# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 3 Energy Efficiency Analysis
### Average Achievable Rate  
Suppose \(\tau_{m}\) and \(\tau_{p}\) are used to denote the average achievable rate of the typical user associated with the MBS and the PBS and expressed as follows, respectively:  
\[\tau_{m}=\mathrm{E}[\log(1+SIR_{m})] \tag{18}\]  
\[\tau_{p}=\mathrm{E}[\log(1+SIR_{p})] \tag{19}\]  
**Theorem 3. **_The average achievable rate of the typical user associated with the MBS is expressed as:_  
\[\tau_{m}=\int_{0}^{\infty}\frac{1}{z}(1-e^{-z})\frac{1}{H(z)}dz \tag{20}\]  
**Proof. ** The deployment of the MBS is modeled by the PPP, and the average achievable rate of the typical user associated with the MBS is written as:  
\[ \begin{array}{l}\tau_{m}^{PPP}=\mathrm{E}(\log(1+SIR_{m})]\\ =\int_{0}^{\infty}f_{m}(\tau_{0})E[\log(1+\frac{\mu_{m}n_{m0}\tau_{0}^{-a}}{x_ {i}\in\Phi_{m}(\{x_{0}\})}\mu_{m}x_{i}^{-a}+\sum\limits_{x_{i}\in\Phi_{p}(\{x_ {0}\})}\mu_{p}\mu_{i}x_{i}^{-a})]dr_{0}\end{array}  \tag{21}\]  
where:  
\[ \begin{array}{l}E[\log(1+\frac{\mu_{m}n_{m0}\tau_{0}^{-a}}{x_{i}\in\[PERSON]_{m}( \{x_{0}\})}\mu_{m}x_{i}^{-a}+\sum\limits_{x_{i}\in\[PERSON]_{p}(\{x_{0}\})}\mu_{p} \mu_{i}x_{i}]^{-a})]\\ =E_{\[PERSON]_{m},\[PERSON]_{p}}[\log(1+\frac{\mu_{m}\tau_{0}^{-a}}{x_{i}\in\[PERSON]_{m}( \{x_{0}\})}\mu_{m}|x_{i}|^{-a}+\sum\limits_{x_{i}\in\[PERSON]_{p}(\{x_{0}\})}\mu_{p} \mu_{i}|x_{i}]^{-a})]\\ =E_{\[PERSON]_{m},\[PERSON]_{p}}[\log(1+\frac{1}{\sum\limits_{x_{i}\in\[PERSON]_{m}( \{x_{0}\})}\mu_{m}|x_{i}|^{-a}+\sum\limits_{x_{i}\in\[PERSON]_{p}(\{x_{0}\})}\mu_{p} \mu_{i}|x_{i}]^{-a}})]\\ =E_{\[PERSON]_{m},\[PERSON]_{p}}[\log(1+\frac{1}{\sum\limits_{x_{i}\in\[PERSON]_{m}( \{x_{0}\})}\tau_{0}^{a}|x_{i}|^{-a}+\sum\limits_{x_{i}\in\[PERSON]_{p}(\{x_{0}\})} \frac{p}{\mu_{m}}\tau_{0}^{a}|x_{i}|^{-a}})]\end{array}  \tag{22}\]  
Let \(I_{m}=\sum\limits_{x_{i}\in\[PERSON]_{m}(\{x_{0}\})}r_{0}^{a}|x_{i}|^{-a}+\sum \limits_{x_{i}\in\[PERSON]_{p}(\{x_{0}\})}\frac{l_{P}}{\mu_{m}}r_{0}^{a}|x_{i}|^{-a}\); according to Lemma 1 in reference [50], Formula (21) can be written as:  
\[ \begin{array}{l}E_{\Phi_{m},\Phi_{p}}[\log(1+\frac{1}{I_{m}})]\\ =\int_{0}^{\infty}\frac{1}{z}e^{-z}[E[e^{-z(I_{m}-1)}]-E[e^{-zI_{m}}]]dz\\ =\int_{0}^{\infty}\frac{1-e^{-z}}{z}E[e^{-zI_{m}}]dz\end{array}  \tag{23}\]  
where:  
\[ \begin{array}{l}E\;[e^{-zI_{m}}]=E[e^{-z\sum\limits_{x_{i}\in\Phi_{m}(x_{0})} \tau_{0}^{a}|x_{i}|^{-a}+\sum\limits_{x_{i}\in\Phi_{p}(\{x_{0}\})}\frac{p_{P}} {\mu_{m}}\tau_{0}^{a}|x_{i}|^{-a}}\\ \stackrel{{(a)}}{{\geq}}E\;[\prod\limits_{x_{i}\in\Phi_{m}(\{x_{0} \})}e^{-z\tau_{0}^{a}|x_{i}|^{-a}}]\cdot E\;[\prod\limits_{x_{i}\in\{\Phi_{p}( \{x_{0}\})\}ppP}e^{-z\frac{p_{P}}{\mu_{m}}\tau_{0}^{a}|x_{i}|^{-a}}]\\ =\exp(-2\pi\lambda_{m}\int_{\tau_{0}}^{\infty}(1-e^{-zx^{-a}}\tau_{0}^{a})dx) \cdot\exp(-2\pi\lambda_{p}\int_{\tau_{0}}^{\infty}(1-e^{-z\frac{p_{P}}{\mu_{m}} \tau_{0}^{a})}ydy)\end{array}  \tag{24}\]where (_a_) the interference generated by \(\Phi_{p}^{PPP}\) with the same density is used to replace the interference generated by \(\Phi_{P}^{PHP}\) and the lower bound of the achievable rate \(\tau_{m}\) is obtained. After a series of algebraic operations, we obtain:  
\[E\ [e^{-zI_{m}}]=\exp(-\pi\lambda_{m}r_{0}^{2}(H(z)-1))\cdot\exp(-\pi\lambda_{p} r_{0}^{2}(\frac{\mu_{p}}{\mu_{m}})^{\frac{2}{2}}(H(z)-1)) \tag{25}\]  
where:  
\[H(z)=1+z^{\frac{2}{2}}\int_{z^{-2/\alpha}}^{\infty}1-e^{-v^{-\alpha/2}}dv \tag{26}\]  
By substituting Formulas (22)-(26) into Formula (21), we obtain the expression of \(\tau_{m}\). Next, the average achievable rate of typical users associated with the PBS is derived. When the deployment of the PBS is modeled by a PHP, the exact expression of the average achievable rate of users is extremely complex. In this section, the method based on the MISR is used to derive the approximate easy-to-handle expression from the relationship between the coverage and the average achievable rate proposed in [14]. \(\square\)  
**Theorem 4**.