# Efficient Data Gathering in 3D Linear Underwater Wireless Sensor Networks Using Sink Mobility
## 4 Framework and Formal Definition of the Problem
### Graphical Analysis  
In our proposed scenario, there are two paths for data transmission from nodes to the MS. The first is direct transmission from the node to the MS and via CN. End-to-end delay is greater when the MS receives data from the node via the CN. End-to-end delay is minimum when nodes send data through direct transmission. The total end-to-end delay, denoted by \(D\), is the combined delay of direct (\(D-MS\)) and multi-hop transmission (\(D-CN\)). We define an objective function:  
\[Minimize\quad D \tag{9}\]  
\[\text{subject to: }D=(D-MS)+(D-CN) \tag{10}\]  
\[87.7\leqslant(D-MS)+(D-CN)\leqslant 172.8 \tag{11}\]  
\[0\leqslant(D-MS)\leqslant 87.7 \tag{12}\]  
\[0\leqslant(D-CN)\leqslant 172.8 \tag{13}\]  
\[0\leqslant(D-MS)+(D-CN)\leqslant 260.5 \tag{14}\]  
Equation (9) aims to minimize the end-to-end delay of the network. Equation (10) defines the nature of the delay, _i.e._, an objective function and a two-dimensional linear programming problem. Constraints in Equation (11) provide lower and upper bounds of the path, respectively. The constraints defined in Equation (12) and (13) are the upper bounds of \(D-MS\) and \(D-CN\) independently. Equation (14) defines the upper and lower bounds of the network jointly considering both types of delay. Figure 6 shows the set of feasible solutions. There are lines, \(L1\), \(L2\), \(L3\) and \(L4\), intersecting each other, and the region formed by their intersection is the set of all feasible solutions.  
The minimum value of each vertex can be obtained as:  
\[\text{at }P1(0,0):D=0\text{ s}\] \[\text{at }P2(87.7,0):D=50\text{ s}\] \[\text{at }P3(0,172.8):D=140\text{ s}\] \[\text{at }P4(87.7,172.8):D=190\text{ s}\]  
The minimum value of \(D\) is 0 s, which shows that transmissions are not started yet. It is the initialization phase. The next value of \(D\) is 50 s. This indicates direct transmission to the MS. Similarly, the next value is 140 s for multi-hop transmission through the CN. The last value is the maximum delay _i.e._, 190 s; all of values lie within this bound. This shows that the values of delay at each point lie within the boundaries of the illustrated region.  
During the NLT, the MS and CNs move in the defined RC and collect data periodically, which reduces delays and increases throughput. In Figure 7, the feasible region for throughput is shown; it is represented by units of Kbps. \((TP=(TP-MS)+(TP-CN))\):  
\[Maximize\quad TP \tag{15}\]  
\[\text{subject to: }0\leqslant(TP-MS)+(TP-CN)\leqslant 300 \tag{16}\]  
\[0\leqslant(TP-MS)\leqslant 55 \tag{17}\]  
\[0\leqslant(TP-CN)\leqslant 245 \tag{18}\]  
The objective function is defined in Equation (15). According to the bounds provided in Equations (16)-(18), Figure 7 shows the intersecting lines \(L1\), \(L2\), \(L3\) and \(L4\), resulting in a bounded region that shows the set of feasible solutions. Values on each vertex are obtain as:  
\[\text{at }P1(0,0)=0\text{ Kbps},\] \[\text{at }P2(55,0)=55\text{ Kbps},\] \[\text{at }P3(55,245)=55+245=300\text{ Kbps},\] \[\text{at }P4(0,245)=245\text{ Kbps}\]  
The validity of the feasible region is proven. The total throughput during a trip of the MS and CN lies within the boundaries of the illustrated region.  
Figure 6: Delay: feasible region.