## 3 Periodic Wireless Sensor Network Model
### Energy Profile of a Network Node  
The second variable associated with time \(t\) is the binary variable \(e^{i}_{kt}=0,1\) taking the value \(1\) when the node \(v_{i}\) at time \(t\in F\) is in the energy state \(k\in 1,\ldots,K\). Otherwise, the variable takes the value \(0\). \begin{table}
\begin{tabular}{l l} \hline \(\mathcal{V}\) & set of nodes (vertices) in the network \\ \(v_{i}\) & \(i\)-th node in the network \(v_{i}\in\mathcal{V}\) \\ \(F\) & time frame \\ \(Q_{i}\) & buffer capacity of node \(v_{i}\) \\ \(q^{i}_{t}\) & number of data packets in the buffer of node \(v_{i}\) at time \(t\in F\) \\ \(e^{i}_{kt}\) & \(e^{i}_{kt}\in\mathbb{B}\) equals \(1\) if the node \(v_{i}\) at time \(t\in F\) is in the energy state \(k\in\{1,\ldots,K\}\), \(0\) otherwise \\ \(\xi^{i}\) & current energy level in the node \(v_{i}\) \\ \(A^{i}_{jk}\) & \(A^{i}_{jk}\in\mathbb{B}\), equals \(1\) when the transmission from the node \(v_{j}\in S_{D}\), which is in the energy state \(k\in S_{D}\), otherwise equals \(0\) \\ \(B^{i}_{jk}\) & \(B^{i}_{jk}\in\mathbb{B}\) equals \(1\) if the signal received at the node \(v_{i}\) cannot be decoded correctly, \(0\) if no interference \\ \(\phi^{i}_{k}\) & energy consumption level in node \(v_{i}\) \\ \(\gamma\) & SINR threshold \\ \(S^{i}_{E}\) & energy model of node \(v_{i}\) \\ \(I_{total}\) & objective function in minimization of total energy consumption \\ \(I_{max}\) & objective function in minimization of the maximal energy consumed by a node \\ \(\mathbb{B}\) & set of binary numbers \{0,1\} \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of Notation. It is assumed that the energy resource available to the device forming the network at any given moment is limited to the charge level of the battery supplying a node of the network. It is also assumed that each node in the network can operate in one of \(k=1,2,\ldots,K\) energy states (\(K\geq 3\)) associated with different energy consumption and transmitted signal power levels. The first index \(k=1\) is reserved for the sleep state, in which the radio module is not involved in communication--it is turned off. The index \(k=2\) denotes the listening state, during which a network node \(v_{i}\) is prepared to receive data from another node \(v_{j}\). The remaining energy states (\(2<k\leq K\)) are used during data transmission with different RF power levels assigned to different energy states. As the transmission signal's power level increases, the range of wireless transmission increases, which may cause changes in the topology of the considered ad-hoc network. Each of the energy states of the node \(v_{i}\) has been assigned an energy consumption level \(\phi_{k}^{i}\) generated by the radio communication module and a power level of the radio signal \(Pt_{k}^{i}\) transmitted by the device in the state \(k\). For the sleep state and the listening state, the power level of the transmitted signal is equal to 0 W. It is additionally assumed that as the state index \(k\) increases, the power level of the transmitted signal increases: \(Pt_{3}^{i}<\ldots<Pt_{K}^{i}\). With the transmission of a signal, there is a corresponding energy consumption \(\phi_{k}^{i}\) measured in units of [mA]. Using the introduced designations, the energy model \(S_{E}^{i}\) of the device \(v_{i}\) can be formulated:  
\[S_{E}^{i}=\{s^{i},e_{k}^{i}\quad:k=1,\ldots,K_{i}\}, \tag{1}\]  
where the binary variable \(e_{k}^{i}=\{0,1\}\) takes the value 1 when the node \(v_{i}\) at time \(t\) is in the energy state \(k\in\{1,\ldots,K\}\). Otherwise, the variable takes the value 0. \[e_{k}^{i}=\left\{ \begin{array}{ll}1&\quad\text{when node $v_{i}$ is in energy state $k$}\\ 0&\quad\text{when node $v_{i}$ is not in energy state $k$}\end{array} \right. \tag{2}\]  
A node \(v_{i}\) can only operate in one energy state at any given time \(t\).