## 4 Numerical Results
### Performance Comparisons  
In this subsection we compare our lifetime optimization method with the so-called greedy algorithm from [21]. No recharge is assumed, as required by the greedy algorithm. In this algorithm, which is the best among those analyzed in that paper, the balance between minimizing the wasted energy and minimizing the energy spent to report the sensory information to the sink node is achieved by means of exploring both channel state information and residual energy information in real time. The net result is that each sensor node is allowed to transmit in bursts under the control of the sink node (or other central element), while the remaining nodes stay silent. The energy consumptions of the nodes thus vary in sharp steps, but in such a way that the overall energy is saved on average, prolonging the network life. Figure 11: Residual energies obtained by the proposed optimization method for \((w_{1},w_{2})=(1,0)\), \(F=10\), \(B_{\min}=0.001\) and \(\rho=0.98\), with \(N=10\) and \(B_{\max}=0.1\) (**a**); and \(N=100\) and \(B_{\max}=1\) (**b**). The non-optimized network is also considered for reference. The greedy algorithm works as follows: the sensor node exclusively selected for transmission at a given moment is the one with maximum energy-efficiency index [21]  
\[\gamma_{n}=e_{n}-E_{r}(c_{n}), \tag{10}\]  
where \(e_{n}\) is the residual energy in the sensor node \(n\) at the beginning of a transmission and \(E_{r}(c_{n})\) is the required reporting energy as a function of the channel gain \(c_{n}\) from the \(n\)-th sensor node to the sink node. Adapted to our notation, \(E_{r}(c_{n})\) corresponds to the element \(B_{n,f}(k)\) of the maximum consumption matrix \(\mathbf{B}(k)\) at each \(k\) and \(f\). The residual energy \(e_{n}\) starts with \(s_{n,1}(1)\), i.e., the initial energy in the \(n\)-th sensor battery, and is updated for each \(k\) and \(f\) according to energy expenditure provided by the greedy algorithm, that is \(e_{n}\gets e_{n}-X_{n,f}(k)B_{n,f}(k)\), with \(X_{n,f}(k)\) being the activity state, which is one for \(n=\arg_{i}\max\gamma_{i}\), and zero otherwise. From this formulation, it is evident that the greedy algorithm uses current (real-time) channel state information \(c_{n}\) and residual energy information \(e_{n}\), whereas our method only uses past residual energy information. Notice that the term activity state is used here in the context of the greedy algorithm only to represent the _on_ (activity state 1) and _off_ (activity state 0) states of the sensor nodes' transmitters as determined by the algorithm, thus establishing consistency with the term activity level defined in this paper. Figure 12 shows the residual energies (**a**) and corresponding sensor nodes' activity states (**b**) obtained from the greedy algorithm of [21]. As in the case of Figures 6 and 7, to plot Figure 12 we have also adopted \(K=400\) optimization events, each one spanning \(F=1\) frame; \(N=10\) sensor nodes; identical initial energies \(\mathbf{s}_{1}(1)=10\times 1\); no recharge; death energy equal to 5% of the maximum; correlation coefficient of the consumptions given by \(\rho=0.98\); and minimum and maximum consumptions \(B_{\min}=0.01\) and \(B_{\max}=1\). Again, a small number of sensor nodes was assumed for not polluting the graphs; higher values of \(N\) are considered later on in this section. The same single realization of the random consumption used to plot Figure 6 has been adopted here again. From Figure 12, one can firstly see the burst-silent communication nature of the greedy algorithm, which happens independently of the number of sensor nodes. Observing the residual energies of the non-optimized network, it can be seen that the node numbered 2 has the steepest energy expenditure during the first 120 frames, implying that it is the farthest away from the sink node, on average, during the corresponding interval. As a consequence, this node was granted with fewer transmissions in this interval, as can be better noticed from the graph (**b**).