## 4 Performance of SERENA and TDMA/CA
### Simulations in an Ideal Environment for a Data Gathering Application  
We now consider a data gathering application and report the results obtained in the same conditions as the general application, but with a tree-based coloring. All the configurations tested have a maximum tree depth of 8. Node \(0\) is the sink, the other nodes are at a distance less than or equal to 8 from it in the tree. We observe first that unsurprisingly the coloring of the same configuration is not the same as previously. This result points out the importance of the coloring rules. For instance, in a configuration with 49 nodes and density 8, node 0 that had color 21 in the previous general coloring has now color 0. Its one-hop neighbors, nodes 1, 11, 16, 21, 29, 35, 38, 43 and 44 had colors 17, 5, 2, 25, 6, 11, 13, 24 and 14 respectively have now colors 8, 2, 4, 9, 1, 13, 15, 6 and 7. Figure 11 depicts the number of colors as a function of the number of nodes for different densities. As previously, the number of colors is smaller than the number of nodes, but here in all configurations tested. The benefit is around 57% for 49 nodes and a density 8 and reaches 72% for 100 nodes and a density of 8. The average number of nodes sharing the same slot is higher than in the general case. Moreover, we notice that the number of colors is smaller than in the general case. The increase with the number of nodes is significantly smaller than in the general case. This can be explained by the fact that the set \(\mathcal{N}(N)\) computed for the tree coloring is smaller than the set of nodes up to three-hop from \(N\). Although the rule requiring the color of a child to be higher than the color of its parent tends to mitigate that, the number of colors is smaller than in a general coloring. To prove that, we have computed for each node, the number of nodes meeting one of the four cases depicted in Figure 3. This number is smaller than the number of three-hop neighbors, explaining the smaller number of colors. For instance, for 49 nodes and a density 8, its average value is 1 whereas the average number of 3-hop neighbor is about 13. More generally, this number is less than 10 as long as the number of nodes is less than or equal to 49. The average number of nodes sharing the same slot is depicted by Figure 12. It reaches 3.6 for 100 nodes and a density of 8. The spatial reuse of the bandwidth is better than in the general case. Figure 10: End-to-end delays for data-gathering traffic obtained with a general coloring. We consider again the topology depicted in Figure 8 with 49 nodes and a density 8 and evaluate the end-to-end delays with the tree-based coloring. Figure 13 shows the gain we obtain by using a tree-based coloring. The average end-to-end delay that a packet undergoes is \(2.94\) seconds which is less than the duration of a cycle. It is interesting to notice that all the traffic generated is received in less than \(2*4\) seconds (the maximum value of end-to-end delay obtained is \(5.99\) seconds). Indeed, frames are generated without taking into consideration the start of the color slot. Hence, in the worst case scenario, a frame is generated just after the end of the color slot during which it should have been transmitted. In that case, this packet is kept in queue until the next color slot of the next cycle. Thus, packets may suffer  
Figure 11: Number of colors in a data gathering application. Figure 12: Average number of nodes sharing the same slot. an end-to-end delay higher than one cycle which explains why the delay can exceed \(4\) seconds with the tree-based coloring. The delivery rate is equal to 100%. We also evaluate the percentage of time a node is awake in the active period to send its frames and receive the frames from its neighbors. Results are depicted in Figure 14. As expected, this percentage decreases with the number of nodes and increases with the density. These results give an idea of the energy saving that can be done.