# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 3 Energy Efficiency Analysis  
### Coverage Probability  
In the downlink HetNets with limited interference, the coverage probability is defined as the probability that the SIR received by the user is larger than the threshold value. Because users are associated with a certain tier at most, the coverage probability can be taken as the sum of the probabilities of several mutually exclusive events. In the network model of this paper, the coverage probability can be given by:  
\[P_{C}=P_{Cm}A_{m}+P_{Cp}A_{p}=P_{C\_m}+P_{C\_p} \tag{2}\]  
where \(A_{m}\) and \(A_{p}\) are the probabilities that a typical user is associated with the MBS and the PBS, respectively, and \(P_{Cm}\) and \(P_{Cp}\) are the coverage probabilities of the MBS and the PBS, respectively. \(P_{Cm}\) is denoted as the probability that the \(SIR_{m}\) provided by the service MBS for the typical user is greater than the threshold value \(\theta_{m}\), i.e., \(P_{Cm}=P(SIR_{m}>\theta_{m})\). \(P_{Cp}\) is denoted as the probability that the \(SIR_{p}\) provided by the service PBS for the typical user is greater than the threshold value \(\theta_{p}\), i.e., \(P_{Cp}=P(SIR_{p}>\theta_{p})\). In this paper, it is assumed that the given thresholds of the MBS and the PBS are the same, which is denoted as \(\theta\). The \(SIR_{m}\) received by a typical user from the MBS is expressed as:  
\[ \begin{array}{l}SIR_{m}=\frac{\sum\limits_{x_{1}\in\Phi_{m}(\{x_{0}\})}\frac {\mu_{m}h_{m0}|x_{0}|^{-\alpha}}{\mu_{m}|x_{1}|^{-\alpha}+\sum\limits_{x_{1}\in \Phi_{p}(\{x_{0}\})}\mu_{p}h_{p}|x_{1}|^{-\alpha}}}{\sum\limits_{x_{1}\in\Phi_ {p}(\{x_{0}\})}\frac{h_{m0}}{\mu_{1}|x_{0}|^{\alpha}|x_{1}|^{-\alpha}}+\sum \limits_{x_{1}\in\[PERSON]_{p}(\{x_{0}\})}\frac{P}{\mu_{p}h_{p}|x_{0}|^{\alpha}|x_{ 1}|^{-\alpha}}}=\frac{h_{m0}}{l_{1}}\\ \end{array}  \tag{3}\]  
The \(SIR_{p}\) received by a typical user from the PBS is expressed as:  
\[ \begin{array}{l}SIR_{p}=\frac{\mu_{p}h_{p0}|x_{0}|^{-\alpha}}{\sum\limits_{x _{1}\in\[PERSON]_{m}(\{x_{0}\})}\frac{\mu_{m}h_{m0}|x_{1}|^{-\alpha}}{\mu_{m}|x_{1}| ^{-\alpha}+\sum\limits_{x_{1}\in\Phi_{p}(\{x_{0}\})}h_{p}|x_{1}|^{-\alpha}}}{ \sum\limits_{x_{1}\in\Phi_{p}(\{x_{0}\})}h_{p}|x_{0}|^{\alpha}|x_{1}|^{-\alpha} }=\frac{h_{p0}}{l_{2}}\\ \end{array}  \tag{4}\]  
Based on the SIR threshold model, the coverage probability is equivalent to the complementary cumulative distribution function (CCDF) of the SIR. It has been proven that taking the PPP as a reference model and adopting the SIR threshold scaling method can provide a good approximation to the SIR distribution of the general point process model; in other words, scaling the threshold value \(\theta\) of the SIR distribution of the Poisson model network to \(\theta/G\) can obtain the SIR distribution of the non-Poisson network. \(G\) is called the gain factor, which is defined as the ratio of the MISR of a typical user under the PPP model \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Definition** \\ \hline \(\lambda_{m},\lambda_{p}\) & Deployment density of the MBS/PBS \\ \(\Phi_{m},\Phi_{p}\) & Point process for modeling MBS/PBS location \\ \(\theta\) & Given threshold of the BS \\ \(\mu_{m},\mu_{p}\) & MBS/PBS transmit power \\ \(A_{m},A_{p}\) & The probability that a typical user is associated with the MBS/PBS \\ \(\tau\) & Repulsion radius \\ \(P_{Cm},P_{Cp}\) & The coverage probability of the MBS/PBS \\ \(f_{m}(\tau_{0}),f_{p}(\tau_{0})\) & The distance distribution between the service MBS/PBS and the typical user \\ \(\tau_{m},\tau_{p}\) & The average achievable rate of the typical user associated with the MBS/PBS \\ \(N_{m},N_{p}\) & The number of users served by the MBS/PBS \\ \(P_{m0},P_{p0}\) & MBS/ PBS static power consumption \\ \(\eta_{EE}\) & Energy efficiency \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of notations. to the MISR of a typical user under the target model, and its mathematical expression can be written as \(G=\frac{MISR_{PPP}}{MISR}\).