## 2 Problem Statement and Formulation
In fact, there is no reason for having \(\textbf{b}_{1}(1)\
eq\textbf{b}_{2}(1)\
eq\cdots\
eq\textbf{b}_{F}(1)\). This would be impractical, since it would demand the prediction of the nodes movement during the first \(F\) frames. Then, if the sensor nodes positions are known at the initial deployment of the network, \(\textbf{b}_{1}(1)\) can be estimated by means of simple link budgets. The transmit powers considered in this estimation are those enough for producing the target error probability at the sink node (the calculations regarding this link budget are exemplified in Section 4.1.4). Alternatively, the elements of \(\textbf{b}_{1}(1)\) can be, for instance, the energy consumptions calculated from an average expected distance between the sensor nodes and the sink node. The subsequent maximum consumption matrix is computed based on the energy consumed by the sensor nodes during the previous \(F\) frames, taking into account past battery recharges and mobility. Figure 2: Time slotted approach for optimizing the activity levels of the mobile sensor nodes. * From above, each row of the maximum consumption matrix can be formed from correlated values of a random variable that will simulate variations in the residual energies as illustrated in Figure 3. The correlation level will determine the rate of variation in the energy consumptions from one frame to the next due to the previously-described combined effects. Specifically, let the auxiliary parameters \(B_{\text{min}}\) and \(B_{\text{max}}\) denote the minimum and maximum energy consumptions if the sensor node transmitter is on during the whole interval of a frame (which only happens if its activity level is one), when positioned at minimum and maximum distances from the sink node, respectively. Then, to mimic sensor nodes deployed at random positions, yielding different distances from the sink node and, thus, different consumptions, the elements of \(\mathbf{b}_{1}(1)=\mathbf{b}_{2}(1)=\cdots=\mathbf{b}_{F}(1)\) are assumed to be uniformly distributed in \([B_{\text{min}},B_{\text{max}}]\). A larger ratio \(N/B_{\text{max}}\) implies less sparse sensor nodes with respect to each other, i.e., a more dense network. The correlation coefficient between the energy consumptions in two consecutive frames is denoted by \(\rho\). As an example, Figure 4 shows two realizations of the maximum consumptions of five sensor nodes under two values of \(\rho\). The method adopted for generating the correlated uniform random variates shown in this figure is from the [31] (Chapter 7) and [32], assuming a triangular correlation function to guarantee the same correlation coefficient between the consumptions during any two neighbor frames. A low mobility network is represented in the graph (**a**) of this figure, whilst a relatively higher mobility is considered in the graph (**b**). In this last case, notice from the dashed line that the maximum consumption of a sensor node goes from \(B_{\text{max}}\) to \(B_{\text{min}}\) within 10 frames, which represents a quite large node speed in practice, despite the apparently high value of \(\rho\). * It is defined that the lifetime of a network is the time interval during which all sensor nodes are in full operation. In other words, the instant at which the first sensor node fails with high probability (does not work properly) or fails permanently (ends its operation or die) due to insufficient energy will determine the network lifetime. This definition has been adopted in several references, as for instance in [13; 21; 23; 24; 33; 34; 35; 36; 37]. The value of \(t\) at which the residual energy of any sensor node of the network becomes less than or equal to a given fraction of its maximum stored energy \(\max\{\mathbf{s}_{1}(1)\}\) is defined as the death instant, \(t_{\text{d}}\), and the corresponding residual energy is defined and the death energy \(s_{\text{d}}\).