## 2 Network Environment and Problem Formulation
### Problem Formulation  
The major objective of this paper is to construct a trajectory \(J_{\min}^{\frac{\alpha-T}{2}}=(x_{a}^{\min}\,,x_{a-1}^{\min}\,,\ldots,x_{1}^{ \min}\,)\) for a mobile sink to collect data from each sensor. The design of the mobile sink's trajectory should meet the energy-balanced requirement that all sensors are energy-balanced when the mobile sink completes the movement of trajectory \(J_{\min}^{\frac{\alpha-T}{2}}\) in each round. The following shows the problem formulations of our work. The problem considered in this paper can be formulated as an integer linear programming labeled from Equations (6) to (11). Let \(e_{ij}\) denote the energy consumption required for sensor \(s_{j}\) after the mobile sink has traversed track \(k_{i}\) for one sweep repetition. Let \(e_{i,\min}^{\infty}\) denote the total energy consumption required for sensor \(s_{j}\) when the mobile sink completes the movement of trajectory \(J_{\min}^{\frac{\alpha-T}{2}}\) in each round. The value of \(e_{i,\min}^{\infty}\) can be calculated by Equation (4):  
\[e_{j,\min}^{\text{com}}=e_{1,j}x_{1}^{\min}+e_{2,j}x_{2}^{\min}+...+e_{(n-1), j}x_{n-1}^{\min}+e_{n,j}x_{n}^{\min}=\sum_{i=1}^{n}e_{i,j}x_{i}^{\min} \tag{4}\]  
To meet the energy-balanced requirement, an energy-balanced index \(f(e_{1,\min}^{\text{com}}\,,e_{2,\min}^{\text{com}}\,,\ldots,e_{n,\min}^{ \text{com}}\,)\), which is defined according to Jain's Fairness Index [16], is used to measure the degree of energy balancing, where \(m\) denotes the number of sensors in \(M\). The energy-balanced index \(f(e_{1,\min}^{\text{com}}\,,e_{2,\min}^{\text{com}}\,,\ldots,e_{n,\min}^{ \text{com}}\,)\) that is normalized between 0 and 1 can be formulated by Equation (5): \begin{table}
\begin{tabular}{l l} \hline \(\mathcal{Y}_{i,b}\) & Number of repetitions of sweeps for mobile sink \(b\) moving along track \(k_{i}\) when mobile sink completes the movement of the constructed trajectory \(J_{\min}^{\frac{\alpha-T}{2}}\) in each round. \\ \(\mathbf{t}^{\min}\) & Time duration for mobile sink traversing each track one repetition of sweep when it moves along the trajectory \(J_{\min}^{\frac{\alpha-T}{2}}\). \\ \(\mathbf{t}\) & A user predefined value. \\ \(\rho\) & Size of each packet. \\ \(g_{i}^{\prime}\) & Total amount of packets generated by each sensor in every time period \(t\). \\ \(z_{i}^{\prime}\) & Total amount of packets forwarded by sensor \(s_{i}\) in every time period \(t\). \\ \(N(s_{i})\) & The set of sensor \(s_{i}\)’s neighbors. \\ \(z_{i,j}^{r}\) & Total amount of packets received by sensor \(s_{i}\) from its neighbor \(s_{j}\) in every time period \(t\). \\ \hline \end{tabular}
\end{table} 
Table 1: Additional notations. \[f\Big{(}e_{1,\min}^{com},e_{2,\min}^{com},...,e_{m,\min}^{com}\Big{)} =\frac{\left(\sum_{\ell=1}^{m}e_{\ell,\min}^{com}\right)^{\frac{1}{2}}}{m\times \sum_{\ell=1}^{m}\Big{(}e_{\ell,\min}^{com}\Big{)}^{2}} \tag{5}\]  
In case that all \(e_{\ell,\min}^{com}\) have the same value where \(\forall_{s_{j}\in S}\), the result of the fairness index equals to 1, which is the optimal value. An energy-balanced index of a mechanism approaching 1 indicates that the mechanism provides better fairness in terms of energy balancing. Therefore, as shown in objective Function (6), this paper aims at maximizing the energy-balanced index \(f(e_{1,\min}^{com},e_{2,\min}^{com},...,e_{m,\min}^{com})\) while satisfying Constraint (7) to Constraint (11):  
data in track \(k_{collect}=k_{i}\). Because the average data transmission rate \(\overline{C_{i}}\) in track \(k_{collect}=k_{i}\) can be measured by:  
\[\overline{C_{i}}=\frac{1}{\left|S_{i}\right|}\sum_{\forall s_{j}\in S_{i}}c_{b,j},\]  
the \(t_{i}^{rec}\) can be simply calculated by:  
\[t_{i}^{rec}=\frac{\rho m}{C_{i}}.\]  
To ensure that the mobile sink has sufficient time to successfully receive all data generated by all sensors, the \(t^{min}\) should be greater than or equal to \(t_{i}^{rec}\). Hence, the lower bound on the time duration of the mobile sink collecting data is \(t_{i}^{rec}\).