function (equation (16)), while the D2D user selectively accesses the cellular user’s channel and controls its own transmit power according to its utility function (equation (18)).The solution to the Stackelberg game is to analyze the existence and uniqueness of the Nash equilibrium. The tree constraint functions of (20) and (21) have P˜m,outcp˜k,P˜−k, P˜m,outD1⟶2p˜mk,P˜−mk, and P˜m,outD2⟶1p˜mk,P˜−mk. The performance problem of Nash Equilibrium can be expressed as follows.Theorem 1.
The subgame of D2D to the user (follower) is the existence of Nash equilibrium and is unique.Proof.
For each D2D pair,∀m∈M, the interrupt constraint functions P˜m,outD1⟶2p˜mk,P˜−mk and P˜m,outD2⟶1p˜mk,P˜−mk are strictly convex functions and are monotonically decreasing in p˜mk. The power vectors Pk=pmk;∀m∈M in the Hessian matrices P˜m,outD1⟶2 and P˜m,outD2⟶1 are both non-negative. Therefore, problem (21) is obviously an optimization problem of strictly convex constraints, NE always exists, and there is a unique optimal solution.Theorem 2.
The subgame of a cellular user (leader) is the existence of Nash equilibrium and is unique.Proof.
For each cellular user, the∀k∈K, the k-th cellular user’s interrupt constraint function is P˜m,outcp˜k,P˜−k; the interrupt constraint function is a strict convex function and is about the p˜mk monotonically decreasing function. Therefore, problem (20) is an optimization problem of strict convex constraints, and there is a unique optimal solution.
The Nash equilibrium of the Stackelberg game consists of the Nash equilibrium of the follower subgame and the Nash equilibrium of the leader subgame and analyzes the existence and uniqueness of the Nash equilibrium of the Stackelberg game. The Nash equilibrium of the leader’s subgame exists and is unique. For a given power allocation scheme, the Nash equilibrium of the follower subgame exists and is unique.


## 5. Stackelberg Game Solution

Since the optimal power control problems of (20) and (21) have constraints, if the complexity of solving the subproblem is higher with the distributed algorithm, we use the Lagrangian dual decomposition method to solve problem (20). And (21) optimizes the problem with the most transmit power.


### 5.1. D2D Solves the User’s Optimal Transmit Power

D2D minimizes its cost utility by observing the interference price vectorc set by all cellular users and controlling its own transmit power. Thus, the Lagrangian function of equation (22) can be expressed as(22)Lmp˜m,p˜−m,c,λm,ωm=∑k=1K1+αmep˜mk+∑k=1Kck1+αmep˜mkLmkhmk2+∑k=1Kλm⋅γ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm−log11−ξmk+∑k=1Kωmγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm−log11−ξmk,where λm and ωm are the dual variables of the constraints. The optimal solution for the Lagrangian dual function is minLmP˜m,P˜−m,c,λm,ωm. According to the convex optimization theory, LmP˜m,P˜−m,c,λm,ωm is a concave function with respect to λm and ωm; therefore, the optimal dual variable that can be obtained by the subgradient method is as follows:(23)λm∗=λm∗+εγ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm−log11−ξm+,ωm∗=ωm∗+ϕγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm−log11−ξm+,where ε and ϕ represent small enough steps and are positive.Since theLmP˜m,P˜−m,c,λm,ωm function is a concave function on λm and ωm, (15) iteratively converges to the optimal λm and ωm. Substituting the obtained optimal λm and ωm into (23), we derive it to obtain(24)∂Lmp˜m,p˜−m,c,λm,ωm∂p˜mk|λm∗,ωm∗=0.By solving (24), we can derive the optimal solution p˜mk∗ in (22). Then, by the exponential change of the pmk∗=expp˜mk∗, the optimal transmit power pmk∗ of D1 in the m-th D2D pair and the optimal transmit power αmpmk∗ of D2 can be obtained. Therefore, we can get the optimal transmit power of each D2D to the user.


