## 4 Cross Layer Adaptation of Check Intervals
### Tree-Based Topologies
Table 1 shows in the first column the values of DC and in the second column their corresponding check intervals according to Equation (3). The subsequent columns show the delay that is accumulated by each hop as well as the value of the next check interval (\(\tau_{new\_check}\)) with \(order(i_{k})=1\) \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**DC** & \(\tau_{check}\) & \multicolumn{2}{c}{\(p=-1\)} & \multicolumn{2}{c}{\(p=1\)} & \multicolumn{2}{c}{\(p=5\)} & \multicolumn{2}{c}{\(p=10\)} \\ \cline{3-10} (\%) & **(ms)** & **Delay** & \(\tau_{new\_check}\) & **Delay** & \(\tau_{new\_check}\) & **Delay** & \(\tau_{new\_check}\) & **Delay** & \(\tau_{new\_check}\) \\ \hline
1 & 495 & \(-\)4.95 & 490.05 & 4.95 & 499.95 & 24.75 & 519.75 & 49.5 & 544.5 \\
2 & 245 & \(-\)2.45 & 242.55 & 2.45 & 247.45 & 12.25 & 257.25 & 24.5 & 269.5 \\
4 & 120 & \(-\)1.2 & 118.8 & 1.2 & 121.2 & 6 & 126 & 12 & 132 \\
10 & 45 & \(-\)0.45 & 44.55 & 0.45 & 45.45 & 2.25 & 47.25 & 4.5 & 49.5 \\
20 & 20 & \(-\)0.2 & 19.8 & 0.2 & 20.2 & 1 & 21 & 2 & 22 \\
30 & 11.6 & \(-\)0.11 & 11.49 & 0.11 & 11.71 & 0.58 & 12.18 & 1.16 & 12.76 \\
40 & 7.5 & \(-\)0.075 & 7.425 & 0.075 & 7.575 & 0.375 & 7.875 & 0.75 & 8.25 \\
50 & 5 & \(-\)0.05 & 4.95 & 0.05 & 5.05 & 0.25 & 5.25 & 0.5 & 5.5 \\
100 & 0 & \(-\) & 0 & \(-\) & 0 & \(-\) & 0 & \(-\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Configuration of delays for different values of DC and \(p\). The values of delays and \(\tau_{new\_check}\) are all expressed in milliseconds. given different values of \(p\) (\(-1\), \(1\), \(5\), and \(10\)). These values are the same as those that we used to compute the accumulated delay at each hop as explained in the previous section. Note that the values of \(\tau_{new\_check}\) increases proportionally with the number of hops in the path. Configuring the DC to be 100% is equivalent to having the radio always on, so the values of \(p\) are irrelevant in this case. A negative value of \(p\) implies a decrease of the check interval and an increase of DC. We used the Avrora simulator [37] to test the applications. Avrora provides a complete simulation framework for evaluating applications written in TinyOS 1.x/2.x over AVR platforms (which is the name of a known hardware platform family) such as Mica2 and MicaZ, and it allows the configuration of two radio models: (1) indoors, which is based on the free space path loss formula; and (2) outdoors, where the user fixes the maximum radio range. We considered the outdoor scenario for our experiments. Table 2 shows the simulation parameters used in our experiments. Each application was tested on two different network topologies: (1) two linear topologies of 10 and 20 nodes; and (2) two tree-based topologies as shown in Figure 6, a binary tree of 10 nodes and a tree with 22 nodes where each node has up to three children (see Figure 6 for details). We repeated each \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Bit rate & 250 Kbps \\ Radio range & 30 m \\ Rx current & 18.8 mA \\ Tx current & 17.4 mA (at 0 dBm) \\ Sending interval & 10 s \\ Data length & 40 B \\ Battery capacity & 2500 mAh \\ Voltage & 3 v \\ DUTY\_ON\_TIME\({}^{2}\) & 5 ms \\ \hline \hline \end{tabular} We assume DUTY\_ON\_TIME equal to 5 ms as defined in BoX-MAC-2.

\end{table} 
Table 2: Parameter values used in the simulations. Figure 6: Deployments of 10- and 22-nodes tree topology. experiment 5 times and each experiment took 600 virtual seconds. The 96% confidence interval was at most 0.01 for all experiments. For each experiment we determine the network lifetime, the end-to-end delay and the packet loss ratio. We compare the results with those obtained using CTP/BoX-MAC when no communication delays are accounted for. ### Connectivity

In CTP each node in the network sends beacon messages to its immediate neighbors. CTP uses these beacons to construct and maintain routing trees such that a node is not able to communicate if it does not have a parent in the routing tree.