# The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process
###### Abstract
The authors of [32] studied the SE and EE of a two-tier heterogeneous network, in which the MBS and the PBS are modeled by two independent PPPs. The results show that deploying PBSs in macro cells is an effective method to improve the SE of the network. When the density of PBSs does not exceed the optimum, the EE of the whole network can be effectively improved. The authors of [33] deduced the local delay and EE of heterogeneous networks when BSs are modeled by the PPP and the PCP. Due to the existence of intra-cluster interference, the local delay and EE of BS deployment modeled by the PCP are greater than those modeled by the PPP. In [34], the distribution of small cells is modeled by the MCP. A general formula for the lower bound of the average reachability rate is derived and the optimal number of cell clusters to maximize EE is studied. BS sleep is also an important technology used to reduce energy consumption in HetNets. The authors of [35; 36; 37] improved the energy efficiency of the network through BS on/off switching and transmission power scaling. The authors of [38] proposed a SBS sleep scheme, which determines whether the SBS is in sleep state according to the system throughput. The simulation results show that, compared with the traditional scheme, this scheme can effectively reduce the total power consumption of the network and improve the energy efficiency of the system. [PERSON] et al. studied the joint sleep strategy based on the MBS density and the SBS density [39]. The authors of [40] designed an optimization model to schedule the four states of the base station (activation, waiting, deep sleep, and shutdown) by maintaining the minimum quality of service required by users, so as to minimize power consumption. The simulation results show that the proposed model hasbetter performance than the existing methods. By reducing the percentage of active small cells, the power consumption is reduced to about 10%. Optimizing the transmit power of base stations is also a feasible way to improve the EE of HetNets. The authors of [41] developed a joint interference and power management mechanism to reduce signal interference and save energy of user equipment and improve LTE-A performance. The authors of [42] used game theory to solve the optimal unit selection and uplink power control problems of open-access two-tier femtocell networks. The authors of [43] studied the EE of two-tier HetNets, in which the BS was modeled as the PPP, the impact of the BS transmit power on the EE was analyzed, and an algorithm used to find the best PBS transmit power to maximize EE was proposed. On this basis, the authors of [44] proposed to maximize the network EE through the joint optimization of the PBS density and transmit power. The authors of [45] studied the EE of two-tier HetNets based on the spatial repulsion of BSs in the same tier, in which the deployment of MBSs is modeled by the \(\beta\)-Ginibre point process, and proposed an EE optimization algorithm to determine the optimum transmit power of PBS to improve the EE of HetNets. In this paper, the performance of HetNets is studied from the inter-tier correlation. Different from the modeling deployment in [43; 44; 45], the MBS deployment is modeled by PPP, and the PBSs are deployed outside the exclusion zone of the given radius of the MBS. In this case, the deployment of the PBS follows the PHP distribution. Introducing repulsion can reduce the interference to the MBS when the PBS is transmitted. This paper studies the energy efficiency of two-tier heterogeneous cellular networks and optimizes the EE by adjusting the PBS transmit power. First, the coverage probability and the average achievable rate of two-tier PPP-PHP HetNets are derived based on the ASPPP method. Secondly, according to the definition of the network energy efficiency and the energy consumption model of the base station, the expression of the network energy efficiency in relation to the base station transmit power is derived.