## 4 The Considered Nodes' Synchronization and Data Transmission Protocol
### Results and Comparisons  
The performances of the wake-up time optimization scheme proposed in this Section are analyzed in terms of average consumed battery capacity and per-node consumed battery capacity for each SYNCH operation. More specifically, we consider the power consumption values reported in Section 3,i.e., a current drawn of 98 mA in transmission mode and 66 mA in receiving and listening mode. Hence, neglecting the energy spent for overhearing, we get for a given packet duration of 2.1 s:  
\[C_{tx} =\frac{98\times 2.1}{3600}\quad\text{mAh}=0.0572\quad\text{mAh} \tag{16}\] \[C_{lst} =\frac{66\times 2.1}{3600}\quad\text{mAh}=0.0385\quad\text{mAh} \tag{17}\]  
As for the synchronization error statistics, we consider the clocking system reported in Section 3. In this setting, we compare the performances in terms of consumed battery during SYNCH operations of the proposed optimized wake-up time procedure with a non-optimized one where the wake times are determined assuming perfect synchronization among nodes, i.e., \(R_{s}=(s-2)T_{p}\). Finally, the set of weights \(w_{s}\) of the optimized scheme are set to one. The cases \(N=5\) and \(N=10\) are shown in Figure 8a,b, while the cases \(N=20\) and \(N=50\) are shown in Figure 9a,b. In all cases, we report the per-node and the average battery capacity depletion during SYNCH operations for the optimized and non-optimized schemes. In order to definitely assess the validity of the analysis, we have also carried out a simulator that implements the proposed SYNCH and DATA protocols depicted in Figure 7. The average battery capacity depletion during SYNCH operations obtained through simulations, reported as diamond markers in Figures 8a and 9b, show a perfect matching with the analysis. These results demonstrate the validity of the proposed wake-up optimization scheme that allows to clearly reduce the average consumed battery power per node with respect to the non-optimized scheme in all the considered scenarios. In particular, it is shown that the average reduction of battery capacity depletion is more evident with the increase of the number of nodes, going from a minimum of \(10\%\) for \(N=5\) to around \(50\%\) for \(N=50\). As for the per-node battery capacity depletion during SYNCH operations, it is worth noting that in the non-optimized case it increases with the order of the nodes in the transmission chain. This is because for each pair of transmitting and receiving nodes the time when SYNCH transmission can be successfully carried out depends on the maximum between the two respective synchronization errors, since both nodes must be active at the same time. Accordingly, despite the clock error is unbiased, the maximum between two independent clock errors it is not and, hence, the wake-up time of the receiver tends to be more and more in advance with respect to the transmission time of the transmitter. This _lagging-behind_ effect ultimately increases the idle listening periods of the nodes with the increase of their order. Of course, this effect is not present in the optimized scheme that allows to achieve a quasi-equalization charge of consumed battery power among all nodes, thus, allowing to noticeably reduce the consumed battery power of high order nodes. In particular, considering the case \(N=50\), the most power hungry node (node \(S-1\)) in the optimized case saves around \(65\%\) of battery power with respect to the non-optimized one. It is worth noting that in all cases node \(S\) consumes less battery power than node \(S-1\). This is because node \(S+1\) is always active and, hence, node \(S\) transmits the SYNCH packet only once. Moreover, it is also worth noting that in the optimized scheme the low order nodes can consume more battery power than in the non-optimized scheme (e.g., this is the case of node 2 in all the considered scenarios).