## 4 Network Interference Probability Analysis
### Heterogeneous Node Deployment Probability Analysis
#### 4.2.1 The Node Distance is \(r/2\leq d\leq r\)
Based on Theorem 1, the coverage area before adjusting the transmission range is:  
\[s_{r} =2\pi r^{2}-(\pi r^{2}-1.93r^{2})\ \, \tag{31}\] \[=5.07r^{2}\]  
As shown in Figure 4, based on Theorem 4, the coverage area after adjusting the transmission ranges is:  
\[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]0 \[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]1  
In Equation (32), there are three parts: (1) \(\pi{r_{1}}^{2}\) is the area of circle \(u\); (2) \(\pi{d_{0}^{2}}-(3.23r_{1}d_{0}-1.57r_{1}{}^{2}-0.44d_{0}^{2})\) is the area of circle \(n\) after removing the overlapping area between circle \(u\) and circle \(n\); (3) \(\pi{d_{1}^{2}}-(3.23d_{0}d_{1}-1.57d_{1}^{2}-0.44d_{0}^{2})\) is the area of circle \(v\) after removing the overlapping area between circle \(n\) and circle \(v\). In Equation (32), since \(d_{0}\) and \(d_{1}\) are all variables, for simplifying the calculation, we assume that the distance \(d_{0}\) is equal to the transmission range of node \(u\), i.e., \(r_{1}=d_{0}\). This assumption is reasonable. Since the distance equal to the transmission range means that the node which is farthest from the source node will be chosen as the relay node. For instance, as shown in Figure 4, when the node \(u\) sends data packet to node \(v\), then the node \(n\) which is in the boundary of the coverage area will be chosen the relay node; therefore, the transmission range of node \(u\) is equal to distance (\(u\), \(n\)), i.e., the distance (\(u\), \(n\)) is the maximum value among all the neighbors' distances. Thus, when the transmission ranges are changed, the coverage area can be rewritten as:  
\[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]2 \[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]3 \[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]4  
Similar to the analysis in Section 4.1, the boundary condition is \(\rho s_{r}=\rho s_{r_{1}}\), which can be expressed as:  
\[f=-1.57\cdot d^{2}+3.23\cdot rd-0.44\cdot r^{2}, \tag{29}\]5  
The transmission range \(r_{1}\) and node distance \(d_{1}\) should satisfy the constraints shown in Equation (17). In addition, in Equation (33), \(r_{1}\) is the transmission range after adjustment and equal to the maximum value between distance (\(u\), \(n\)) and distance (\(n\), \(v\)). For instance, as shown in Figure 4, the distance (\(u\), \(n\)) is larger than the distance (\(n\), \(v\)), so \(r_{1}\) is equal to distance (\(u\), \(n\)) and larger than \(d_{1}\) (the situation that \(r_{1}\) is smaller than \(d_{1}\) will be investigated following). Moreover, in this scenario, we assume \(r_{1}/2\leq d_{1}\leq r_{1}\), so the constraints are shown as follows:  
\[\left\{ \begin{array}{l}0<r_{1}<r\\ 0<d_{1}<r\\ r_{1}/2<d_{1}<r_{1}\\ r_{1}+d_{1}>r\end{array} \right.\, \tag{35}\]  
In Figure 14, the area of constraint \(r_{1}/2\leq d_{1}\leq r_{1}\) in Inequality (35) is between the line \(OF\) and \(OA\); the line \(OF\) represents \(d_{1}=r_{1}/2\) and the line \(OA\) represents \(d_{1}=r_{1}\). Therefore, the feasible region when \(d_{1}<r_{1}\) is _KHBA_ in Figure 14. As discussed in Section 4.1, the role of \(r1\) and \(d_{1}\) are equal in Equations (33) and (34). So when distance (\(n\), \(v\)) is larger than distance (\(u\), \(n\)), the feasible region should be symmetrical with _KHBA_ based on line \(d_{1}=r_{1}\) (line \(OA\)), which is _IKEA_ in Figure 14. According to Equation (34), when \(d_{1}>r_{1}\), the transmission range equals to \(d_{1}\), then the constraint \(r_{1}/2\leq d_{1}\leq r_{1}\) in Inequality (35) will be \(d_{1}/2\leq r_{1}\leq d_{1}\), which is between line \(OD\) and line \(OA\); moreover, according to Equation (34), the boundary condition is \(d_{1}=0.445\cdot\sqrt{4.5r^{2}-4r_{1}{}^{2}}+0.3r_{1}\) for feasible region _IKEA_ in Figure 14. Thus, the whole feasible region in this scenario is _EIHB_ which is shown in Figure 14.  
The coordinates of the points in Figure 14 are shown in Table 2.