## 4 Cross Layer Adaptation of Check Intervals
### Tree-Based Topologies
Therefore, each node must find a parent with ETX less than \(\infty\). The root is the only node with ETX initialized to 0; network connectivity extends from the root to the leaves. Given this fact, we must ensure that the check intervals do not lead to isolated nodes which yield unconnected networks. This problem can arise for some extremely low values of DC and it becomes worse when the check interval is shifted too much, _i.e._, when the value of \(p\) is extremely large. If the value of DC is not appropriate, the activity period in a node may be insufficient to allow CTP to establish a valid route in an instant of time between pairs of nodes if the beacon messages are not received by the neighbors. Note that beacon messages are continuously sent by the nodes during their lifetime, at increasing sending intervals. Thus, although some beacon messages are lost, it does not mean that a node cannot find a route in the future. We illustrate this issue in Figure 7 by showing the connectivity of CLAC against the connectivity of BoX-MAC for each of the experiments. We measure connectivity by counting the number of nodes successfully linked to a parent. On the left hand side we show the results for a 10-node linear topology using different values of DC and \(p\). We observe that for low values of DC (between 1% and 10%) the network is not totally connected for either protocols. For values of DC equal to 20% the network is connected only for the two smallest values of \(p\) (\(-1\) and \(1\)), while for the two largest values of \(p\) none of the networks is connected. For values of DC greater than 20% all of the nodes

Figure 7: Connectivity of BoX-MAC and CLAC in a 10-node (left) and in a 20-node (right) linear topologies. in the network successfully find a valid route to the sink regardless of the strategy. We observe that connectivity is directly proportional to the duty cycle: the larger is the DC, the higher is the connectivity. For negative values of \(p\) we effectively increase the DC since we reduce the check interval and, as a result, we improve the connectivity with respect to BoX-MAC. On the contrary, for large values of \(p\) and small DC, the connectivity is low. This is because increasing DC increases the probability of receiving beacon messages, since the beacon messages are sent at arbitrary times (even at times different from the scheduled time for sending the data); when DC is low the beacons may not be received and, consequently, routes may not be determined. The right hand side of Figure 7 shows the connectivity for BoX-MAC and CLAC for a 20-nodes linear topology. The graphic shows a similar trend as the 10-node case. We can observe that the increase in network size results in a lower connectivity in CLAC (for DC \(\leq\) 20%). It is also interesting to note that the increase of DC, when \(p=-1\), does not guarantee total network connectivity. Figure 8 shows the connectivity achieved for tree topologies. CLAC improves the connectivity when compared to linear topologies, as observed for DC=20%, for which all networks are completely connected. In the case of the 10-node tree (left hand side) for values of \(p=-1\) and \(p=1\) and DC equal to 10%, the network is also connected (as in BoX-MAC), which is an improvement over the linear topology. When we increase the size of the tree (right hand side), the network connectivity decreases slightly for DC = 10%. The reason is that in tree topologies a node is affected by the load of its children; in particular, a heavy load at a node generated by its children may impair its communication with some of them. We can conclude that the neighboring size is an important factor for achieving connectivity, and that the connectivity loss in CLAC as compared to BoX-MAC is mainly due to the loss of beacon messages, whose arrival times cannot be known a priori. Once the topology is established, CLAC may then adapt the delays due to data packets as proposed in this work. Figure 8: Connectivity of BoX-MAC and CLAC in 10-node (left) and 22-node (right) tree topologies.