## 5 Evaluation: Influence of the Main RPL Parameters on Network Convergence Performance
### Influence of the Minimum Interval on the Network Convergence Process  
In this subsection we study the effect of the minimum interval, \(I_{min}\), on the network convergence process. We evaluate the impact of \(I_{min}\) on the network convergence time, number of DIO message transmissions and number of collisions during the network convergence process. We also analyze the relationship among these performance parameters. The RPL specification states that the default value for \(I_{min}\) is 8 ms [5]. However, in order to investigate the influence of \(I_{min}\) on network convergence performance, we evaluate different \(I_{min}\) values in the same scenarios and for the same redundancy constant values considered in the previous subsection. Figure 10 shows the average network convergence time when \(I_{min}\) is set to 4 ms, 8 ms and 16 ms in the scenarios and for the settings presented in Section 4. As it can be seen in Figure 10, decreasing the minimum interval yields a reduction of network convergence time regardless of the network size, the network density or the redundancy constant, for the range of \(I_{min}\) values considered. The reason is that when \(I_{min}\) decreases, the length of Trickle intervals decreases as well, therefore nodes send their DIO messages earlier and more frequently, and as result a DODAG forms faster. Another important consideration that can be inferred from Figure 10 is that halving the \(I_{min}\) value decreases the network convergence time by a factor smaller than two. As we already mentioned, when \(I_{min}\) decreases, the number of DIO message transmissions during network convergence increases; as a consequence, the number of collisions increases as well (see Figures 11-13). The number of collisions increase does not avoid the network convergence time decrease by decreasing \(I_{min}\), but it is the reason why this decrease is smaller than the \(I_{min}\) decrease. Figure 11: Average number of DIO messages transmitted and collisions in the small network scenario for different average node degrees, as a function of the redundancy constant, \(k\), for different values of \(I_{min}\). Figure 10: Influence of \(I_{min}\) on the network convergence time in the small, medium and large network scenarios with different average node degrees as a function of the redundancy constant, \(k\), when the \(I_{min}\) value is set to 4 ms, 8 ms and 16 ms. (**a**) Small, (**b**) medium, and (**c**) large network size scenarios. Analyzing in deeper detail the number of DIO messages transmitted during network convergence (Figures 11a, 12a and 13a), we observe that as both the redundancy constant and the network density grow, the influence of \(I_{min}\) becomes greater. This is due to a lower amount of DIO message suppressions in the network. As a consequence of a greater number of DIO message transmissions, the number of collisions also grows as \(I_{min}\) decreases (see Figures 11b, 12b and 13b). The relative  
Figure 12: Average number of DIO messages transmitted and collisions in the medium network scenario for different average node degrees, as a function of the redundancy constant, \(k\), for different values of \(I_{min}\). Figure 13: Average number of DIO messages transmitted and collisions in the large network scenario for different average node degrees, as a function of the redundancy constant, \(k\), for different values of \(I_{min}\). influence of \(I_{min}\) on the number of collisions does not vary with the redundancy constant, except for a high network density, whereby as the redundancy constant grows, the influence of \(I_{min}\) becomes smaller. This happens because for a large \(I_{min}\) value (e.g., \(I_{min}\) = 16 ms), the number of collisions is low for low redundancy constant values. However, as the redundancy constant grows, and when the network is dense, the probability of collision significantly increases due to the greater number of DIO message transmissions in the network.