## 4 Numerical Results
### Influence of System Parameters
#### 4.1.1 Effect of the Objective Function Weights  
Figure 6 shows the residual energies attained with the proposed optimization method under different weights in the objective function defined in Equation (3), namely \((w_{1},w_{2})=(1,0)\) (**a**), \((w_{1},w_{2})=(0,1)\) (**b**) and \((w_{1},w_{2})=(1,2)\) (**c**). The non-optimized network is considered in all graphs for reference. We have set \(N=10\), \(K=400\), \(F=1\), \(\rho=0.98\), no recharge, \(B_{\min}=0.001\), \(B_{\max}=1\), and identical initial energies \(\mathbf{s}_{1}(1)=S_{\max}\times\mathbf{1}\), with \(S_{\max}=10\). A single realization of the energy consumptions was adopted. In the majority of graphs from here, for the sake of clarity we intentionally did not use lines with symbols; only colors were used to identify different sensor nodes. From the graph (**a**) of Figure 6, it can be inferred that all sensor nodes are allowed frequent communication with the sink node in a way that they spend approximately the same amount of energy as time elapses, that is, nodes will die altogether. Moreover, the unused residual energies at the network death moment, i.e., the wasted energy, is minimum due to the equalized consumptions. As will be shown ahead, for \(F>1\), smaller \(\rho\) or both, the adoption of \((w_{1},w_{2})=(1,0)\) would not equalize the consumptions precisely, yet reducing the network lifetime. This is due to the use of past information to optimize the current block of \(F\) frames: lower correlations combined with longer blocks represent past information less reliable with respect to the moment when it is used. From the graph (**b**) of Figure 6, which considers \((w_{1},w_{2})=(0,1)\), it can be observed that the optimization allows for burst-like transmissions combined with silent periods and continuous transmissions. The burst-like transmissions correspond to steep changes of the residual energies, whilst the intervals during which the residual energies do not change are associated with disabled (silent) nodes. Notice that this form of weighting achieves an increased network lifetime with respect to \((w_{1},w_{2})=(1,0)\). As will be demonstrated later on in this paper, this effect is even more pronounced when the number of sensor nodes is larger. In the graph (**c**) of Figure 6, the weights applied to the objective function were \((w_{1},w_{2})=(1,2)\), resulting in a mixed situation in which consumptions are not perfectly equalized anymore and only some small burst-like transmissions and silent periods occur. No penalty and no improvement is observed in the network lifetime with respect to the one achieved when \((w_{1},w_{2})=(1,0)\), which happens for a moderate to small number of sensor nodes. If \(N\) is large, both \((w_{1},w_{2})=(1,0)\) and \((w_{1},w_{2})=(1,2)\) would bring a lifetime penalty, which will be smaller for \((w_{1},w_{2})=(1,2)\). In all situations considered in Figure 6, the lifespan improvements were equal to or greater than \(375/115\approx 3.27\) times. Even larger lifetime records and, thus, lifetime improvements can be unveiled 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, if the consumptions are decreased for the same initial energy or if the number of sensor nodes is increased. This happensbecause the slopes associated with the residual energies in the optimized and non-optimized networks become more separated from each other as \(t\) increases. Then, in a real network in which the expected lifetimes are much higher than those simulated here, one might expect considerably larger lifetime improvements achieved with the proposed method. Another relevant conclusion obtained from Figure 6 refers to the fact that the energy levels tend to be more continuous when \((w_{1},w_{2})=(1,0)\) and more step-like when \((w_{1},w_{2})=(0,1)\).