## 3 The Proposed Energy-Balanced Data Collection (EBDC) Mechanism
### Trajectory Construction Phase  
In this phase, the trajectory \(J_{\min}^{x=x}\) of the mobile sink will be scheduled using the information obtained in the previous two phases, such that the energy consumptions of all sensors can be totally balanced. \begin{table}
\begin{tabular}{c c c} \hline
**Case** & **Relation** & **Result** \\ \hline
1 & \(k_{x}>k_{y}\) & \(e_{x>y}=E_{x>y}/\left|S_{y}\right|\) \\
2 & \(k_{x}<k_{y}\) & \(e_{x<y}=E_{x<y}/\left|S_{y}\right|\) \\
3 & \(k_{x}=k_{y}\) & \(e_{x>y}=E_{x>y}/\left|S_{y}\right|\) \\ \hline \end{tabular}
\end{table} 
Table 2: The energy consumption of sensor \(s\). Let \(e_{j}^{\text{\tiny{max}}}\) denote the total energy consumption required for any sensor belonging to set \(S_{j}\) when the mobile sink moves one round along the trajectory \(J_{\text{\tiny{min}}}^{\text{\tiny{EB-$T$}}}=(x_{n}^{\text{\tiny{min}}},x_{n-1}^{ \text{\tiny{min}}},\ldots,x_{n}^{\text{\tiny{min}}})\). Recall that \(x_{i}^{\text{\tiny{min}}}\) denotes the number of sweep repetitions of that mobile sink as it moves along track \(k_{i}\). The \(e_{j}^{\text{\tiny{max}}}\) can be derived by Equation (24):  
\[e_{j}^{\text{\tiny{total}}}=\sum_{\text{\tiny{$\epsilon$}}=1}^{\text{\tiny{u}}} e_{i\text{\tiny{$\epsilon$}}}x_{i}^{\text{\tiny{min}}},\text{ where }\delta\in\{\!\!\!\!\!\succ,\!\!\!\!\succ,\!\!\!\!\succ\} \tag{24}\]  
As shown in objective function (6), the goal of this paper is to maximize the energy-balanced index \(f(e_{1,\text{\tiny{min}}}^{\text{\tiny{com}}},e_{2,\text{\tiny{min}}}^{\text{ \tiny{com}}},\ldots,e_{n,\text{\tiny{min}}}^{\text{\tiny{com}}})\). To accomplish this, the energy consumptions of any two sensors should be identical when the mobile sink completes the movement of trajectory \(J_{\text{\tiny{min}}}^{\text{\tiny{EB-$T$}}}\) for one round. Consequently, according to Equation (24), we can obtain Equation (25):  
\[ \begin{bmatrix}e_{1\text{\tiny{l}}}&e_{2\text{\tiny{l}}}&\ldots&e_{n\text{ \tiny{l}}}\\ e_{1\text{\tiny{l}}2}&e_{2\text{\tiny{l}}2}&\ldots&e_{n\text{\tiny{l}}2}\\ \vdots&\vdots&\ddots&\vdots\\ e_{1\text{\tiny{$\epsilon$}}\alpha}&e_{2\text{\tiny{$\epsilon$}}\alpha}&\ldots &e_{n\text{\tiny{$\epsilon$}}=n}\end{bmatrix} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!degree of energy balancing, another crucial factor considered in WSNs is the network lifetime. Herein, the network lifetime is measured by the time interval starting from the time that sensors have been deployed to the time that a coverage hole appears. To verify the performance of the network lifetime, this section further compares the proposed _EBDC_ mechanism against the _RMDC_ scheme proposed in [11]. The _RMDC_ scheme is considered as approach to compare because _RMDC_ outperforms related schemes [5, 6, 7, 8, 9, 10]. In general, the related data collection schemes can be mainly classfied into _fixed sink schemes_[5, 6, 7] and _mobile sink schemes_[8, 9, 10, 11]. Unlike the fixed sink schemes [5, 6, 7] which are based on a fixed sink, _RMDC_ employed a mobile sink to collect data. Hence, the _RMDC_ has a better performance than the existing fixed sink schemes [5, 6, 7] in terms of network lifetime. Recall that the mobile sink applying the efforts described in [8, 9] has to pass through the transmission range of each sensor, thereby leading to a long data collection latency of each sensor. On the contrary, the mobile sink which applies the _RMDC_ scheme does not need to pass through the transmission range of each sensor.