## 3 The Proposed Energy-Balanced Data Collection (EBDC) Mechanism
### Trajectory Construction Phase
The \(e_{max}^{EBDC}\) can be formulated by Equation (29):  
\[e_{t_{unit}}^{EBDC}=\frac{e_{n}^{total}}{t_{round}^{EBDC}}=\frac{e_{1\times n} x_{1}^{\min}+e_{2\times n}x_{2}^{\min}+...+e_{(n-1)\times n}x_{n-1}^{\min}+e_{n=n }x_{n}^{\min}}{t\times\sum_{i=1}^{n}x_{i}^{\min}} \tag{29}\]  
Since the energy consumption of sensor \(s_{l}\) can be derived by the total energy consumption in track \(k_{n}\) divided by the total number of sensors in track \(k_{n}\), Equation (30) can be obtained by substituting Equations (20) and (23) into Equation (29):\[e_{lunit}^{EBDC}=\!\!\left(\frac{E_{1\times\alpha}x_{1}^{\min}+E_{2\times\alpha}x_ {2}^{\min}+...+E_{(n-1)\times\alpha}x_{n-1}^{\min}+E_{n=\alpha}x_{n}^{\min}}{t \times\sum_{i=1}^{\alpha}x_{i}^{\min}}\!\!\left(\frac{1}{|S_{n}|}\right)\right) \tag{30}\]  
Furthermore, the total energy consumption in track \(k_{n}\) can be evaluated by the multiplication of the total number of packets and the energy consumption required for transmitting a packet, Equation (31) can be obtained by substituting Equations (19) and (22) into Equation (30):  
\[e_{lunit}^{EBDC}=\!\left(\frac{P_{n}^{1\times\alpha}x_{1}^{\min}+P_{n}^{2\times \alpha}x_{2}^{\min}+...+P_{n-1}^{(n-1)\times\alpha}x_{n-1}^{\min}+P_{n}^{n= \alpha}x_{n}^{\min}}{t\times\sum_{i=1}^{\alpha}x_{i}^{\min}}\!\!\left(\frac{e _{unit}}{|S_{n}|}\right)\right) \tag{31}\]  
Since each sensor generates \(p\) packets in each time period \(t\), based on the Equations (18) and (21), the \(e_{lunit}^{EBDC}\) can be further derived by Equation (32):  
\[e_{lunit}^{EBDC}=\!\!\left(\frac{x_{1}^{\min}\!\sum_{i=1}^{\alpha }\!\!\left[S_{i}\right]\!+\!x_{2}^{\min}\!\!\left[S_{i}\right]\!+\!...+\!x_{n- 1}^{\min}\!\!\sum_{i=1}^{\alpha}\!\!\left[S_{i}\right]\!+\!x_{n}^{\min}\!\! \left[S_{i}\right]\!}{t\times\sum_{i=1}^{\alpha}x_{i}^{\min}}\!\!\left(\frac{p \times e_{unit}}{|S_{n}|}\right)\right) \tag{32}\] \[=\!\!\left(\frac{\sum_{j=1}^{n-1}x_{j}^{\min}+x_{n}^{\min}\sum_{i =1}^{\alpha}\!\!\left[S_{i}\right]}{\sum_{j=1}^{\alpha}x_{j}^{\min}}\!\!\left( \frac{p\times e_{unit}}{t}\right)\right)\]  
Equation (32) indicates that the energy consumption of sensor \(s_{l}\!\in\!S_{n}\) is highly impacted by the parameters, including the number of sweep repetitions performed by the mobile sink and the total number of sensors in each track. On the other hand, the energy consumption of \(RMDC\) scheme is analyzed below. Recall that the key idea of \(RMDC\) scheme is that mobile sink moves along the boundary track of \(M\) for collecting data generated by all sensors in \(M\). For simplicity, we discuss the energy consumption of sensor \(s_{l}\) that is located in the boundary track \(k_{n}\). To facilitate the analysis, herein, the _round of data collection in \(RMDC\) scheme_ is initially introduced. As shown in Figure 1, when a mobile sink completes the movement of boundary track \(k_{n}\) one sweep repetition in the clockwise direction, the mobile sink is said to move along the boundary of \(M\) in one round. Let \(S\) denote the set of all sensors in \(M\). Let \(S_{n}\) denote the set of boundary sensors, each of which can communicate with mobile sink when the sink passes through its transmission range. Since the mobile sink always moves along the boundary track \(k_{n}\), only sensors belonging to set \(S_{n}\) are able to play the relay roles to deliver the data generated by the sensors belonging to sets \(S-S_{n}\) to mobile sink. Let \(P_{round}^{RMDC}\) denote the workloads of sensors belonging to set \(S_{n}\) by applying \(RMDC\) scheme. The value of \(P_{round}^{RMDC}\) can be evaluated by Equation (33):  
\[P_{round}^{RMDC}=p\times\sum_{i=1}^{n}\left|S_{i}\right| \tag{33}\]  
Let \(E_{round}^{RMDC}\) denote the total energy consumption required for all sensors belonging to set \(S_{n}\) in each round. The value of \(E_{round}^{RMDC}\) can be calculated by Equation (34):  
\[E_{round}^{RMDC}=P_{round}^{RMDC}\times e_{unit} \tag{34}\]  
As a result, we have \(a-c>b-c\).