# Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks
## 1 Introduction
As an alternative, wireless energy transfer technology allows low-power nodes to harvest energy from their received radio frequency (RF) signals to recharge their batteries and prolong their lifetimes [14]. In [15], three policies of wireless power transfer are proposed to guarantee secure communications in large scale cognitive cellular networks. Considering direct communication in underlying downlink cellular networks, the harvested energy in the RF is exploited to prolong the lifetime of nodes link and obtain the maximum the sum-rate by joint resource block and power allocation [16]. In [17], the authors analyze the outage probability and the corresponding optimal offloading bias in an EH HetNet which can provide energy for communication. The optimal EE of cellular communication and device-to-device (D2D) communication hybrid network is investigated by jointly the time allocation, spectrum allocation and the power control when D2D transmitters obtain energy by wireless power transfer [18]. The second key problem is high energy efficiency transmission schemes in WPCNs. Compared to the natural renewable sources available for EH technology, BS using WPT can offer a more controllable and relatively stable energy source [19; 20; 21; 22; 23]. In [19], the authors studied the switching between acting as an information relay and an energy harvesting node. In [20], the authors focus on designing appropriate transmission policies to improve the global EE in sensor networks with simultaneous wireless information and power transfer (SWIPT). An optimal maximum throughput approach for energy beamforming, receive beamforming, and time-slot allocation jointly optimization is proposed in [21]. The Ginibre model is adopted in [22] to analyze the performance of self-sustainable communications over cellular networks considering the RF energy harvesting rate and the energy outage probability. A unified framework is proposed in [23] to investigate the impact of SWIPT on the system performance with both time splitting and power splitting schemes. However, the optimal tradeoff between receiving information and harvesting energy from BSs in WPCNs has not been reported yet. With the help of WPT technology, transmitting information and harvesting energy can be added to the new fairness criterion. Another fundamental issue of multi-tier heterogeneous WPCNs is to associate a sensor with a particular serving BS. In practical systems, the received power based user association rule is the most commonly adopted one [24], where a sensor will choose to associate with the specific BS providing the maximum received signal strength (max-RSS). This user association policy is made according to the quality of service (QoS) requirements of the devices with the goal of maximizing the capacity of communication. As far as the problem formulation is to maximize the EE of networks, the max-RSS user association decisions may not be optimal. Considering that BSs in macrocells have a significantly higher transmit power than those in small cells, the access network energy consumption is typically higher when a user is associated with a macrocell. On the other hand, resource allocation with the consideration of fairness has become an important issue in communication networks [25]. For a traditional user fairness problem, each user should be allocated with a certain amount of radio resources via a careful design of scheduling. In multi-tier heterogeneous WPCNs, the fairness problem arises not only in scheduling within a traditional cell but also in the user association decision among BSs in different tiers. Thus, the optimal energy efficiency proportional fairness of multi-tier heterogeneous WPCNs by adjusting the user association policy should also be investigated. To address the above issues in this paper, we propose a framework for modeling and evaluating the downlink energy efficiency proportional fairness of multi-tier heterogeneous WPCNs. Our model takes into account the information transmission for active nodes and WPT for inactive nodes at BSs.