## 4 System Model and Problem Formulation
### Network Model and Assumptions
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(N\) & The number of sensor nodes \\ \(g\) & Number of groups in the network \\ \(V\) & Optimal number of data collector vehicles (DCVs) \\ \(h\) & The total number of cluster heads (CHs) \\ \(\hat{C}\) & A set of all cluster heads (CHs) \\ \(r_{k}\) & Transmission range of sensor nodes \\ \(r\) & Total number of overlapped regions \\ \(s\) & Total number of optimal points \\ \(T_{m}\) & Total movement time of the data collector vehicle \\ \(\Omega_{OP}\) & A set of all optimal location points \\ \(OP_{i}\) & \(i\)th subset of \(\Omega_{OP}\) \\ \(v\) & Moving speed of a DCV \\ \(nOP_{i}\) & Number of optimal points in \(OP_{i}\) \\ \(OP_{inOR_{i}}\) & Number of optimal points assigned to \(DCV_{i}\) \\ \(OP_{i}\) & An optimal location point, \(1\leq j\leq s\) \\ \(R\) & A set of all regions \\ \(L(\theta)\) & The length of traveling path \(\theta\) \\ \(L_{path(\theta i)}\) & Total path length traveled by \(DCV_{i}\) \\ \(T_{d}\) & Total time spent by a DCV in one trip \\ \(T_{so}\) & Stopping time at an optimal point \\ \(T_{opt}\) & Optimal time \\ \(T_{c}\) & Deadline time constraint for data collection \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Notations with their descriptions. Figure 4: Network model and components for LS-WSN. The DCV starts its network tour from the BS and traverses along the designated OPs or DCPs to collect all the sensed data and finally returns to the BS for data offloading. Upon arrival at each region, the DCV communicates with the CH to collect all previously aggregated data. CH selection is dynamic, as every region reselects the CHs to minimize the hotspot problem and the data request flooding problem typical of multi-hop clustering. Data offloaded at the BS can be accessed by waste managers/authorities for decision-making purposes, including real-time routing and scheduling opertaions. For instance, waste collectors can be promptly scheduled and deployed for waste collection operations. Let \(\hat{C}\) denote a set of \(h\) cluster heads (CHs) in a 2D Euclidean plane, i.e., \(\hat{C}=c_{1},\ c_{2}\dots,\ c_{h}\) with \((x_{i},y_{i})\) coordinates of CH \(\hat{C}_{i}\) and \(1\leq i\leq h\). The neighborhood area of a CH (\(\hat{C}_{i}\)) is denoted by a circular disc (\(P_{k}\)), the center of which is \(k\) and the radius of which is \(r_{k}\), where \(r_{k}\) is a non-negative value. That is, \(P_{k}\) is the set of cluster heads (including CH \(\hat{C}\)) with coordinates \((x,y)\) such that:  
\[(x-x_{i})^{2}+(y-y_{i})^{2}\leq{r_{k}}^{2} \tag{1}\]  
That is, \(\{P_{k}|(x-x_{i})^{2}+(y-y_{i})^{2}\leq{r_{k}}^{2}|\), and \(\hat{C}\) is a CH with coordinates \((x,y)\)}. We note that any two discs, such as \(P_{k_{i}}\) and \(P_{k_{j}}\), may or may not overlap with each other. The DCVs visit the neighborhood regions of the CHs including the overlapped CH regions and optimal location points. It is also assumed that a DCV can receive data from a sensor lying within the vicinity and transmission range of the sensor (e.g., a 50 m range). Ignorance of this location point leads to a longer travel path, which results in large delays and an increase in energy consumption. Hence, it is expedient to determine the optimal DCPs for the DCVs to optimize energy consumption for big data collection in LS-WSNs. Let \(\hat{R}\) denote a set of r regions such that \(\hat{R}=\left\{\hat{R}_{1},\ \hat{R}_{2},\dots\hat{R}_{r}\right\}\) and \(\Omega_{OP}\) denote a set of \(s\) optimal points, i.e., \(\Omega_{OP}=\left\{OP_{1},\ OP_{2},\ \dots,\ OP_{3}\right\}\). Let \(D\left(OP_{j},\ OP_{j+1}\right)\) represent the distance between optimal points (\(OP_{j},\ OP_{j+1}\in\Omega_{OP}\)), and let \(T\left(OP_{j}\right)\) be the time spent by a DCV stopping at \(OP_{j}\). It is observed that the total time spent by a DCV on its tour is the sum of the traveling time and the stopping time at OPs in \(\Omega_{OP}\).