# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 3 Energy Efficiency Analysis
### Coverage Probability
 \tag{10}\]  
where \((a)\) the interference generated by \(\Phi_{P}^{PPP}\) with the same density is used to replace the interference generated by \(\Phi_{P}\), and the lower bound of coverage probability is obtained, \((b)\) according to the probability generating function (PGF) [48] of the PPP network, \((c)\) the channel gain obeys Rayleigh fading, i.e., \(h_{mi}\sim\exp(1)\) and \(h_{pi}\sim\exp(\ 1)\), and (\(d\)) let \(T(\alpha,\theta)=1+\theta^{2/\pi}\int_{\theta^{-2/\alpha}}^{\infty}\left(1-(1+u^ {-\alpha/2})^{-1}\right)du\). By substituting Formulas (9) and (10) into Formula (8), the coverage probability \(P_{Cm}\) of the MBS in the two-tier HetNets can be obtained. \(\square\)  
**Theorem 2**. : _The coverage probability of the typical user serviced by the PBS in two-tier HetNets is given by:_  
\[P_{Cp}(\theta)=\frac{\lambda_{p}+\lambda_{m}(\frac{h_{pi}}{h_{p}})^{2/\alpha}} {\lambda_{m}(\frac{h_{pi}}{h_{p}})^{2/\alpha}T(\alpha,\theta)+\lambda_{p}T( \alpha,\frac{\theta}{C_{p}})} \tag{11}\]  
**Proof**. : \(P_{Cp}\) is denoted as the probability that the \(SIR_{p}\) provided by the service PBS for the typical user is greater than the threshold value \(\theta_{p}\), which can be expressed as:  
\[ \begin{array}{l}P_{Cp}(\theta)=P(SIR_{p}>\theta)=\int\limits_{r_{0}>0}P(h_{ p0}>\theta\ I_{2}|r_{0})f_{p}(r_{0})dr_{0}\\ =\int\limits_{r_{0}>0}E_{I_{2}}[e^{-\theta\ I_{2}}]f_{p}(r_{0})dr_{0}=\int \limits_{r_{0}>0}L_{I_{2}}(\theta)f_{p}(r_{0})dr_{0}\end{array}  \tag{12}\]  
where \(f_{p}(r_{0})\) denotes the distance distribution between the service PBS and the typical user. \[f_{p}(r_{0})=2\pi r_{0}(\lambda_{p}+\lambda_{m}(\mu_{m}/\mu_{p})^{2/\alpha}) \exp(-\lambda_{p}\pi r_{0}^{2}-\lambda_{m}\pi(\mu_{m}/\mu_{p})^{2/\alpha}r_{0} ^{2}) \tag{13}\]  
\[ \begin{array}{l}L_{I_{2}}(\theta)=E_{2}[e^{-\theta(\sum\limits_{r_{0}\in \Theta_{m}(\lambda_{p})}\frac{\rho_{m}}{\mu_{p}}h_{mi}|x_{0}|^{2}|x_{1}|^{- \alpha}+\sum\limits_{r_{0}\in\Theta_{p}(\lambda_{p})}h_{mi}|x_{1}|^{2}|x_{1}|^{ -\alpha}+\sum\limits_{r_{0}\in\Theta_{p}(\lambda_{p})}h_{mi}|x_{1}|^{2}|x_{1}| ^{-\alpha}]}\\ \stackrel{{(d)}}{{\sim}}E_{\Phi,\mu_{mi}}[\prod\limits_{x_{1}\in \Theta_{m}(\lambda_{p})}\exp(-\theta h_{mi}r_{0}^{2}|x_{1}|^{-\alpha})]\cdot E _{\Phi,\mu_{p},\mu_{p}}[\prod\limits_{x_{i}\in\Theta_{p}(\lambda_{p})}\exp(- \frac{\theta}{C_{p}}h_{pi}r_{0}^{2}|x_{1}|^{-\alpha})]\\ \stackrel{{(b)}}{{=}}\exp(-2\pi\lambda_{m}f_{0}^{\infty}\left(1-L _{h_{mi}}(\theta\cdot\frac{\theta_{m}}{\mu_{p}}r_{0}^{2}|x|^{-\alpha})\right) xdx)\cdot\exp(-2\pi\lambda_{p}f_{0}^{\infty}\left(1-L_{h_{pi}}(\frac{\theta}{C_{p}}r_{0}^{ 2}|y|^{-\alpha})\right)ydy)\\ \stackrel{{(c)}}{{=}}\exp(-\pi\lambda_{m}r_{0}^{2}(\frac{h_{pi}}{ \mu_{p}})^{2/\alpha}(T(\alpha,\theta)-1))\cdot\exp(-\pi\lambda_{p}r_{0}^{2}(T (\alpha,\frac{\theta}{C_{p}})-1))\end{array}  \tag{14}\]  
where (\(a\)) the MISR-based approximation method is used, i.e., to replace \(\theta\) with \(\frac{\theta}{C_{p}}\), and the distribution of user-received SIR in the network modeled by PHP can be accurately approximated by the PPP network, (\(b\)) according to the PGF of the PPP network, (\(c\)) the channel gain follows Rayleigh fading, i.e., \(h_{mi}\sim\exp(1)\), and \(h_{pi}\sim\exp(\ 1)\), and \(T(\alpha,\theta)=1+\theta^{2/\alpha}f_{\theta^{-2/\alpha}}^{\infty}\left(1- \left(1+u^{-\alpha/2}\right)^{-1}\right)du\), which can be derived. By substituting Formulas (13) and (14) into Formula (12), we obtained the coverage probability of the PBS in two-tier HetNets. In [49], it is assumed that the user adopts open-access mode, i.e., whereby the user is allowed to associate with the BS of any tier. The probability that a typical user is associated with a particular tier depends on the density of the BS and the transmit power of the BS.