### 5.2. Cellular User (Leader) Optimal Transmit Power Solution

In order to minimize the cost utility of cellular users, each cellular user needs to adaptively update the market interference price set to the user to the D2D based on the transmit power response of the userP˜m∗ (or Pm∗) by D2D. The subgame problem of the leader in the optimization problem (20) can be decomposed into two subproblems: by correcting the price factor ck, the optimal transmit power of each cellular user k can be obtained, and then the optimal ck is obtained. The solution of the optimal transmit power of the cellular user k can be obtained by the Lagrangian dual decomposition method.Therefore, for the subgame problem of the leader in (20), the corresponding Lagrangian function can be obtained:(25)Lkp˜k,p˜−k,ck,νk=ep˜k−ck∑m=1Mep˜mk∗Lmkhmk2+νkγ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mk∗Lmkep˜kLk−log11−ξk,where the dual variable vk represents the constraint in (20); the optimal solution for the Lagrangian dual function is minLkp˜k,p˜−k,ck,νk.Lmp˜m,p˜−m,c,λm,ωm is a concave function for vk. Similarly, we can use the gradient method to get the optimal dual variable:(26)νk∗=νk∗+θγ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mk∗Lmkep˜kLk−log11−ξk+,where θ is a sufficiently large positive real number. The iteration in (26) will converge to the optimal vk∗. Thus, we can derive the optimal solution (p˜k∗) of the problem in (20) by solving the following problem. The solution of the problem is as follows.Then, we can get the transmit powerpmk∗=expp˜mk∗ of the cellular user k by the exponential change of the formula pk∗.Based on the analysis of the optimal transmit power of D2D to the user subgame and the cellular user subgame, it is discussed how the cellular user sets the optimal price according to the transmit power policy response of all D2D pairs.If them-th D2D interference price to the user is greater than the payment threshold, the D2D pair of users will stop renting the channel of the k-th cellular user. In other words, for each D2D pair user, if it is transmitted on the channel of the k-th cellular user, its cost utility cannot be greater than the maximum cost threshold Umk,maxep˜kk∗. Thus, the cost for the m-th D2D pair of users should satisfy the constraint:(27)1+αmep˜mk∗+ck1+αmep˜mk∗Lmkhmk2≤Umk,maxep˜mk∗.Therefore, we can get the constraint of the priceck, expressed as follows:(28)ck≤Zm=Umk,maxep˜mk∗−1+αmep˜mk∗1+αmep˜mk∗Lmkhmk2.As can be seen from (28), if the cellular subscriber wants the m-th D2D pair to communicate on its channel, the price set by the cellular subscriber k should not be higher than the upper limit price in (28). Therefore, the optimal interference price ck∗ is ck∗=Zm.According to the above analysis, the k-th cellular user will minimizeQKep˜k by paying the threshold vector Z = (Z1, Z2, …, ZM) and then select the appropriate overall price ck∗.In order to achieve Nash equalization and to converge to the optimal transmit power, we propose the following iterative power control algorithm.(i)
Step 1: lett = 0; for all cellular users k, initialize its transmit power pkt and price ckt; for all D2D pairs, initialize Pmt(ii)
Step 2: for each D2D user, calculate the price vector paid by the m-th D2D to all cellular users(iii)
Step 3: updateλk∗ and ωk∗ according to formula (23), respectively(iv)
Step 4: for a given cellular userpkt and ckt, each D2D pair user m is solved according to equation (24) to obtain Pm∗t+1. If ckt>ZKt, D2D terminates the channel using cellular user k for user m(v)
Step 5: for each cellular userk, update vk∗ according to formula (18)(vi)
Step 6: at a given D2D user transmit powerPm∗t+1, the cellular user k updates its transmit power pm∗t+1 by solving equation (19)(vii)
Step 7: according to formulas (28) and Z, the cellular user updates its price ck∗t+1(viii)
Step 9: lett = t + 1 repeat steps 2 to 8 until convergence


