## 5 Evaluation: Influence of the Main RPL Parameters on Network Convergence Performance
### Influence of the Redundancy Constant on the Network Convergence Process
For example, when the node degree is 15, in the small network size scenario, a network convergence time decrease factor of 8.3 can be achieved by using \(k=15\) in comparison with the result obtained for \(k=1\). However, for the same node density, the network convergence time decrease factor that can be achieved by comparing the use of \(k=15\) with \(k=1\) in the large network scenario is 14.5. Figures 5-7 show the Cumulative Distribution Function (CDF) of the network convergence time for all the network size and density scenarios considered. The results illustrated by these figures evidence the dramatic impact of the redundancy constant on network convergence time. Increasing the redundancy constant value causes the percentage of DODAGs formed in a given time to increase as well. For example, in the medium network scenario, when the node degree is 5 (Figure 6a), and \(k\) is equal to 1, more than 80% of the DODAGs are formed in less than 120 s, while the same ratio of DODAGs are formed in less than 18 s for \(k\geq 2\). As the network density grows, network convergence times decrease as well. Still in the medium network scenario, when the node degree is 10 (Figure 6b), and \(k\) is set to 1, 80% of the DODAGs are formed in less than 2 s, whereas the same ratio of DODAGs are formed in less than 0.6 s for \(k\geq 2\). In the same scenario, for a node degree of 15 (Figure 6c), 90% of the DODAGs are formed for any value of \(k\) in less than 0.6 s.  
On the other hand, the reader may observe the wave shape that can be appreciated in Figures 5-7, especially when the node degree is low. This behavior is due to the interval duplication of the Trickle algorithm, and reflects the distribution of the number of intervals needed to completely form a DODAG, as well as the size of the interval in which a DODAG is formed. Another useful performance result is the time it takes for a node to join a DODAG. Figure 8 shows the CDF of the DODAG join time for a node in the considered network scenarios. When the average node degree is 5 and \(k\) is equal to 1, around 80% of the nodes have joined the DODAG in 0.1 s, 1.1 s and 15 s in the small, medium and large network size scenarios, respectively. These times are significantly smaller than the average network convergence times shown in Figure 4. Therefore, a relatively small fraction of the nodes require a large amount of time to join a DODAG. Whether this is a critical aspect for the user depends on the application requirements the network is deployed for. As shown in Figures 5-8, the difference between the curves for \(k=1\) and for \(k=2\) is greater than the difference between the curve for \(k=2\) and any curve for \(k>2\). This result illustrates that the network convergence time and join time improvement that can be achieved by increasing the redundancy constant decreases as the redundancy constant grows. Figure 5: CDF of network convergence time in the small network size scenario. The average node degrees are 5, 10 and 15 in figures (**a**), (**b**) and (**c**), respectively. Figure 6: CDF of network convergence time in the medium network size scenario. The average node degrees are 5, 10 and 15 in figures (**a**), (**b**) and (**c**), respectively. Figure 8: CDF of nodes join time for an average node degree of 5: (**a**) small, (**b**) medium, and (**c**) large network size scenarios. Figure 6: _Cont._  
Figure 7: CDF of network convergence time in the large network size scenario. The average node degrees are 5, 10 and 15 in figures (**a**), (**b**) and (**c**), respectively. In order to provide insight on the influence of the redundancy constant on network convergence performance, we also evaluated the average number of DIO message transmissions and collisions that occur from the first DIO transmission by the root until the instant in which all nodes join the DODAG, for all the network size and density scenarios considered.