# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## 4 Numerical Results
### Performance Evaluation: Trends and Discussion  
Before numerically analyzing the overall performance of the proposed scheme, we first analyze the conditional coverage probabilities of Equations (16)-(19) to obtain a better understanding of the  
Figure 2: Validation of the analytical results for the overall coverage probability via Monte Carlo simulations with D = 0.5 km and \(\beta=4\) dB.  
overall performance trends. In Figure 3, it can be seen that the coverage probabilities of MIUs and femto users (i.e., \(\mathcal{P}_{M,in}\) and \(\mathcal{P}_{F}\)) remained invariant with an increasing cooperation threshold, as \(\mathcal{P}_{M,in}\) and \(\mathcal{P}_{F}\) were independent of the cooperation threshold, whereas, as the radius of the inner region \(D\) increased, \(\mathcal{P}_{M,in}\) and \(\mathcal{P}_{F}\) decreased. This was because more macro users were allocated within the inner region; thus the interference from the macro tier increased. Meanwhile, the performance of femto users was degraded because more FBSs were deactivated. It can also be observed that the coverage probability of macro outer users (i.e., \(\mathcal{P}_{M,out}\)) increased with an increasing cooperation threshold, as more users were offloaded to the femto tier. However, the coverage probability of the CoMP users (i.e., \(\mathcal{P}_{C}\)) initially increased as the cooperation threshold increased, given an appropriate value of \(D\), but it decreased beyond a certain prescribed distance value. This resulted from the fact that the non-uniform (cell-edge) deployment of FBSs can enhance the performance of CoMP users. When the prescribed distance exceeds a certain value, too many FBSs will be deactivated, which degrades the performance of CoMP users.  
Figure 4 on the next page depicts the effect of increasing the radius of the inner region on the overall coverage probability and the average rate for different cooperation thresholds. It can be seen that the performance of the network can be improved by introducing the prescribed distance \(D\) (the radius of the inner region) for MBSs. The gains are obtained by the non-uniform deployment of FBSs. That is, the coverage area of FBSs increases and more users are offloaded to FBSs, as FBSs are deployed farther away from the MBSs. Hence, the more efficient deployment of FBSs enhances the performance of the network. For a given cooperation threshold, both the overall coverage probability and the average rate initially increase as the prescribed distance increases, but they decrease beyond a certain prescribed distance value. Hence, an optimal prescribed distance exists. The descending trend of the coverage probability and average rate is mainly due to more and more FBSs being deactivated, which degrades the network performance. This means that selecting an appropriate value of \(D\) indeed matters for the network performance.  
The variation of the overall coverage probability and the average rate with an increasing cooperation threshold for different prescribed distances is also shown in Figure 4. It is clear that a higher coverage probability and average rate can be achieved by deploying the LA-CTC scheme, given an appropriate value of \(D\). This improvement is due to the joint impact of offloading MEUs to FBSs and the CoMP transmission for the offloaded users. Therefore, the optimal network performance can be achieved by properly selecting the parameters of the LA-CTC scheme.  
Figure 3: Effect of the radius of the inner region on the conditional coverage probabilities of four types of users for different cooperation thresholds.