## 5.1. D2D Solves the User’s Optimal Transmit Power

D2D minimizes its cost utility by observing the interference price vectorc set by all cellular users and controlling its own transmit power. Thus, the Lagrangian function of equation (22) can be expressed as(22)Lmp˜m,p˜−m,c,λm,ωm=∑k=1K1+αmep˜mk+∑k=1Kck1+αmep˜mkLmkhmk2+∑k=1Kλm⋅γ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm−log11−ξmk+∑k=1Kωmγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm−log11−ξmk,where λm and ωm are the dual variables of the constraints. The optimal solution for the Lagrangian dual function is minLmP˜m,P˜−m,c,λm,ωm. According to the convex optimization theory, LmP˜m,P˜−m,c,λm,ωm is a concave function with respect to λm and ωm; therefore, the optimal dual variable that can be obtained by the subgradient method is as follows:(23)λm∗=λm∗+εγ¯mkN0ep˜mkLm+xmD1⟶2+∑i≠mMlog1+γ¯mk1+αiep˜ikLimep˜mkLm−log11−ξm+,ωm∗=ωm∗+ϕγ¯mkN0αmep˜mkLm+xmD2⟶1+∑i≠mMlog1+γ¯mk1+αiep˜ikLimαmep˜mkLm−log11−ξm+,where ε and ϕ represent small enough steps and are positive.Since theLmP˜m,P˜−m,c,λm,ωm function is a concave function on λm and ωm, (15) iteratively converges to the optimal λm and ωm. Substituting the obtained optimal λm and ωm into (23), we derive it to obtain(24)∂Lmp˜m,p˜−m,c,λm,ωm∂p˜mk|λm∗,ωm∗=0.By solving (24), we can derive the optimal solution p˜mk∗ in (22). Then, by the exponential change of the pmk∗=expp˜mk∗, the optimal transmit power pmk∗ of D1 in the m-th D2D pair and the optimal transmit power αmpmk∗ of D2 can be obtained. Therefore, we can get the optimal transmit power of each D2D to the user.


## 5.2. Cellular User (Leader) Optimal Transmit Power Solution

In order to minimize the cost utility of cellular users, each cellular user needs to adaptively update the market interference price set to the user to the D2D based on the transmit power response of the userP˜m∗ (or Pm∗) by D2D. The subgame problem of the leader in the optimization problem (20) can be decomposed into two subproblems: by correcting the price factor ck, the optimal transmit power of each cellular user k can be obtained, and then the optimal ck is obtained. The solution of the optimal transmit power of the cellular user k can be obtained by the Lagrangian dual decomposition method.Therefore, for the subgame problem of the leader in (20), the corresponding Lagrangian function can be obtained:(25)Lkp˜k,p˜−k,ck,νk=ep˜k−ck∑m=1Mep˜mk∗Lmkhmk2+νkγ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mk∗Lmkep˜kLk−log11−ξk,where the dual variable vk represents the constraint in (20); the optimal solution for the Lagrangian dual function is minLkp˜k,p˜−k,ck,νk.Lmp˜m,p˜−m,c,λm,ωm is a concave function for vk. Similarly, we can use the gradient method to get the optimal dual variable:(26)νk∗=νk∗+θγ¯kN0ep˜kLk+∑m=1Mlogγ¯k1+αmep˜mk∗Lmkep˜kLk−log11−ξk+,where θ is a sufficiently large positive real number. The iteration in (26) will converge to the optimal vk∗. Thus, we can derive the optimal solution (p˜k∗) of the problem in (20) by solving the following problem. The solution of the problem is as follows.Then, we can get the transmit powerpmk∗=expp˜mk∗ of the cellular user k by the exponential change of the formula pk∗.Based on the analysis of the optimal transmit power of D2D to the user subgame and the cellular user subgame, it is discussed how the cellular user sets the optimal price according to the transmit power policy response of all D2D pairs.If them-th D2D interference price to the user is greater than the payment threshold, the D2D pair of users will stop renting the channel of the k-th cellular user. In other words, for each D2D pair user, if it is transmitted on the channel of the k-th cellular user, its cost utility cannot be greater than the maximum cost threshold Umk,maxep˜kk∗. Thus, the cost for the m-th D2D pair of users should satisfy the constraint:(27)1+αmep˜mk∗+ck1+αmep˜mk∗Lmkhmk2≤Umk,maxep˜mk∗.Therefore, we can get the constraint of the priceck, expressed as follows:(28)ck≤Zm=Umk,maxep˜mk∗−1+αmep˜mk∗1+αmep˜mk∗Lmkhmk2.As can be seen from (28), if the cellular subscriber wants the m-th D2D pair to communicate on its channel, the price set by the cellular subscriber k should not be higher than the upper limit price in (28). Therefore, the optimal interference price ck∗ is ck∗=Zm.According to the above analysis, the k-th cellular user will minimizeQKep˜k by paying the threshold vector Z = (Z1, Z2, …, ZM) and then select the appropriate overall price ck∗.In order to achieve Nash equalization and to converge to the optimal transmit power, we propose the following iterative power control algorithm.(i)
Step 1: lett = 0; for all cellular users k, initialize its transmit power pkt and price ckt; for all D2D pairs, initialize Pmt(ii)
Step 2: for each D2D user, calculate the price vector paid by the m-th D2D to all cellular users(iii)
Step 3: updateλk∗ and ωk∗ according to formula (23), respectively(iv)
Step 4: for a given cellular userpkt and ckt, each D2D pair user m is solved according to equation (24) to obtain Pm∗t+1. If ckt>ZKt, D2D terminates the channel using cellular user k for user m(v)
Step 5: for each cellular userk, update vk∗ according to formula (18)(vi)
Step 6: at a given D2D user transmit powerPm∗t+1, the cellular user k updates its transmit power pm∗t+1 by solving equation (19)(vii)
Step 7: according to formulas (28) and Z, the cellular user updates its price ck∗t+1(viii)
Step 9: lett = t + 1 repeat steps 2 to 8 until convergence


