## 6 Data Fitting and Simulation Analysis
This is because there are repulsion radiuses between BSs in MHCP-modelled networks, BSs are not completely randomly distributed, while the BSs in PPP-modelled networks are completely randomly distributed. Therefore, the interference between BSs in PPP-modelled networks is larger than that in MHCP-modelled networks, and the coverage performance of MHCP-MHCP networks is significantly higher than that of PPP-PPP networks. It should be noted that, since the ASAPP method is used to derive the coverage of the MHCP-MHCP network, we assume that the equivalent SIR threshold is greater than 1, i.e., \(\beta_{k}/G_{k}>1\) (0 dB). So, when the SIR threshold is relatively small, there is a certain gap between the theoretical approximation and simulation value. When the SIR threshold rises to make the equivalent SIR threshold greater than 0 dB, the theoretical approximation values are consistent with the simulation values. Figure 4 compares the EE of the MHCP-MHCP network with that of the PPP-PPP network. We observe the relationship between the EE and the transmit power of the PBS. The transmit power of the MBS is fixed to 20 W, and the transmit power of the PBS is variable. Both the target SIR for the MBSs and the target SIR for PBSs are set to 10 dB. It can be seen that in the whole PBS transmit power region, there are only subtle differences between simulation values and theoretical approximated values, which proves the correctness of our theoretical derivation. From Figure 4, we can see that the EE of the MHCP-MHCP network is about \(1.2\times 10^{6}\) to \(1.3\times 10^{6}\) bits/Joule higher than that of the PPP-PPP network under the same parameters, which means that the EE of the MHCP-MHCP network is much higher than that of the PPP-PPP network. This is due to the relative regularity of base station deployment in the MHCP-MHCP network, which greatly reduces the mutual interference between BSs. Compared with the PPP-PPP network, the MHCP-MHCP network can improve downlink minimum achievable data rates with the same power consumption. It can also be seen that EE shows a trend of rising first and then declining. This is because, with the increase of the transmit power of the PBS, more users choose to access PBS, so EE is on the rise. However, with the further increase of the transmit power of PBS, the interference between the BSs is also increased,  
Figure 3: Coverage probability for the two-tier PPP-PPP network and two-tier MHCP-MHCP network. which leads to the decline of the MUE's downlink rate, so the EE is gradually reduced. Therefore, EE is a convex function of the transmit power of the PBS and there exists an optimal transmit power of PBS which can maximize the EE. Furthermore, it has been pointed out that EE can be improved by optimizing BS density [29]. We also compare EE of the MHCP-MHCP network with that of the PPP-PPP network which uses the optimization algorithms proposed in [29]. It can be seen from Figure 4 that even if the fixed BS density is adopted, the EE of the MHCP-MHCP network is still higher than that of the PPP-PPP network adopting the optimal BS density. Figure 5 shows the relationship between the EE and the transmit power of the MBS. The transmit power of the PBS is fixed to 0.13 W, and the transmit power of the MBS is variable. It can be seen that the EE of the MHCP-MHCP network is about \(1.2\times 10^{6}\) to \(1.3\times 10^{6}\) bits/Joule higher than that of the PPP-PPP network under the same parameters. Similar to Figure 4, the EE shows a trend of rising first and then declining, so it is a convex function of the transmit power of the MBS. Therefore, there exists an optimal transmit power of MBS which can maximize the EE. Figure 4: Network energy efficiency (EE) versus picocell base station (BS) transmit power. Figure 5: Network energy efficiency versus macrocell BS transmit power.