# Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks
## 4 Utility Optimization
### Property of the Problem  
The object function of Problem (20) is also non-convex, we will investigate the properties of the problem in this subsection. We can solve the considered problem according to the following Theorems. Theorem 1: _For any tier in the downlink multi-tier heterogeneous WPCNs, there exists an optimal receive threshold \(\theta_{k}^{*}\) for maximizing the energy efficiency proportional fairness utility function in tier \(k\)._  
Proof: According to (19), it can be easily proved that  
\[\frac{\partial^{2}\log\left(\log\left(1+\theta_{k}\right)\right)}{\partial \theta_{k}^{2}}=-\left[\frac{1}{\log^{2}\left(1+\theta_{k}\right)}+\frac{1}{ \log\left(1+\theta_{k}\right)}\right]\frac{1}{\left(1+\theta_{k}\right)^{2}} <0. \tag{21}\]  
Hence, \(\log\left(\log\left(1+\theta_{k}\right)\right)\) is a concave function of \(\theta_{k}\). Meanwhile, we can obtain that \(\log\left[\mathbb{E}\left(C_{k}\right)\right]\) is an affine function of \(\theta_{k}\) from (10). Substituting (10) and (15) into (19) yields  
\[\log\left(P_{t,k}+P_{s,k}-\eta\left(1-\mathbb{E}\left(C_{k}\right) \right)P_{EH,k}\right) \tag{22}\] \[=\log\left(P_{t,k}+P_{s,k}-\eta\left(1-\exp\left(-\frac{2\pi \theta_{k}l^{2}P_{k}}{\alpha-2}\frac{\lambda_{k}^{\frac{\delta}{2}+1}}{A_{k}^{ \frac{\delta}{2}+1}}\sum_{j=1}^{K}\frac{A_{j}^{\alpha+1}}{P_{j}\lambda_{j}^{ \frac{\delta}{2}}}\right)\right)\sum_{j=1}^{K}\frac{2\pi^{2}\lambda_{j}P_{t,j} ^{2/\alpha}}{\alpha\sin\left(2\pi/\alpha\right)}\right),\]  
is a convex function of \(\theta_{k}\). Therefore, \(U_{EE,k}\left(\theta_{k}\right)\) is a concave function for \(\theta_{k}\). The optimal receive threshold can be obtained via taking the first order derivative of \(U_{EE}\) with respect to \(\theta_{k}\) as  
\[\frac{\partial U_{EE}(\theta_{k})}{\partial\theta_{k}}=0 \tag{23}\]  
Hence, we could get the optimal receive threshold by the expression as follows  
\[\bar{P}+\beta\exp\left(-h\left(A_{k}\right)\theta_{k}\right)=\bar{P}h\left(A_ {k}\right)\left(1+\theta_{k}\right)\log\left(1+\theta_{k}\right), \tag{24}\]  
where \(\bar{P}=P_{t,k}+P_{s,k}-\eta\sum_{j=1}^{K}\frac{2\pi^{2}\lambda_{j}p_{j}^{ \frac{\delta}{2}}}{\alpha\sin\left(\frac{2\pi}{\alpha}\right)}\) which is independent of \(A_{k}\), \(\beta=\eta\sum_{j=1}^{K}\frac{2\pi^{2}\lambda_{j}p_{j}^{\frac{\delta}{2}}}{ \alpha\sin\left(\frac{2\pi}{\alpha}\right)}\), and \(h(A_{k})\) can be obtained in the Appendix A.  
Theorem 1 shows that if in any tier where the user association probability of nodes \(A_{k}\) is fixed, we can find an optimal receive threshold \(\theta_{k}^{*}\) to obtain the tradeoff of nodes between receiving information and harvesting energy. Note that the receive threshold includes two special cases: (i) when \(\theta_{k}=0\), the receive threshold is so low that all nodes can receive information regardless of the QoS requirement from information transmission; (ii) when \(\theta_{k}\rightarrow\infty\), the receive threshold is so high that all nodes should harvest energy regardless of the distance between nodes and BSs. From (13), we can obtain the optimal receive threshold \(\theta_{k}^{*}>0\). Therefore, the optimal receive threshold \(\theta_{k}^{*}\) brings to the balance for node to receive information and harvest energy. To reveal the relationship between energy efficiency proportional fairness utility function and the receive threshold in a two-tier heterogeneous WPCNs, we show some numerical results for different transmit powers and densities of BSs in two-tier heterogeneous WPCNs in Figure 2. Numerical and analytical results show that there exists an optimal threshold \(\theta^{*}\) for maximizing the utility function in each tier. We can obtain the relationship between optimal receive threshold \(\theta_{k}^{*}\) and the user association probability \(A_{k}\) as the following lemma.