## 6 DIS-Trickle: A Mechanism for Accelerating Network Convergence
### DIS-Trickle Evaluation  
In this subsection, we evaluate DIS-Trickle and discuss the results obtained. We assume the default value of \(I_{min}=8\) ms, analyze the influence of the redundancy constant on network convergence when DIS-Trickle is used, and consider the same range of network density and size scenarios evaluated in Section 5. We analyze the network convergence time, total number of RPL (_i.e._, DIO and DIS) messages transmitted, and collisions during network convergence. We also study the impact of DIS-Trickle parameters on performance. As can be seen in Figure 15, using DIS-Trickle decreases network convergence time by 2 to 3 orders of magnitude, regardless of the network density and size. Actively requesting DIO messages constitutes a dramatically better strategy for non-DODAG nodes than passively waiting for DIO message reception, in terms of joining quickly a DODAG. Interestingly, the network convergence time improvement yielded by DIS-Trickle does not always lead to performance degradation in terms of the total number of RPL (_i.e._, DIO and DIS) messages sent. In dense networks, the RPL message overhead during network convergence does not grow, and even decreases, when DIS-Trickle is used (see Figure 16). However, in sparse networks (e.g., when the average node degree is 5), DIS-Trickle requires a greater amount of total RPL message transmissions than that needed in absence of DIS-Trickle; impact of this phenomenon increases with network size. For a deeper analysis of the number of RPL messages sent during network convergence, we next study the number of DIO and DIS messages sent, separately. Figure 14: Calculation of the DIS response time. The figure illustrates the maximum delay between the transmission of a DIS message and the reception of a DIO message in response, where node A is interested in joining a DODAG and node B is already a DODAG member. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Initial delay (ms) & 200 \\ Interval length, \(I_{DIS}\) (ms) & 30 \\ Number of doublings & 0 \\ DIS-Trickle redundancy constant, \(k_{DIS}\) & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: DIS-Trickle parameter configuration. Figure 17 shows that the average number of DIO messages transmitted decreases when DIS-Trickle is used. This reduction is more significant as network density decreases. The reason is that, as shown in subsection 5.1, in a sparse network, when DIS-Trickle is not used, the network convergence time is very high, and during this time a high number of DIO messages are transmitted until a DODAG is formed. However, using DIS-Trickle allows non-DODAG nodes to join a DODAG in a shorter time, and thus fewer DIO messages are required to complete the DODAG construction. On the other hand, when DIS-Trickle is used, the number of DIS message transmissions decreases as the redundancy  
Figure 16: Influence of using DIS-Trickle on the total number of RPL messages (_i.e._, DIO and DIS messages) transmitted in the small, medium and large networks, for different average node degrees as a function of the redundancy constant, \(k\), when the \(I_{min}\) value is set to 8 ms. (**a**) Small network size, (**b**) medium network size, and (**c**) large network size. Figure 15: Influence of using DIS-Trickle on network convergence time in the small, medium and large network scenarios, for different average node degrees, NDeg, as a function of the redundancy constant, \(k\), when the \(I_{min}\) value is set to 8 ms. (**a**) Small network size, (**b**) medium network size, and (**c**) large network size. constant grows (see Figure 18). This happens because as \(k\) increases, a greater number of DODAG nodes are able to send their DIO messages, therefore non-DODAG nodes have more chances to receive DIO messages, making it unnecessary to send DIS messages. Another important result is that the number of DIS message transmissions increases as network density decreases, since a lower amount of DIO messages are transmitted naturally by DODAG members.