## 5 Problem Description, Formulation, and Solution  
Since our proposed model intersects between both network lifetime and data collection efficiency, this section will be divided into two segments: (i) battery charging, and (ii) data collection. We are looking at a network with N number of sensor-nodes distributed across network area as shown in Figure 4. In order to recharge a sensor node's battery, a wireless mobile battery charger (MBC) which is also used as a mobile data collector (MDC) opportunistically traverses the network for the purposes of data collection and battery charging. Since the movement is opportunistic, it is worth noting that the charger may be doing only charging when there is no data to collect from the sensor-nodes and vice versa. We will assume that the MBC takes off from a service station (S) with a traveling speed of V (m/s). On arrival at a sensor node \(\mathbf{r}\), the MBC spends \(\mathbf{\tau_{r}}\) quantity of time to wirelessly recharge the battery of the node. Let \(\mathbf{E_{t}}\) represent the rate of transmitted energy of the MBC. Having charged the sensor-node \(\mathbf{r}\), at the time of \(\mathbf{\tau_{r}}\), the MBC travels along its route to charge the next node. We will also assume that the MBC has enough energy to charge every sensor node in the network. A sensor node is assumed to have been fully charged when its battery capacity is \(\mathbf{E_{max}}\) and a minimum operational energy of \(\mathbf{E_{min}}\) is allowed. A network lifetime is threatened when the energy of the sensor node falls below the minimum operational energy \(\mathbf{E_{min}}\). The data collected by the MDC or data mule is transmitted to base station or access point (AP). A sensor node \(\mathbf{r}\) transmits its generated data to the MDC at a rate of \(\mathbf{\mathcal{R}_{r}}\) (bits/s), so that \(\mathbf{r}\in N\). Each sensor node a certain amount of energy for data transmission. Here, we will adapt the energy consumption analysis for data mule model in [4,9]. We assume that the mobile battery charger (MBC) returns to the service to be serviced (maybe to replace or recharge its battery) before making another opportunistic movement to the network. Let \(\mathbf{R}=(\mathbf{\pi}_{0},\mathbf{\pi}_{1},\mathbf{\pi}_{3},\ldots,\mathbf{\pi}_{N})\) denote the physical route traversed by the wireless mobile battery charger (MBC) over a trip cycle beginning from and ending at the service (S) station, and \(\mathbf{r_{th}}\) node covered by MBC along a predetermined route (R) is \(\mathbf{\pi_{r}}\), for \(1\leq\mathbf{r}\leq|\mathbf{N}|\). If \(\mathbf{\mathcal{D}_{\mathbf{\pi}_{0}}}\mathbf{\pi}_{1}\) denotes the distance between the MBC and the first sensor node visited along R and \(\mathbf{\mathcal{D}_{\mathbf{\pi}_{k}}}\mathbf{\pi}_{k+1}\) denote the distance between the \(\mathbf{n_{th}}\) and \((\mathbf{n}+1)_{th}\) sensor nodes, \(\mathbf{t_{i}}\) denotes the time of arrival of the MBC at node \(\mathbf{r}\), then we obtain:  
\[\mathbf{t_{\mathbf{\pi}_{i}}}\ =\ \mathbf{\tau}+\ \sum_{\mathbf{k}=0}^{i-1}\frac{\mathbf{\mathcal{D} _{\mathbf{\pi}_{k}}}\mathbf{\pi}_{k+1}}{\mathbf{V}}+\ \sum_{\mathbf{k}=1}^{i-1}\mathbf{\tau_{\mathbf{\pi}_{k}}} \tag{1}\]  
Let \(\mathbf{D_{R}}\) denote the physical distance of route \(\mathbf{R}\) and \(\mathbf{\tau_{r}}=\frac{\mathbf{D_{R}}}{\mathbf{\Psi}}\) being the time spent by MBC to travel over a distance \(\mathbf{D_{R}}\). Then, the renewable cycle time putting into consideration the vacation time of the battery charg \(\mathbf{\tau_{\mathbf{\tau_{\mathbf{\tau_{\mathbf{\tau_{\mathbf{\tau_{\bm  
\[ \begin{split}&\text{But}\,\mathbf{e}_{i}(\mathbf{a}_{i})=\mathbf{E}_{t}-(\ \mathbf{a}_{i}-\mathbf{\tau})\mathbf{R}_{i}\geq\mathbf{E}_{min}\,\text{and}\,\mathbf{e}_{i}(\mathbf{a} _{i})=\ \mathbf{E}_{max}-(2\mathbf{\tau}-\mathbf{a}_{i}-\mathbf{\tau}_{i})\mathbf{R}_{i}-(\ \mathbf{a}_{i}-\mathbf{\tau})\mathbf{R}_{i}\\ =\mathbf{E}_{max}-(\mathbf{\tau}-\mathbf{\tau}_{i})\mathbf{R}_{i}\end{split}