# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 3 Energy Efficiency Analysis
### Coverage Probability
The MISR of a typical user is defined as:  
\[MISR=\mathrm{E}\bigg{\{}\frac{I}{\mathrm{E}_{h}(S)}\bigg{\}}=\mathrm{E}\Bigg{\{} \frac{\sum\limits_{x_{i}\in\Phi_{m}\backslash\{x_{0}\}}\mu_{m}h_{mi}|x_{i}|^{- \alpha}+\sum\limits_{x_{i}\in\Phi_{p}\backslash\{x_{0}\}}\mu_{p}h_{p}|x_{i}|^{ -\alpha}}{\mu_{k}h_{k0}|x_{0}|^{-\alpha}}\Bigg{\}} \tag{5}\]  
where \(E_{h}(s)\) is the average received signal power of a typical user and \(I\) is the sum of all interference powers. Based on this method, this section deduces the coverage probability of the PPP-PHP HetNets. The authors of [46] show that the calculation result of the MISR in the PPP network is \(MISR_{PPP}=2/(\alpha-2)\). Assuming that the density of the MBS and the PBS is constant, and the relationship between \(MISR_{PHP}\) and the path loss factor \(\alpha\) is obtained by simulation and data fitting, \(MISR_{PHP}=14.31\times\alpha^{-1.99}\)[47]. The gain factor \(G_{p}\) of the HetNets modeled by the PHP can be written as:  
\[G_{p}=\frac{MISR_{PPP}}{MISR_{PHP}}=\frac{2/(\alpha-2)}{1.43\times\alpha^{-1.99}} \tag{6}\]  
Next, we use the method based on the MISR to derive the coverage probability of the MBS and the PBS in two-tier HetNets, and then obtain the total coverage probability of the HetNets according to the user association probability. **Theorem 1**. : _The coverage probability of the typical user serviced by the MBS in two-tier HetNets is expressed as:_  
\[P_{Cm}(\theta)=\frac{\lambda_{m}+\lambda_{p}{(\frac{\mu_{p}}{\mu_{m}})}^{2/ \alpha}}{\lambda_{m}T(\alpha,\theta)+\lambda_{p}{(\frac{\mu_{p}}{\mu_{m}})}^{ 2/\alpha}T(\alpha,\theta)} \tag{7}\]  
**Proof. **\(P_{Cm}\) is denoted as the probability that the \(SIR_{m}\) provided by the service MBS for the typical user is greater than the threshold value \(\theta_{m}\), which can be expressed as:  
\[ \begin{array}{l}P_{Cm}(\theta)=P(SIR_{m}>\theta)=\int\limits_{r_{0}>0}P(h_{ m0}>\theta\;I_{1}|r_{0})f_{m}(r_{0})dr_{0}\\ =\int\limits_{r_{0}>0}E_{I_{1}}[e^{-\theta\;I_{1}}]f_{m}(r_{0})dr_{0}=\int \limits_{r_{0}>0}L_{I_{1}}(\theta)f_{m}(r_{0})dr_{0}\end{array}  \tag{8}\]  
where \(f_{m}(r_{0})\) denotes the distance distribution between the service MBS and the typical user. \[f_{m}(r_{0})=2\pi r_{0}(\lambda_{m}+\lambda_{p}{(\mu_{p}/\mu_{m})}^{2/\alpha} )\exp(-\lambda_{m}\pi r_{0}^{2}-\lambda_{p}\pi(\mu_{p}/\mu_{m})^{2/\alpha}r_{0 }^{2}) \tag{9}\]  
\[ \begin{array}{l}L_{I_{1}}(\theta)=E_{I_{1}}[e^{-\theta\;I_{1}}]=E_{[1}[e^{- \theta\;(\sum\limits_{x_{i}\in\Phi_{m}\backslash\{x_{0}\}}h_{mi}|x_{i}|^{- \alpha})+\sum\limits_{x_{i}\in\Phi_{p}\backslash\{x_{0}\}}\frac{\pi_{p}}{\mu_{ m}}\mu_{p}^{\alpha}|x_{i}|^{-\alpha}}]\\ \stackrel{{(a)}}{{\geq}}E_{\[PERSON]_{m},h_{mi}}[\prod\limits_{x_{i}\in \[PERSON]_{m}\backslash\{x_{0}\}}\exp(-\theta\cdot h_{mi}\mu_{0}^{\alpha}|x_{i}|^{ -\alpha})]\cdot E_{\[PERSON]_{p},h_{mi}}[\prod\limits_{x_{i}\in\[PERSON]_{p}\backslash\{ x_{0}\}}\exp(-\theta\frac{\pi_{p}}{\mu_{m}}h_{p}r_{0}^{\alpha}|x_{i}|^{- \alpha})]\\ \stackrel{{(b)}}{{=}}\exp(-2\pi\lambda_{m}\int_{r_{0}}^{\infty}(1 -L_{h_{mi}}(\theta\cdot\theta_{0}^{\alpha}|x|^{-\alpha}))xdx)\cdot\exp(-2\pi \lambda_{p}\int_{r_{0}}^{\infty}(1-L_{h_{mi}}(\frac{\pi_{p}}{\mu_{m}}r_{0}^{ \alpha}|y|^{-\alpha}))ydy)\\ \stackrel{{(c)}}{{=}}\exp(-2\pi\lambda_{m}\int_{r_{0}}^{\infty}( 1-\frac{1}{1+\theta\frac{\pi_{p}^{2}}{\mu_{m}^{\alpha}}\lambda x})xdx)\cdot\exp (-2\pi\lambda_{p}\int_{r_{0}}^{\infty}(1-\frac{1}{1+\theta\frac{\pi_{p}^{2}}{ \mu_{m}^{\alpha}}\lambda y|^{-\alpha}})ydy)\\ \stackrel{{(d)}}{{=}}\exp(-\pi\lambda_{m}r_{0}^{2}(T(\alpha,\theta )-1))\cdot\exp(-\pi\lambda_{p}r_{0}^{2}(\frac{\pi_{p}}{\mu_{m}})^{2/\alpha}( T(\alpha,\theta)-1))\end{array}