## 5 Problem Description, Formulation, and Solution
 \tag{5}\]  
Some constraints may arise in several scenarios where for instance, (1) the charging rate of MBC is too small or the energy consumption rate of the sensor node is too large; and (2) the interval of time of visit of MBC to a node for battery recharging is too large. However, these are common physical parameters that tend to limit the MBC from achieving a lifetime renewable network for a WSN. To determine the energy consumption for the data mule model when it collects data from the sensor nodes, we adapt the energy consumption analysis in [9]. Two cases are considered for the total energy consumption: (1) the energy consumption for data initiation (the data mule broadcasts the DREQ to the nodes and subsequently floods the network. This is denoted by the length of DREQ as \(\mathbf{L}_{DREQ}\) and (2) the energy consumption for data collection (the nodes transmit their data to the mobile collector. The length of data collection is denoted by \(\mathbf{L}_{DATA}\)). The energy consumptions for the duo have been derived in [9] and are given as:  
\[\mathbf{E}_{init}(\mathbf{r})=\mathbf{N}\mathbf{L}_{DREQ}\times\left\{\mathbf{E}_{tx}(\mathbf{r})+ \frac{\mathbf{r}^{2}}{\mathbf{R}\mathbf{\epsilon}^{2}}(\mathbf{N}_{c}-1)\mathbf{E}_{\mathbf{rx}}\right\} \tag{6}\]  
where \(\mathbf{E}_{init}(\mathbf{r})\) is the energy consumption for data initiation, \(\mathbf{N}\) is the total number of IoT nodes in the network and \(\mathbf{E}_{tx}(\mathbf{r})\) is the energy consumption by a node for a transmission covering the neighborhood of r radius and is denoted by [9]:  
\[\mathbf{E}_{tx}(\mathbf{r})=\mathbf{e}_{tx}+max\{\mathbf{e}_{min},\ \mathbf{e}_{out}\mathbf{r}^{ \alpha}\} \tag{7}\]  
where \(\alpha\) denotes the path attenuation factor, \(\mathbf{e}_{tx}\) is the energy consumption of the transmitter, \(\mathbf{e}_{out}\) is the output of the antenna, and \(\mathbf{e}_{min}\) is the energy radiated irrespective of the range of transmission; \(\mathbf{E}_{tx}\) is the energy consumption of a node for receiving, and \(\mathbf{r}\) is the optimal node transmission radius given by:  
\[\mathbf{R}_{c}\sqrt{\frac{\mathbf{logN}}{\mathbf{N}_{c}}}\]  
Hence, the total energy consumption for all the nodes for data initiation is given by:  
\[\mathbf{\varepsilon}_{INIT}(\mathbf{r})=\mathbf{N}\ \times\ \mathbf{E}_{init}(\mathbf{r}) \tag{8}\]  
Similarly, the energy consumption for data transmission from the nodes to the mobile collector (data mule) depends on the number of nodes responding to the DREQ with available data to transmit in respect of the request from the data mule. Given a large number of nodes \(\mathbf{N}\) (\(\mathbf{N}\rightarrow\infty\)), the energy consumption for data collection as modeled in [9] is given by:  
\[\mathbf{\varepsilon}_{collect}(\mathbf{r})=\mathbf{\lambda}_{c}\mathbf{N}_{c}\mathbf{L}_{DATA} \frac{2\mathbf{R}_{c}}{3\mathbf{r}}\left\{\mathbf{E}_{tx}(\mathbf{r})+\ \frac{\mathbf{r}^{2}}{\mathbf{R}\mathbf{ \epsilon}^{2}}(\mathbf{N}_{c}-1)\mathbf{E}_{\mathbf{rx}}\right\} \tag{9}\]  
where \(\mathbf{\lambda}_{c}\) is the average collection response rate of the node. Then, the total energy consumption for all sensor-nodes for data collection is given by:  
\[\mathbf{\varepsilon}_{COLLECT}\left(\mathbf{r}\right)=\mathbf{N}\ \times\ \mathbf{\varepsilon}_{collect}(\mathbf{r}) \tag{10}\]Total energy consumption for the network is the summation of the energy consumption for data initiation and data collection:  
\[\varepsilon_{N}\left(r\right)=\ \varepsilon_{\text{{INIT}}}(r)+\ \varepsilon_{\text{{COLLECT}}} \left(r\right) \tag{11}\]  
WPT Empirical Model: We adapt the empirical model for wireless power transfer in [5] to determine the effect of varying distances on the power transfer efficiency. Wireless mobile charging system performance is dependent on several factors including efficiency of the wireless power transfer, energy consumption of the sensor for data collection and transmission, feasible route locations, travelling time constraints, [5], etc.