Elsevier

Ad Hoc Networks

Volume 133, 1 August 2022, 102891
Ad Hoc Networks

Power allocation in D2D enabled cellular network with probability constraints: A robust Stackelberg game approach

https://doi.org/10.1016/j.adhoc.2022.102891Get rights and content

Abstract

In this paper, we develop a Stackelberg game with a non-uniform pricing in a two-tier network environment, where multiple device-to-device (D2D) users reuse the uplink spectrum resources of the cellular user (CU). Assuming that the base station (BS) is protected by pricing the interference from D2D users, we propose a Stackelberg game framework where BS and D2D users are treated as the leader and followers respectively. Besides, the channel state information (CSI) uncertainty in the actual communication environment is considered to be suppressed the interference fluctuation effects. Specifically, the probability constraint is included to optimize the user utility. The complex probability constraint is transformed to the deterministic form of the convex second-order cone constraint by a novel method. Furthermore, a norm approximation method is proposed to reduce the computational complexity. Finally, the analytical solution of Nash equilibrium is acquired to generate the best response to the leader and all followers. Simulation results show that the proposed scheme is more effective comparing to the algorithms without the CSI probability constraint.

Introduction

The evolving fifth-generation (5G) communication system will be a highly heterogeneous network composed of macrocells and multiple small cells [1]. With the multi-tier communication architecture of 5G networks, multiple small cells are underlaid on the macrocells to improve the throughput of the system. Traditional multi-tier cellular networks are however congested by a large number of communicating devices when the data traffic demand rapidly increases. Therefore, advanced device-to-device (D2D) communication technologies have been developed. Since sharing cache files through D2D communication is a promising approach to alleviate the traffic pressure of cellular networks, the spectrum resources of the cellular network are shared by the cellular user (CU) and D2D users to transmit signals [2], [3]. However, D2D links generate several interferences in the communication system, when the cellular links are multiplexed. Therefore, it is necessary to manage the interference of D2D users. Besides, significant channel state information (CSI) feedback and data exchange overhead exist in cooperative D2D communication systems. Non-cooperative communication systems were developed to reduce the required CSI. It is a challenging issue to determine the equilibrium point where the utility of all users are maximized, especially in a scenario with imperfect CSI. The game theory can be used to develop strategies that enable independent and competing players to make optimal decisions in a non-cooperative competitive situation. Therefore, game theory has been applied in D2D communication. Stackelberg game is adopted since it matches the commonly used two-tier network system. CU in the upper cellular network is treated as the leader and the D2D users in the lower network are treated as followers. In the utility function of game players, the pricing of interference is treated as the restriction between the leader and followers. Solving the optimization problem attempts to determine the optimal strategy combination of the participants. Furthermore, we propose a robust Stackelberg game approach to realize effective interference management and maximize the utility of all users.

To manage the interference effectively, some power control strategies coordinate the inter-layer and cross-layer interference by using the full instantaneous CSI. Kim et al. [4] introduced the concept of D2D belt to ensure the orthogonality between cellular link and D2D link in the frequency domain. The interference between D2D links is eliminated by assigning a non-overlapping subframe to D2D links. Kuruvatti et al. [5] proposed three power control schemes by integrating the interference aware power control, blind power control, and threshold-based power control. The power control scheme can be used to improve the throughput of the base station and ensure the performance and effectiveness of the D2D performance. Based on cooperative D2D communication, the power control methods are developed in the two-tier networks. Alghamdi [6] presented a design on clustering and safety message dissemination for 5G-based V2X and D2D communication, which is focused on the QoS improvement of D2D users. Lee et al. [7] treated the cooperative D2D users as relays of CU, and used the optimal spectrum and power allocation to maximize the total average achievable rate. Sun et al. [8] proposed an enhanced Stackelberg game model which is involved a single leader and multiple followers. The model uses the discount factor to tackle the quality of service (QoS) constraints of both CU and D2D users. An algorithm for user association, resource blocks allocation and power control was developed when the energy harvesting relays are considered in D2D relay assisted network [9]. In this approach, A centralized solution was developed by the time sharing strategy and a distributed low complexity solution was developed by the stable matching theory.

