# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
###### Abstract  
In order to decrease energy consumption caused by the dense deployment of pico base stations (PBSs) in heterogeneous cellular networks (HetNets), this paper first analyzes the energy efficiency (EE) of two-tier HetNets and then proposes a method to maximize the network EE by adjusting the PBS transmit power. The two-tier HetNets are modeled by the Poisson point process (PPP) and the Poisson hole process (PHP), and then the coverage probability of the macro base station (MBS) and the PBS in the two-tier HetNets is derived based on the mean interference to signal ratio (MISR). According to the user association probability, the coverage probability of the PPP-PHP HetNets is obtained. Then, the tractable expression of the average achievable rate is deduced on the basis of the relationship between the coverage probability and the average achievable rate. Finally, the expression of EE is derived and the EE optimization algorithm is proposed based on the PBS transmit power. The simulation results show that the PPP-PHP network is superior to the PPP-PPP network in terms of coverage probability and EE, and the network EE can be effectively improved by setting an appropriate PBS transmit power. on stochastic geometry has become more necessary. The Poisson point process (PPP) has been widely used in the modeling and analysis of cellular networks due to its mathematical tractability. However, the PPP assumes that the BSs are independent, which deviates from the actual network deployment. Therefore, the analysis results are more ideal. To solve the modeling of network deployment, some studies propose using the non-Poisson point process with spatial exclusion or aggregation to model the different spatial correlations of realistic HetNets. The Ginibre point process (GPP) [14; 15] and the Matem hard-core point process (MHCPP) [16] are used to fit the exclusivity between the BSs of the same tier. The Matem cluster process (MCP) [17; 18] and the Poisson cluster process (PCP) [19; 20] are used to fit the clustering characteristics of small base stations. The Poisson hole process (PHP) is used to fit the repulsion between MBSs and small base stations (SBSs) [21; 22]. It is very difficult to accurately calculate the signal-to-interference ratio (SIR) distribution of cellular networks through the non-Poisson point process. In the cellular network modeled by the GPP, the calculation of the SIR is relatively simple, but the combination of the infinite sum and the integral also appears in the precise expression of its complementary cumulative distribution function, and the network analysis based on other non-Poisson point processes is more complex, which makes its performance analysis more challenging. To solve this problem, references [23; 24; 25] proposed that the SIR based on the non-Poisson point process network can be accurately approximated by the SIR distribution of the PPP network; this method is also called \"PPP-based approximate SIR analysis\" (ASPPP), which provides an effective analysis method for analyzing the performance of the actual network. For future mobile communication systems, research on network structures and transmission mechanisms to save energy and improve energy efficiency (EE) is a widespread concern in academic and industrial circles [26; 27; 28; 29]. The following section summarizes the current status of improving network EE from the aspects of BS deployment, BS sleep, and power allocation. The authors of [30] indicated that the transmit power of the MBS, the distance between the BSs, and the number of BSs have an impact on the EE of the system. The proper selection of these parameters can ensure significant energy savings. In addition, adding PBSs in HetNets can help improve network EE. Moreover, [31] analyzed the EE of two-tier heterogeneous wireless networks from the perspective of PBS deployment, and studied whether deploying PBSs can increase system capacity and save energy from the aspects of area spectrum efficiency and area power consumption.