# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
## 2 System Model  
The two-tier HetNets, composed of MBSs and PBSs, are considered, as shown in Figure 1.  
Figure 1: Model of two-tier heterogeneous cellular network.  
The deployment of the MBS follows the PPP \(\Phi_{m}\) distribution, and its density is \(\lambda_{m}\), and the deployment of the PBS follows the PHP \(\Phi_{p}\) distribution, and its density is \(\lambda_{p}\). The generation process of the PHP can be expressed as follows: suppose \(\widetilde{\Phi}_{p}\) is an independent uniform PPP with density \(\widetilde{\lambda}_{p}\), let \(\Xi_{r}=\cup\{x\in\Phi_{m}:b(x,r)\}\), where \(\Xi_{r}\) is the union of all disks in \(\Phi_{m}\), with \(x\) as the center and \(R\) as the radius. The Poisson hole process \(\Phi_{p}\) is expressed as \(\Phi_{p}=\left\{x\in\widetilde{\Phi}_{p}:x\
otin\Xi_{r}\right\}=\widetilde{ \Phi}_{p}\backslash\Xi_{r}\), i.e., the point process made up of the remaining points after removing all points in the \(\Xi_{r}\) area in \(\widetilde{\Phi}_{p}\), and its density is \(\lambda_{p}=\widetilde{\lambda}_{p}\exp(-\lambda_{m}\pi r^{2})\). The base station distribution diagram of the HetNets is shown in Figure 2, where \(\lambda_{m}=0.001\) m\({}^{-2}\), \(\widetilde{\lambda}_{p}=0.01\) m\({}^{-2}\), \(r=5\) m, and the network area is 10,000 m\({}^{2}\).  
The transmit powers of the MBS and the PBS are denoted by \(\mu_{m}\) and \(\mu_{p}\), respectively. The network model is stationary because the entire Poisson point process is moved and the position of the typical user is at the center origin; according to the Palm distribution and the Slivnyak theorem in stochastic geometry, the conditional distribution at this time still follows the Poisson point process of the same density and all users have the same statistical characteristics, so only the typical users at the origin are considered. Considering the open-access mode, the average maximum received power is used as the criterion for the associated base station and the received power of the user-associated service base station can be expressed as \(\mu_{k}h_{0}|x_{0}|^{-\alpha}\), where the path loss factor \(\alpha>2\),\(k\in\{m,p\}\), \(x_{0}\) is the location of the service base station in \(\Phi_{k}\), which also represents the distance between the typical user and the service BS, and \(h_{k0}\) is the channel gain. This paper assumes that the fast fading is Rayleigh fading, i.e., \(h_{k0}\sim\exp\) (1), and \(\mu_{k}\) indicates the transmit power of the \(k\)-th tier base station associated with a typical user. In light of general frequency reuse, the signals received by a typical user from base stations other than the service base station are regarded as interfering signals. In heterogeneous networks with limited interference, the noise power can be ignored. Then, the user SIR associated with the service base station located at \(x_{0}\in\Phi_{k}\) is:  
\[SIR_{k}=\frac{\mu_{k}h_{k0}|x_{0}|^{-\alpha}}{\sum\limits_{x_{i}\in\Phi_{m} \backslash\{x_{0}\}}\mu_{m}h_{mi}|x_{i}|^{-\alpha}+\sum\limits_{x_{i}\in\Phi_{ p}\backslash\{x_{0}\}}\mu_{p}h_{pi}|x_{i}|^{-\alpha}} \tag{1}\]  
We list the main notations used in the paper in Table 1.  
Figure 2: Two-tier HetNets BS distribution diagram.