## 6 Simulation and Discussion
### Probability Analysis in Homogenous Network  
The PDF of the neighbor distance (shown in Equation (18)) can be found in Figure 21. The Figure 21 indicates that the distribution of the neighbor node distance is Rayleigh Distribution. In Figure 21, with the increasing of the node density, the maximum probability increases while the maximum neighbor distance decreases. The reason is that when the node density increases, more nodes are deployed in the same area, so the neighbor distances decrease. For instance, in Figure 21, when the node densities are 0.1, 0.01, and 0.005, the maximum neighbor node distances are 5, 15, and 20, respectively. The similar variation trend can be found in the neighbor node distances which have the maximum probabilities.  
Figure 20: The error between the boundaries of energy-efficiency and interference-optimality.  
Equation (17) is the probability of interference-optimality by adjusting the transmission range in homogenous WSNs, which relates to both the original transmission range and the node density. The simulation results under different initial transmission ranges and node densities are shown in Figure 22.  
In Figure 22, the node densities vary from 0.005 to 0.1 and the original transmission ranges vary from 0 to 25. With the increasing of the original transmission ranges, the probabilities that the network is interference-optimal increase to 1 when the original transmission ranges are larger than the maximum neighbor distance that is shown in Figure 22. For instance, in Figure 22, when the node density is 0.1, with the increasing of the original transmission range, the probability increases when the transmission range is smaller than 5 and levels off to 1 when the transmission range is larger than 5. The same variation trend can be found when the node densities are 0.01 and 0.005. The trend in Figure 22 is consistent with the simulation results of Figure 21. In Figure 21, when the node density is 0.1, the maximum neighbor distance is 5 (when the probability is equal to 0 in Figure 21); meanwhile, in Figure 22, when the node density is 0.1 and the transmission range is larger than 5, the probability of interference-optimality by adjusting the transmission range levels off to 1. The reason is that under the specific node density, if the original transmission range is larger than the maximum neighbor node distance, then, once the transmission ranges are reduced, the optimal transmission range can be found with probability 1.  
Figure 21: The probability density function of (17).  
Figure 22: The probability under different node density and original transmission range in homogeneous networks.  
Under different node densities, the probabilities of interference-optimality by adjusting the transmission ranges are different. As shown in Figure 22, when the node density increases, the probability increases under the same original transmission range. For instance, when the original transmission range is 10, the probabilities are 0.51, 0.85, and 1 when the node densities are 0.05, 0.01, and 0.1, respectively. In addition, the optimal transmission ranges under different node densities are equal to the maximum neighbor node distances shown in Figure 21.  
Figure 23 describes the probability under specific transmission range, in which the original transmission range and node density are fixed to 10 and 0.01, respectively. From Figure 23, we can conclude that the maximum probability is approximately 0.83 when the transmission range is 0.85\(r\). Therefore, the maximum probability is 0.83 and the optimal transmission range is 0.85\(r\). This conclusion means that when the transmission range is 0.85\(r\), the probability of interference-optimality is the highest. This conclusion is consistent with the theoretical analysis in Section 4.1.