Since cooperative D2D communication systems led to a large amount of data exchange overhead, the game theory can be used to make optimal decisions in a non-cooperative competitive situation. Therefore, some existing works have adopted game theory to optimize wireless networks. A game-based approach is proposed to solved inter-layer interference problem, which considered both the interference suppression and power allocation at the same time [10]. Zhang et al. [11] proposed a two-level ordered Stackelberg game to describe the asymmetric competition between CUs and D2D pairs. Shi et al. [12] proposed a Stackelberg model which uses the incentive mechanism to price the contribution of D2D transmitters. The Stackelberg model attempts to solve the conflict of interests between the operators and D2D transmitters; the model also maximize the profit by optimizing the caching strategy and incentive price. To encourage users to cooperate, Yu et al. [13] proposed a power allocation scheme based on the prices. The Stackelberg game attempts to determine the final price and power allocation strategy through the dynamic bidding process. Kang et al. [14], [15] proposed a pricing-based Stackelberg game to maximize the utilities of both leader and followers, while the maximum tolerated interference power is considered as a constraint. Yuan et al. [16] proposed a distributed channel power allocation scheme to solve the problem; the scheme increases the cost of acquiring global CSI when the network scale increases. Ma et al. [17] derived a closed-form solution to minimize the power of D2D and cellular links, and the closed-form solution also satisfies different QoS constraints.

However, none of the aforementioned methods have taken the dynamic characteristics of CSI into account. The changing and uncertain CSI is inevitable in the real world, and the assumption of perfect CSI is impractical [18], [19]. To determine the power control decisions with statistical CSI, the system performances of D2D users and cellular networks are analyzed, when the outage probability is included. To solve the optimization problem for power and mode selection in imperfect CSI, Li et al. [20] derived an approximate closed-form expression to maximize the sum ergodic capacity for all imperfect CSI. A lower bound-based power control method is proposed to reduce the complexity. Gorantla and Mehta [21] proposed a scheme to feedback the state information of D2D pairs of each subchannel in a multi-cell scenario, which attempts to ensure that the outage probability of D2D user is kept lower than the predetermined threshold. Kuang et al. [22] proposed a tractable convex optimization problem to maximize the average energy efficiency of all D2D links, and to satisfy both the constraints of the quality of service of CUEs and the energy harvesting of the D2D links. Zhang et al. [23] proposed k-regular and k-connected(KK) structure against multifaults in ultra-high capacity optical networks to improve spare spectrum resources efficiency. Chung et al. [24] presented an innovative algorithm to minimize total transmission power consumption while satisfying minimum data-rate requirements of all users. Shen et al. [25] devised a distributed optimization method to determine the worst-case robust beamforming solutions. Compared with the aforementioned methods, the analytical form of Stackelberg Equilibrium is calculated in this paper, which reduces the complexity of the algorithm effectively.

In this paper, a power control optimization problem is formulated to minimize the channel estimation error at the D2D receiver, where both the dynamic and fast fading communication environment is considered. This optimization problem is formulated using the Stackelberg game. The optimization problem is also formulated with the probability constraints, in order to ensure that the defect of imperfect CSI does not exist.

The contributions of this paper are summarized as follows:

  • To analyze the Stackelberg game, the probability constraint is formulated on the optimization problem by integrating the interference fluctuation which is caused by the CSI uncertainty. Therefore, the optimization problem is more realistic since this probability constraint is included. Simulation results show the superiority of the proposed algorithm when different conditions are considered.

  • Since the optimization problem with the uncertain probability constraints is nonconvex, we transform the probability constraint as the deterministic constraint which is equivalent to the convex second cone. We also propose a norm approximation scheme to reduce the complexity of the optimization problem.

The rest of this paper is organized as follows, a two-tier network model is established in Section 2. In Section 3 the Stackelberg game is developed. In Section 4, the Nash equilibrium analytical solution is deduced for the Stackelberg game. An interference power control algorithm is proposed based on the probability constraint. Section 6 presents the simulation results and the analysis of system performance. Finally, a conclusion is drawn in Section 7.

Notation: In this paper, vectors are in bold lowercase letters, and matrices are in bold capital letters. Some notations are given in Table 1.

Section snippets

System model

In this paper, we consider a two-tier network with a central base station (BS) which has a service range of R for CU and N D2D pairs. CU often needs to download large files from BS, such as high-resolution videos and pictures, but rarely upload large files. Generally, the uplink of CU is only used to send the information of content request. Hence the network data transmission is less and stabler. Besides, multiple D2D users reuse the spectrum resources of CU at the same time. If downlink is

Problem formulation

In this section, we first present the Stackelberg game of the proposed power allocation scheme based on pricing. Then, the Stackelberg equilibrium of the game is investigated.

Reformulation of probability constraints

Since the multiple random variables gi,j, g0,i, and gi,0 are formulated in (3), (7), these variables are converted as the deterministic expressions to solve this optimization problem. Therefore, all the random variables in the objective function are set to the expected form.

