## 4 Numerical Results
### Performance Comparisons
On the other hand, sensor node 7, which is closer to the sink node on average, is enabled more times because of its advantageous position up to \(t=310\). Moreover, notice from the nodes' activity states that sensor node 7 is enabl  
Figure 12: Residual energies (**a**) and sensor node activity states (**b**) produced by the greedy method of [21]. The vertical slashes on the graph (**b**) indicate _on_ states, which are mutually exclusive for each \(t\). \(t=60\) and \(t=180\), which is consistent with the behavior of the proposed optimization method, as depicted in Figure 7.  
Comparing Figures 6 and 12, one can observe that the lifetime obtained from the proposed optimization method is comparable to or larger than the one achieved with the greedy algorithm. As will be demonstrated ahead, this is true for small \(N\) and any \(w_{1}\) and \(w_{2}\), whereas for large \(N\) the superiority of our method occurs mainly when \((w_{1},w_{2})=(0,1)\). However, the unused residual energies at the network death moment, i.e., the wasted energy, tend to be higher for some sensor nodes in the greedy algorithm than in the proposed method. This higher wasted energy results from some sensor nodes being less frequently activated than others. From the graph (**a**) of Figure 12 and the graph (**b**) of Figure 6, it can be observed that the energy expenditures produced by the greedy algorithm resemble, to a certain extent, the ones provided by our method when \((w_{1},w_{2})=(0,1)\). In other words, when \((w_{1},w_{2})=(0,1)\) the proposed optimization allows for burst-like transmissions combined with silent periods and continuous transmissions, yielding a net result that can overcome the greedy algorithm, mainly for a higher number of sensor nodes. Larger lifetime records and, thus, larger lifetime improvements are unveiled in favor of the proposed method or the greedy algorithm, as the system parameters are chosen such that the network death occurs later, as for instance if the initial energy is increased while the consumptions are kept unchanged or the consumptions are decreased for the same initial energy. Table 1 shows other results comparing the proposed lifetime optimization method and the greedy algorithm of [21], as obtained from 200 Monte Carlo events with random maximum consumption matrices. Given the myriad of possibilities in terms of system parameters, we have adopted those that fairly approximate the conditions imposed on our method and the greedy algorithm, mainly with respect to the efficient use of the information available. Specifically, recalling that the greedy algorithm operates under the assumption of knowing the current residual energy and channel state information, we have chosen \(\rho=0.98\) (see Figure 4), and small optimizations spans \(F=1\) and \(F=5\). By doing so, the past residual energy information used by our method does not exhibit very low correlation with the present one. We have measured the lifetime improvements with \(N=10\) and \(N=100\) sensor nodes to assess scalability. Moreover, we have analyzed the two extreme situations in terms of the weights in the optimization problem (4): \((w_{1},w_{2})=(0,1)\) is the configuration that produces an operation similar to the one achieved by the greedy algorithm, i.e., burst-like transmissions and larger lifetimes, whereas \((w_{1},w_{2})=(1,0)\) privileges the equalized consumptions of the sensor nodes to the detriment of smaller lifetimes. Whenever possible, conclusions are drawn regarding the results that would be obtained if different setups were chosen. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{2}{c}{**Mean Lifetime,**} & \multicolumn{1}{c}{**Lifetime**} & \multicolumn{1}{c}{**\% Mean Lifetime**} \\  & & & **in Frames** & **Std. Deviation** & **Improvement** \\ \hline \multirow{4}{*}{ \begin{tabular}{l} Death energy: 5\% \\ No recharge \\ \(B_{\text{min}}=0.1\) \\ \(B_{\text{max}}=1\) \\ \(\rho=0.98\) \\ \end{tabular}  } &