# Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks
## 5 Numerical Simulations
### Effect of the Transmit Power of BSs  
In Figure 3, we plot the energy efficiency proportional fairness of total two-tier heterogeneous WPCNs with respect to the transmit power of BSs in tier 2 for different values of the tier-2 cell association bias \(B_{2}\). We can see that as the transmit power of BSs in tier 2 changes from 1 dBm to 7 dBm, the maximal energy efficiency proportional fairness decreases. This is because, as the transmit power of tier 2 increases, the energy efficiency proportional fairness of tier 2 decreases according to (7). Therefore, the total energy efficiency proportional fairness of network decreases with the increasing transmit power of BSs in tier 2. Meanwhile, due to the transmit power of BSs in tier 2 being less than that of BSs in tier 1, the energy efficiency proportional fairness of the network increases as the bias of tier 2 increases. For comparison, Figure 3 also shows the energy efficiency proportional fairness of another fixed threshold scheme. For this baseline scheme, we adopt a fixed threshold and small enough \(\theta=0.05\). Therefore, most of the nodes are in the heterogeneous WPCNs transmit information and the harvest energy from the wireless power transfer is small. As can be observed, the energy efficiency proportional fairness of the system with a fixed threshold is substantially lower than those of the proposed optimal scheme. It can be seen that from the energy efficiency of the network, all of the nodes associating with network is not the best strategy since the power consumption of BSs is large. We can choose optimal threshold to let some nodes harvest energy temporarily. In the current HetNet, a lot of low-power BSs is deployed around high-power BSs. The deployment of the network can be described as the scenario that the density of BSs in tier 2 is larger than that of BSs in tier 1 and the transmit power of BSs in tier 2 is less than that of BSs in tier 1. In this case, we can obtain from the Figure 3 that for improving the energy efficiency proportional fairness of total network, when the transmit power of BSs in tier 2 is high, the bias factor of BSs in tier 2 should be large. This is because if the low-power BSs can serve more nodes, the high-power BSs  
Figure 3: Energy efficiency proportional fairness of total two-tier heterogeneous WPCNs with respect to the transmit power of BSs in tier 2 for different values of the tier-2 cell association bias B2. should offload some traffic to low-power BSs to enhance the energy efficiency proportional fairness of total network and balance the traffic in different tiers. For different values of the transmit power of BSs, Figure 4 contains the mean energy efficiency proportional fairness with respect to the density of BSs in tier 1 obtained from the proposed suboptimal scheme and optimal brute-force two-dimension search scheme. We can see that as the transmit power of BSs in tier 2 changes from 0 dBm to 7 dBm, the maximal energy efficiency proportional fairness firstly increases and then decreases. This is because if the transmit power of tier 2 is small, it will serve little nodes and a lot of nodes associated to the BSs in tier 1. Due to the BSs in tier 1 having larger transmit power than BSs in tier 2, the energy efficiency proportional fairness of the network is small. Therefore, the total energy efficiency proportional fairness of network firstly increases with the increase of the transmit power of BSs in tier 2. Meanwhile, when the transmit power of BSs in tier 2 is large enough, nodes associated with BSs in tier 2 cannot obtain higher energy efficiency. In this case, the energy efficiency proportional fairness of the network decreases as the transmit power of BSs in tier 2 increases. As the transmit power of BSs in tier 2 changes from 0 dBm to 7 dBm, the energy efficiency proportional fairness of nodes become a bell-shaped curve when the transmit power of BSs in tier 2 increases.