## 6. Simulation Results and Performance Analysis

This paper uses simulation to analyze the performance of the Stackelberg game power control. Assuming the radius of the cell is 500 meters, the maximum transmit power of the cellular user and D2D to the user isPmmax=24dBm and Pmmax=20dBm, the noise power N0=10−10 w, and the maximum distance of each cellular user from the base station is 300 meters, and the maximum communication distance of each D2D pair is 50 meters. For each D2D pair user, we set αm=−1; self-interference relief coefficient η1=η2=−100dB.Figure 2
Convergence of cellular users (a) and D2D to users (b).
(a)(b)Figure2 shows the interrupt performance of three cellular users and five D2D pairs at different target SINR values. The outage probability of the two users becomes larger as the target SINR threshold increases. However, the terminal probability of the proposed game scheme is significantly lower than the other two power control schemes, especially at the low SINR threshold. This is because the proposed scheme takes into account the user’s minimum SINR requirement and builds an optimization problem under the outage probability, which effectively reduces the probability of interruption of users in the network. The other two schemes do not consider the outage probability into the system optimization problem, so the interrupt performance is lower than the proposed scheme.Figure 3
Cellular user (a) and D2D outage probability to user (b).
(a)(b)Figure3 is a comparison of the three algorithms for cellular users and D2D throughput performance for users. As can be seen from Figure 4, as the signal-to-noise ratio increases, the total throughput of the user also increases. When the signal-to-noise ratio is low, the three algorithms can get better performance, and the performance of the proposed algorithm in this case is almost the same as other algorithms mainly because of the constraints. However, as the signal-to-noise ratio increases, the performance of the proposed scheme is improved and higher than the other two schemes. In addition, it can be seen that the performance of D2D to the user’s throughput performance is not significantly improved in the case of high signal-to-noise ratio. This is because the interference power to the cellular user is less than a certain value, so the transmission power of the D2D to the user is limited. However, compared with the other two schemes, D2D still has higher throughput for users.Figure 4
Cellular user and D2D throughput to users. (a) Cellular user. (b) D2D user.
(a)(b)Figure4 compares the three algorithms for the total cost utility of cellular users and D2D to users. As can be seen from Figure 4, as the background noise increases, the total cost utility of the user also increases. This is because, in the case of large background noise, the user only increases his own transmission power to improve the signal-to-interference ratio of the receiving end to meet the requirement of the outage probability, and the cost increases. The algorithm proposed in this paper can reduce the power cost of the network to a certain extent compared with other algorithms. In addition, it can be seen that the cost required by the greedy algorithm is the largest, because the algorithm causes the user to self-increase the transmission power to satisfy the performance in the network, and ignores the interference caused to other users. The details are shown in Figure 5.Figure 5
Total cost utility of cellular users and D2D to users. (a) Cellular user. (b) D2D user.
(a)(b)


## 7. Conclusions

This paper studies the problem of uplink power control in a full-duplex D2D communication network under interrupt constraints. In order to effectively control the users transmit power, this paper introduces the Stackelberg game power control algorithm, setting up cellular users and D2D users as leaders and followers, respectively. Cellular users and D2D users compete with each other to minimize the cost utility and a new utility function is proposed. Under the condition of channel interruption constraint and power constraint, the optimization problem is constructed and the Lagrange dual decomposition method is used to obtain the user’s transmitting power. Finally, the performance of the proposed game scheme is analyzed and the Nash equilibrium can be converging. Future work will focus on the construction of intelligent algorithm research for D2D competition in the case of multi-users and assigning solutions through user weights.



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*Source: 1786349-2020-11-11.xml*