## 4 Numerical Results
### Performance Comparisons
\begin{tabular}{l} \(K=400\) \\ \(N=10\) \\ \(\mathbf{s}_{1}(1)=\mathbf{10}\) \\ \end{tabular}  & \begin{tabular}{l} \(K=80\) \\ \(F=5\) \\ \end{tabular}  & \begin{tabular}{l} (236; 242; 238) \\ \((230;240;236)\) \\ \end{tabular}  & \ \begin{tabular}{l} (22; 25; 24) \\ \((22;25;23)\) \\ \end{tabular}  & \begin{tabular}{l} (90; 98; 93) \\ \end{tabular}  \\ \cline{2-5}  & & &  
A first important conclusion obtained from Table 1 is that larger lifetimes are achieved by the proposed method when \((w_{1},w_{2})=(0,1)\), no matter the remaining parameters. The advantage over the greedy algorithm in this case is more pronounced when \(N\) is larger, with comparable standard deviations of the lifetimes. When \(N\) is small, the performance achieved by the proposed optimization method does not vary too much for different weights \(w_{1}\) and \(w_{2}\), giving to the network administrator full flexibility in terms of the balance between equalized consumptions and burst-like transmissions. As \(N\) is increased, the lifetime is reduced if the equalized consumptions are privileged, implying that the burst-like operation achieved with \((w_{1},w_{2})=(0,1)\) must be chosen when longer lifetimes is the main concern. Notice, however, that when \(N\) is large, the adoption of \((w_{1},w_{2})=(1,0)\) reduces the variations of the lifetimes, as demonstrated by the smaller standard deviations mainly in the case of \(N=100\). Still referring to Table 1, one can notice that the change from \(F=1\) to \(F=5\) did not produce significant performance degradation in our method, no matter the number of sensor nodes. Recall that, for \(F>1\), past information on consumptions as used by our method become progressively more uncorrelated with the current ones, thus reducing the lifetime improvement efficiency. Then, if results with \(F>5\) were presented in Table 1, yet larger performance degradations would be unveiled. As a final comparison, Figure 13 shows average lifetimes obtained from Monte Carlo simulations of the greedy algorithm and the proposed optimization method for \((w_{1},w_{2})=(1,0)\) and \((w_{1},w_{2})=(0,1)\), with a variable number of sensor nodes, \(N\). The average lifetime of the non-optimized network is also shown for reference. To prevent simulations from having excessively large durations when \(N\) is large, the number of Monte Carlo events was varied linearly from 500 for \(N=10\), to 50 for \(N=100\), benefiting from the reduced lifetime variance as \(N\) increases (see Table 1). For a consistent comparison, we have adopted parameters from [21] adapted to our notation: equal initial energies \(\mathbf{s}_{1}(1)=S_{\text{max}}\times\mathbf{1}\), with \(S_{\text{max}}=5\), and \(B_{\text{min}}=0.01\). The maximum energy consumption was empirically adjusted to \(B_{\text{max}}=5\) so that the performance of the greedy algorithm approximated the one reported in [21] (Figure 1) at the limits \(N=10\) and \(N=100\). From Figure 13, it can be concluded that our method attains comparable performances with respect to the greedy algorithm for a small number of sensor nodes. As this number increases, the performance of our method becomes worse than the greedy algorithm for \((w_{1},w_{2})=(1,0)\) and superior for \((w_{1},w_{2})=(0,1)\). The large lifetime improvements are also evident with respect to the non-optimized network, for any number of sensor nodes and any objective function weights. From the results presented in this subsection, one can conclude that our method is capable of yielding performances comparable to those achieved by the greedy algorti  
Figure 13: Average lifetimes achieved with the greedy algorithm of [21] and with the proposed optimization method for \((w_{1},w_{2})=(1,0)\) and \((w_{1},w_{2})=(0,1)\), under a variable number of sensor nodes, \(N\). The average lifetime of the non-optimized network is also shown for reference.