## 5 Evaluation: Influence of the Main RPL Parameters on Network Convergence Performance
### Influence of the Redundancy Constant on the Network Convergence Process  
One of the RPL parameters that affects network convergence performance to a greatest extent is the redundancy constant, \(k\). As presented in subsection 3.3, this parameter limits the number of transmitted messages per Trickle interval in a given coverage area. In order to analyze the influence of the redundancy constant on the network convergence process, we simulated the network size and density scenarios shown in Table 2 for a range of redundancy constant values between 1 and 15, which includes the default value for this parameter stated by the RPL specification (_i.e.,_\(k=10\)) [5]. For the results presented in this subsection, the rest of RPL parameters were set to the default values stated in the RPL specification (note that \(I_{min}\) was therefore set to 8 ms). We evaluated network convergence time, DODAG join time of a node, number of DIO messages transmitted, and number of collisions. Figure 4 shows average network convergence time in small, medium and large network scenarios, for different node density and \(k\) values. To obtain the convergence time of a network, we calculated the time from the instant in which the DODAG root starts its first Trickle interval until the instant in which all nodes have joined the DODAG. We consider that a non-root node joins a DODAG when it receives a DIO message for the first time. As it can be seen in Figure 4, using low values for the redundancy constant yields high network convergence time in all network size and density scenarios considered. This behavior is mainly due to the message suppression mechanism in the Trickle algorithm. When the root node transmits its first DIO message, the neighbors of the root that receive this DIO message schedule themselves to send their own first DIO message. However, when \(k\) is set to a low value, only a few neighbors of the root will be allowed to transmit their DIO message in the current interval. If the rest of root neighbors (if any) hear a number of DIO messages greater than or equal to \(k\) in the current interval, they will suppress their DIO message transmission. As a result, nodes that are only neighbors of the latter root neighbors have to wait for subsequent intervals to have the opportunity to receive their first DIO  
Figure 4: Average network convergence time in three network size scenarios, for different densities, as a function of the redundancy constant, _k_: (**a**) Small network scenario, (**b**) medium network scenario, and (**c**) large network scenario, respectively. message. The same phenomenon happens as DIO messages propagate through the network, which finally leads to high network convergence time. As the redundancy constant increases up to medium values, the network convergence time decreases. This occurs because, with greater redundancy constant values, the probability that nodes suppress their DIO messages is lower. Therefore, the probability that a node receives its first DIO message earlier is greater. However, network convergence times do not vary significantly as the value of \(k\) increases beyond the values around the node degree. The reason for this behavior is that for such values of \(k\), the number of DIO message suppressions is low, since many nodes have a number of neighbors lower than \(k\), and thus varying \(k\) within the mentioned range does not significantly affect the network convergence time. As is also shown in Figure 4, network convergence time decreases with the node degree since greater network density provides better network connectivity and a greater amount of opportunities for a node to receive DIO messages from its neighbors. By analyzing Figure 4 results in deeper detail, we found that the relative impact of the redundancy constant on network convergence time does not vary significantly with the network size when networks are sparse. However, as the network density increases, the improvement that can be achieved by increasing the redundancy constant grows with the network size.