# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 5 Simulation Results  
In this section, we use the MATLAB software platform to simulate the performance of the two-tier PPP-PHP HetNets. The default simulation parameters of the system model are shown in Table 2. For the transmit power of the MBS and the PBS, we refer to the statistical data in [43]. Figure 3 gives the relationship between the coverage probability and the SIR threshold \(\theta\) of the two-tier HetNets under different path loss factors. The coverage probability decreases with the increase in the SIR, because the coverage probability is defined as the probability that the SIR received by the user is greater than the threshold value. The higher the threshold value, the lower the probability, so the coverage probability also decreases accordingly. When \(\alpha=4\), there is a slight gap between the simulation results of the coverage probability of the PPP-PHP HetNets and the corresponding theoretical deduction results. The gap is due to the approximate interference between the PHP distribution and the PPP distribution, which verifies the correctness of the MISR-based gain method. As seen from Figure 3, as the path loss factor increases, the coverage probability increases. As the path loss factor increases, the cumulative interference \(I_{1}\) and \(I_{2}\) decrease, the \(SIR_{m}\) and \(SIR_{p}\) increase, and the coverage probability also increases. Figure 3 also shows that the coverage probability of the PPP-PHP HetNets is larger than that of the PPP-PPP HetNets under the same path loss factor. Figure 4 shows the relationship between the network energy efficiency and the PBS transmit power in different path loss factors. Figure 4 shows that the network energy efficiency first increases, and then decreases as the transmit power of the PBS increases. As the transmit power of the PBS increases, more users access the PBS and the power consumption of the PBS is small, causing the energy efficiency of HetNets to increase. When the number of users accessing the PBS reaches saturation, the continuous increase in the PBS transmit power will increase the total network power consumption and reduce the network energy efficiency. Therefore, we can trace an optimal PBS transmit power to obtain the maximum network EE. Figure 4 shows that, when the path loss factor is equal to 4, the network energy efficiency is greater than that under the other two path loss factors (\(\alpha=3\) and \(\alpha=3.5\)); thus, there is no monotonic relationship between the network energy efficiency and the path loss factor, which provides an idea of how to optimize the EE from the perspective of the path loss factor. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Symbol** & **Parameter Description** & **Parameter Value** \\ \hline \(\lambda_{m}\) & Deployment density of the MBS & \(10^{-2}\) m\({}^{-2}\) \\ \(\lambda_{p}\) & Deployment density of the PBS & \(0.1\) m\({}^{-2}\) \\ \(\theta\) & Given threshold of the BS & \(0\) dB \\ \(\mu_{m}\) & MBS transmit power & \(40\) W \\ \(\mu_{p}\) & PBS transmit power & \(10\) W \\ \(r\) & Repulsion radius & \(2\) m \\ \(P_{m0}\) & MBS static power consumption & \(1000\) W \\ \(P_{p0}\) & PBS static power consumption & \(50\) W \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 5 shows the relationship between the EE of the two-tier HetNets and the SIR threshold in the cases of \(\alpha=3.5\) and \(\alpha=4\). The quadratic interpolation optimization algorithm is used to obtain the optimal transmit power of the PBS when \(\alpha\) takes different values in the PPP-PPP HetNets and the PPP-PHP HetNets. The proposed algorithm is compared with the golden section method proposed in [43], as shown in Table 3. When the optimal value of the PBS transmit power is obtained, the golden section method iterates 32 times and the running time is 20.37 s. The quadratic interpolation algorithm iterates 30 times and the running time is 10.88 s. The running time is related to the computer hardware configuration.