## 3 The Proposed Energy-Balanced Data Collection (EBDC) Mechanism
### Energy Estimation Phase  
This phase aims to evaluate the energy consumption of each sensor. For the ease of presentation, the following initially introduces several notations. Let \(k_{i}\)\(\delta\)\(k_{j}\) denote the relative location relation of tracks \(k_{i}\) and \(k_{j}\) where \(\delta\in\{<,>,=\}\). The value of \(\delta\) is '\(<\)', '\(=\)', or '\(>\)' if the value of (\(O_{i}-O_{j}\)) is less than, equal to, or greater than zero. The notation \(e_{i\delta}\) denotes the energy consumption required for each sensor located in track \(k_{j}\) when the mobile sink completes the movement of track \(k_{collect}=k_{i}\) for one sweep repetition. Figure 4 depicts an example which illustrates the observation on \(e_{i\delta}\) for different values of \(\delta\). Let sensors \(s_{a}\), \(s_{b}\), and \(s_{c}\) be located in tracks \(k_{1}\), \(k_{2}\), and \(k_{4}\), respectively. Assume that the mobile sink has  
Figure 3: The number of sensors in each track can be evaluated in the _Initialization Phase_. already completed the movement of track \(k_{2}\) for one sweep repetition. Since sensor \(s_{a}\) is located in track \(k_{1}\), the overall workload of \(s_{a}\) is to send its reading to its neighbor in track \(k_{2}\). On the contrary, sensor \(s_{e}\) not only sends its reading to its neighbor in track \(k_{3}\) but also needs to relay the data received from the other sensors located in the outer tracks. As a result, we have \(e_{2}<4>e_{2}>1\). Furthermore, the workload of sensor \(s_{b}\) is larger than that of sensors \(s_{a}\) and \(s_{e}\) since \(s_{b}\) not only sends its reading to the mobile sink but also relays data from all tracks other than track \(k_{2}\) to mobile sink. Consequently, the relations \(e_{2}=2>e_{2}>1\) and \(e_{2}=2>e_{2}<4\) hold. According to this observation, we conclude that sensors located in different tracks have different energy consumptions. Let sensor \(s\) be located in track \(k_{y}\) and mobile sink has already completed the movement of track \(k_{x}\) for one sweep repetition. Based on the relation \(\delta\), the energy consumption \(e_{xy}\) of sensor \(s\) is discussed for the following three cases. **Case 1. **\(k_{x}>k_{y}\)  
Let \(S_{y}\) denote the set of sensors located in track \(k_{y}\) and let \(|S_{y}|\) denote the number of sensors belonging to set \(S_{y}\). Let \(p\) be the number of packets generated by each sensor in each time period \(t\). Let \(P_{y}^{x>y}\) represent the total number of packets delivered by all sensors located in the track \(k_{y}\), for all \(y<x\). The value of \(P_{y}^{x>y}\) can be measured by Equation (15):  
\[P_{y}^{x>y}=\sum\
olimits_{t=1}^{y}\big{|}S_{t}\big{|}\times p \tag{15}\]  
Let \(e_{unit}\) denote the energy consumption required for each sensor to transmit one packet to its neighbor. Let \(E_{x\;>\;y}\) denote the total energy consumption required for all sensors located in track \(k_{y}\) when the mobile sink completes the movement of track \(k_{x}\) one sweep repetition, for all \(y<x\). The value of \(E_{x\;>\;y}\) can be calculated by Equation (16):  
\[E_{x>y}=P_{y}^{x>y}\times e_{unit} \tag{16}\]  
Consequently, the value of \(e_{x>y}\) can be evaluated by Equation (17):  
\[e_{x>y}=E_{x>y}\;/\;\big{|}S_{y}\big{|} \tag{17}\]  
Figure 4: Sensors located in different tracks have different energy consumptions. **Case 2. **\(k_{x}<k_{y}\)  
Let \(P_{y}^{x<y}\) denote the total number of packets delivered by all sensors located in track \(k_{y}\), for all \(y>x\). The value of \(P_{y}^{x<y}\) can be measured by Equation (18):  
\[P_{y}^{x<y}=\sum\
olimits_{l=y}^{n}\left|S_{l}\right|\times p \tag{18}\]  
Let \(E_{x<y}\) denote the total energy consumption required for all sensors located in track \(k_{y}\) when the mobile sink completes the movement of track \(k_{x}\) for one sweep repetition, for all \(y>x\).