## 4 Numerical Results
### Influence of System Parameters
#### 4.1.4 Effect of the Node Density  
As stated in Section 2, a larger ratio \(N/B_{\rm max}\) implies less sparse sensor nodes with respect to each other, i.e., a more dense network. Then, the node density variation can be analyzed by varying \(N\) for a fixed \(B_{\rm max}\) or by varying \(B_{\rm max}\) for a fixed \(N\). Both analyses are made in what follows, demonstrating the scalability of our method. Before presenting the results, let us elaborate a little bit more on the node density, first defining it as the ratio between the number of nodes and the area where they are deployed. Let \(d_{\rm min}\) and \(d_{\rm max}\) be the minimum and maximum distances that a sensor node can be from the sink node, respectively. The minimum distance will define \(B_{\rm min}\), and the maximum distance will define \(B_{\rm max}\). Assuming that the sink node wireless coverage is circular and that a sensor node can be at any position within this circular area, then \(d_{\rm min}=0\) and \(d_{\rm max}\) is the radius of the coverage area. Thus, the node density, in nodes per square meter, can be defined as  
\[D=\frac{N}{\pi d_{\rm max}^{2}}. \tag{6}\]  
In practice, the energy consumption will be proportional to the transmit power necessary for a target error rate in the received data at the sink node, which in turn will depend on the physical layer specifications. Assume that this target error rate is achieved if the received signal power at the sink node is \(P_{\rm target}\). From the log-distance path loss model ([PERSON], 1999; 200; 201),  
\[P_{\rm target}=\frac{P_{\rm ref}(d_{0})}{(d/d_{0})^{\eta}}, \tag{7}\]  
where \(P_{\rm ref}(d_{0})\) is a reference power at a close-in reference distance \(d_{0}\) from the sensor node transmitter, \(\eta\) is the environment-dependent loss exponent and \(d\) is the distance of analysis. Typical values of \(\eta\) range from two in the free-space propagation scenario to four or even more in obstructed areas. The energy consumed by a sensor node at a distance \(d\) from the sink node will be proportional to \(P_{\rm ref}(d_{0})\), that is it will be proportional to \((d/d_{0})^{\eta}P_{\rm target}\). Given that \(P_{\rm target}\) is constant and assuming \(d_{0}=1\) meter without loss of generality, then the energy consumption of a sensor node located at a distance \(d\) from the sink node will be proportional to \(d^{\eta}\). More specifically,  
\[ \begin{array}{l}B_{\rm min}=Cd_{\rm min}^{\eta},\\ B_{\rm max}=Cd_{\rm max}^{\eta},\end{array}  \tag{8}\]  
Figure 10: Residual energies (**a**) and activity levels (**b**) from the proposed optimization method for \((w_{1},w_{2})=(1,0)\), with recharge at \(t=t_{\rm r}=184\) and unequal initial energies. The non-optimized network is also considered on the graph (**a**) for reference. where \(C\) is a hardware-dependent and battery-dependent proportionality constant that accounts for the proper conversion from transmit power into consumed energy. From this result, we see that \(d_{\max}\) is proportional to \(B_{\max}^{1/\eta}\). With this result in Equation (6), it can be concluded that  
\[D\propto\frac{N}{B_{\max}^{2/\eta}}. \tag{9}\]  
Assuming \(\eta=2\) for simplicity, then the network density can be multiplied by any factor, multiplying \(N\) or dividing \(B_{\max}\) by this factor. In Figure 8, \(B_{\min}=0.001\), \(N=10\) and \(B_{\max}=1\) have been considered, yielding a node density \(D\propto 10\). Obviously, in a real network, the actual density will be typically smaller than 10 nodes per square meter, meaning that the above proportionality is typically smaller than one. Moreover, one must recall that we have rescaled the energy values to anticipate the network death, which is consistent with such a high node density. Figure 11 shows the residual energies as produced by the proposed method with a 10-fold increase in the node density with respect to the one considered in the graph (**b**) of Figure 8, which has been chosen as representative for the present analysis.