In the practical fast fading communication, the channel gain is generally in the Rayleigh distribution. The instantaneous CSI gi,j (i,j{0,1,2,,n}) is modeled as gi,j=(di,jα)φi,j,i=j,(di,jβ)φi,j,ij,where di,j is the

Algorithm construction

An iterative game solving process algorithm is proposed to determine the best response of the leader and followers. The Stackelberg game is described as follows.

Step1: The price strategy of BS and power strategies of D2D users are initialized. The initial price vector is μ(0) and the initial power vector is p(0). The initial price is required to satisfy the condition (20), μ(0)i<Wḡi,iḡ0,iIi.

Step2: D2D users update their powers based on the price set by the BS. For D2D users, the total

Simulation results and analysis

In this section, we evaluate the performance of the proposed Stackelberg game algorithm. We compare the performance of the proposed algorithm with other methods. The superiority of RSGA can be validated. Finally, the randomness of the actual channel gain is considered. By comparing the actual outage probability, the feasibility of the proposed scheme can be demonstrated. Table 2 shows the simulation parameters.

Fig. 2, Fig. 3 show the convergence performance of the proposed Stackelberg game

Conclusion

In this paper, the power control strategies of the D2D network are proposed based on the Stackelberg game theory. The probabilistic interference power constraint is formulated by including the uncertainty of channel gain in the actual communication environment. The QoS of the CU user can be satisfied in the uplink. The utility function of the BS and D2D users is formulated based on the Stackelberg game model. The optimal resource allocation scheme is integrated on the power allocation

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is partly supported by National Natural Science Foundation of China under grant 61873223, 61803328, the Natural Science Foundation of Hebei Province under grant F2019203095, and the Beijing Natural Science Foundation under Grant L201002.

Zhixin Liu received his B.S., M.S., and Ph.D. degrees in control theory and engineering from Yanshan University, Qinhuangdao, China, in 2000, 2003, and 2006, respectively. He is currently a professor with the Department of Automation, School of Electrical Engineering, Yanshan University, China. He visited the University of Alberta, Edmonton, AB, Canada, from 2009 to 2010. He is the author or coauthor of more than 80 papers in technical journals and conference proceedings. His current research

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  • Cited by (0)

    Zhixin Liu received his B.S., M.S., and Ph.D. degrees in control theory and engineering from Yanshan University, Qinhuangdao, China, in 2000, 2003, and 2006, respectively. He is currently a professor with the Department of Automation, School of Electrical Engineering, Yanshan University, China. He visited the University of Alberta, Edmonton, AB, Canada, from 2009 to 2010. He is the author or coauthor of more than 80 papers in technical journals and conference proceedings. His current research interests include performance optimization and energy-efficient protocol design in wireless sensor networks, wireless resource allocation in cognitive radio networks and Femtocell networks.

    Zijian Liu is currently working toward the M.S.degree in the School of Electrical Engineering, Yanshan University, China. His current research interests include wireless resources allocation, performance optimization of D2D communication networks.

    Yuan’ai Xie received the B.S. degree in automation from the North China University of Science and Technology, Tangshan, China, in 2016. He is currently working toward the Ph.D. degree in control science and engineering with Yanshan University, Qinhuangdao, China. His current research interests include wireless resource optimization, vehicular network and D2D communication.

    Kit Yan Chan is a Senior Lecturer in the Department of Electrical and Computer Engineering, Curtin University, Australia. He received his Ph.D. in Computing in London South Bank University, U.K., in 2006. He was a full time researcher in The Hong Kong Polytechnic University (2004–2009) and Curtin University (2009–2013). He was the guest editor for IEEE Transactions Industrial Informatics, Applied Soft Computing, Neurocomputing, Engineering Applications of Artificial Intelligence. His research interests include machine learning, pattern recognition and algorithm design, optimization for communication systems.

    Yazhou Yuan received his B.S., M.S. degrees in control science and engineering from Yanshan University, Qinhuangdao, China, in 2009, 2012, and Ph.D degree in Control Theory and Engineering from Shanghai Jiaotong University in 2016. He is now an associate professor in the School of Electrical Engineering, Yanshan University, China. His research interests include resources allocation in wireless networks, industrial internet of things.

    Yi Yang received her B.S., M.S. degrees in management science and engineering from Yanshan University, Qinhuangdao, China, in 2006, 2009, respectively. She is now an assistant researcher with the School of Electrical Engineering, Yanshan University, China. Her research interests include optimization theory and applications.

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