# Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks
## 3 Performance Analysis for Downlink WPCN  
### Coverage Probability  
In this subsection, we will now compute the mean coverage probability of the typical user, which is defined as \(\mathbb{E}\left[\log\left(C_{k}\right)\right]\). When the typical node associates with cell \(C_{i}\) in tier \(k\), the path loss between BS in cell \(C_{i}\) and the typical node is \(L_{kj}=l\), the cumulative distribution function (CDF) of \(SIR\) of the typical node can be obtained as  
\[ \begin{array}{rcl}\mathbb{E}\left[C_{k}\right]&=&\mathbb{E}_{\Phi,h}\left[ \mathbb{P}\left(SIR_{k}>\theta_{k}\right)\right]\\ &\stackrel{{(a)}}{{=}}&\mathbb{E}_{\Phi,h}\left[\mathbb{P}\left( \frac{P_{t,k}h_{k,l}l^{-\alpha}}{\sum_{j=1}^{K}\sum_{y\in\Phi_{j}\setminus i}P _{t,j}h_{j,y}l^{-\alpha}_{j,y}}>\theta\right)\right]\\ &=&\mathbb{E}_{\Phi,h}\left[\mathbb{P}\left(h_{k,i}>\theta l^{\alpha}P_{t,k}^{ -1}\left(\sum_{j=1}^{K}\sum_{y\in\Phi_{j}\setminus i}P_{t,j}h_{j,y}l^{-\alpha} _{j,y}\right)\right)\right]\\ &\stackrel{{(b)}}{{=}}&\mathbb{E}_{\Phi,h}\left[\exp\left(- \theta l^{\alpha}P_{t,k}^{-1}\left(\sum_{j=1}^{K}\sum_{y\in\Phi_{j}\setminus i }P_{t,j}h_{j,y}l^{-\alpha}_{j,y}\right)\right)\right]\\ &\stackrel{{(c)}}{{=}}&\mathbb{E}_{\Phi,h}\left[\prod_{j=1}^{K}L _{lj}\left(\theta l^{\alpha}P_{t,k}^{-1}\right)\right],\end{array}  \tag{8}\]  
where \(L_{lj}\left(s\right)\) is the Laplace transform of \(I_{j}=\sum_{y\in\Phi_{j}\setminus i}P_{t,j}h_{j,y}l^{-\alpha}_{j,y}\). (a) is derived from the expression of \(SIR\) in (3); (b) is due to \(h_{x}\sim\exp\left(1\right)\) is exponentially distributed with unit mean power; (c) is from the Campbell theorem for PPP. The Laplace transform of the total interference power from the BSs in tier \(j\) can calculated as  
\[ \begin{array}{l}\mathbb{E}_{\Phi,h}\left[\prod_{j=1}^{K}L_{lj}\left(\theta l ^{\alpha}P_{t,k}^{-1}\right)\right]\\ =\mathbb{E}_{\Phi,h}\left[\exp\left(\theta l^{\alpha}P_{t,k}^{-1}P_{t,j}\sum_{ y\in\Phi_{j}\setminus i}h_{j,y}l^{-\alpha}_{j,y}\right)\right]\\ =\mathbb{E}_{h}\left[\exp\left\{2\pi\lambda_{j}\theta l^{\alpha}P_{t,k}^{-1}P_{ t,j}\int_{\left(\frac{P_{t,j}B_{j}}{P_{t,k}B_{k}}\right)}^{1/\alpha}_{l}\left[1-L_{ l_{lj}}\left(y^{-\alpha}\right)\right]ydy\right\}\right]\\ \stackrel{{(d)}}{{=}}\exp\left\{2\pi\lambda_{j}\theta l^{\alpha}P_ {t,k}^{-1}P_{t,j}\int_{\left(\frac{P_{t,j}B_{j}}{P_{t,k}B_{k}}\right)}^{1/ \alpha}_{l}\left(1-\frac{1}{1+y^{-\alpha}}\right)ydy\right\}\\ \leq\exp\left[2\pi\lambda_{j}\theta l^{\alpha}P_{t,k}^{-1}P_{t,j}\int_{\left( \frac{P_{t,j}B_{j}}{P_{t,k}B_{k}}\right)}^{\infty}_{l}\left(\frac{P_{t,j}B_{j} }{P_{t,k}B_{k}}\right)^{1/\alpha}_{l}y^{-\alpha}ydy\right]\\ =\exp\left(-\frac{2\pi\lambda_{j}\theta l^{2}}{\alpha-2}\left(\frac{P_{t,j}}{P _{t,k}}\right)^{2/\alpha}\left(\frac{B_{j}}{B_{k}}\right)^{2/\alpha-1}\right), \end{array}  \tag{9}\]  
(d) is derived from associated rule that when the node is associated with the \(k\)-th BS tier, the length of interfering links and that of the serving link has the following relationship \(\left|l_{j,i}\right|\geq\left|l_{k,0}\right|\left(\frac{P_{t,j}B_{j}}{P_{t,k}B _{k}}\right)^{1/\alpha}\), for any \(i\) and \(j\). Considering \(\log\left(x\right)\) is a concave function, we can obtain from the Jensen inequality and (9)  
\[ \begin{array}{rcl}\mathbb{E}\left[\log\left(C_{k}\right)\right]&\leq&\log \mathbb{E}\left[C_{k}\right]\\ &=&-\frac{2\pi\lambda_{j}\theta l^{2}}{\alpha-2}\left(\frac{P_{t,j}}{P_{t,k}} \right)^{2/\alpha}\left(\frac{B_{j}}{B_{k}}\right)^{2/\alpha-1}\\ &\stackrel{{(e)}}{{=}}&-\frac{2\pi\theta_{k}l^{2}P_{k}}{\alpha-2} \frac{\lambda_{k}^{\frac{\alpha}{2}+1}}{A_{k}^{\frac{\alpha}{2}+1}}\sum_{j=1}^ {K}\frac{A_{j}^{\alpha+1}}{P_{j}\lambda_{j}^{\frac{\alpha}{2}}},\end{array}  \tag{10}\]  
(e) is derived from the relationship described in (1). It should be noted that the mean logarithm of coverage probability of the typical user is affected by the receive threshold and user association probability.