# Efficient Data Gathering in 3D Linear Underwater Wireless Sensor Networks Using Sink Mobility
## 4 Framework and Formal Definition of the Problem
### Relationship between Nodes and MS  
UWSNs can be represented by a directed graph \(G=(V,E)\), where \(V\) is the set of vertices (nodes), _i.e.,_\(\mid V\mid=n\), where, \(n=1,2,3,...,300\). \(E\) are the edges that are links between the nodes. We consider the cuboid field of dimensions 500 m \(\times\) 500 m \(\times\) 1000 m, which is further logically divided into four RCs, shown in Figure 3. In one RC, the MS gathers the data from the nodes, and in the remaining three RCs, CNs are appointed to collect sensed data from the nodes. CNs further forward the received data to the MS, shown in Figure 4. The MS and CNs stop at locations defined on their trajectories. The set of stop locations of the MS and CNs is defined as \(S=s_{1},s_{2},...,s_{m}\). If the node \(n_{i}\) is in the transmission radius of the MS, this means that \(n_{i}\) is the neighbor of the MS, and the data transmission is direct between them, as shown in Figure 5. When the MS moves to the next sink stop, it again finds the neighbor nodes by broadcasting a control packet that contains its location information. \(n_{i}\) is a non-neighbor of the MS if it is not in the same RC, where the MS is deployed. Then, it may be a neighbor of any \(C_{i}\). Where \(C_{i}\) belongs to CNs that move in each of the remaining linear RC regions and gather data from \(n^{\prime}_{i}\), \(C_{i}\) relays the data of \(n^{\prime}_{i}\)s to the MS.  
Figure 4: Courier node (CN) to mobile sink (MS) transmission. Figure 3: Network model. To support the proposed model, a set of linear programming equations is given as follows: Objective function:  
\[Maximize\quad T=\sum_{s}t_{s_{i}}+\sum_{t}t_{t}\qquad\forall s_{i}\in S \tag{1}\]  
subject to: \(h(i)\leqslant d(n_{i},n_{j})+h(j)\qquad\forall i\in N,j=1,2,3,4\) (2)  
\[T_{max}\cdot f_{i}\geqslant t_{s}\geqslant T_{min}\cdot f_{i}\qquad\forall i\in N \tag{3}\]  
\[\sum_{j=1}^{4}\sum_{i=1}^{n}d(n_{i},n_{j})f_{i}\leqslant l\qquad\forall i\in N,j \tag{4}\]  
\[T\cdot(\sum_{j=1}^{4}\sum_{i=1}^{n}p_{ij})\leqslant E_{i}\qquad\forall i\in N, j=1,2,3,4 \tag{5}\]  
\[E_{i}(t_{a})\geqslant E_{(min)}^{tx}\qquad\forall i\in N \tag{6}\]  
Figure 4.0  
Figure 4.1  
In Equation (1), \(T\) is the NLT, \(t_{s}\) defines the time duration that the MS or CN spends on a stop for data gathering, and \(t_{t}\) shows the traveling time of the sink between sink stops. Both \(t_{s}\) and \(t_{t}\) are calculated till the death of the first node. In Equation (1), the first summation accounts for the total number of sink stops, while the second summation accounts for the total number of travels between the sink stops. \(t_{s_{j}}\) is the time duration for the sink to stop at a particular stop, and \(t_{t}\) is the traveling time between two stops. Node deployment in the field is random, and the number of nodes on each sink stop may vary. Equation (2) shows the heuristic to find the minimum distance between neighbors and non-neighbors. \(h(i)\) is the cost function, and \(h(j)\) is a heuristic estimate of the distance from node \(i\) to node \(j\). \(d(n_{i},n_{j})\) is the distance between nodes \(i\) and \(j\). In Equation (3), \(T_{m}in\) and \(T_{m}ax\) represent the minimum and maximum stop time of the sink at a location, respectively. The sink stops at a stop until it receives data from the nodes in its sensing range. \(t_{s}\) is the optimal sink stop time period for data  
Figure 4.2 The MS is the neighbor of \(n_{i}\) nodes. gathering. \(f_{i}\) is the flow of information (data sensed by nodes). Equation (4) shows that the MS tour will not exceed \(l\). The energy conservation constraint is represented in Equation (5), which means that the total energy that nodes consume during the network lifetime cannot exceed the initial energy. When a node \(i\) transmits data to the nearby sink \(j\), it consumes power \(p_{ij}\). The transmission of data depends on the residual energy of the node, if it is equal to the minimum energy required for transmission, which is represented in Equation (6).