# An Improved Forwarding of Diverse Events with Mobile Sinks in Underwater Wireless Sensor Networks
## 5 ESDR: Event Segregation Based Delay Sensitive Routing
### Mobility of Mobile Sinks
#### 5.6.1 Synchronized and Uniform Mobility of AUV  
In this type of mobility four courier nodes deployed on four different depths move in elliptical path. The centers of the four ellipse are given in Table 3. The motion is uniform in the sense that speed of each AUV remains same in its sojourn tour on elliptical path. By the term synchronized mean that phase relation between any two AUVs remains same throughout the tour as shown in Figure 3. In this figure two ellipses are shown by solid line and two by dashed line. The both AUVs of solid line ellipses move in same direction and remain in phase with each other, while both the dashed line AUVs are in phase, move in the same direction but opposite to the previous pair of AUV. The green dots in these figures represent the stopping points of AUVs.  
Elliptical path of courier node \(c\) in \(yz\) plane is defined in following equation.  
\[\frac{(z-k_{c})^{2}}{a^{2}}+\frac{(y-h_{c})^{2}}{b^{2}}=1 \tag{22a}\]  
Elliptical path is planned in such a way that two ellipses have some overlapping region as depicted in Figure 3a. This allows better coverage to whole region and also facilitate to respective lower AUV to send its data to upper AUV. If the length of overlapping region on \(z\) axis is \(x\), semi major axis length \(a\) is related with \(x\) by following equation.  
\[a=\frac{3}{8}x+125 \tag{22b}\]  
This relation reveals that for \(x=0\), \(a\) becomes equal to 125 making length of major axis of the each ellipse to \(2a=250\). \(x\) is subsequently changed to 20 and 40 making \(a=133\) and 140. The resulting ellipses are shown in Figure 4. In this type of mobility vertex length \(a=140\) and co-vertex length \(b=50\) are used. In order to calculate time period of a node it is necessary to have knowledge of perimeter (circumference) of an ellipse. There is no straightforward formula to calculate this and following infinite series is utilized.  
\[p=\pi(a+b)\sum_{n=1}^{\infty}\binom{0.5}{n}^{2}h^{n} \tag{22c}\]Figure 4: Adaptive mobility of courier nodes. (**a**) Adaptive mobility of courier nodes; (**b**) Adaptive mobility of courier nodes with deployed nodes.  
Figure 3: Synchronized mobility of AUV. (**a**) Uniform mobility of AUV; (**b**) Uniform mobility of AUV with node deployed nodes.  
The parameter h is defined as follows:  
\[h=\frac{(a-b)^{2}}{(a+b)^{2}} \tag{22d}\]  
We have employed the following formula by [PERSON] [45] to calculate the perimeter of the each elliptical.  
\[p\approx\pi[3(a+b)-\sqrt{(3a+b)(a+3b)}] \tag{23}\]  
Time period of each nodes is dependent on uniform tangential speed of courier node during its sojourn tour. In synchronized and uniform mobility pattern speed of a node \(i\) is \(v_{i}\) making time period:  
\[T_{i}=\frac{p}{v_{i}} \tag{24}\]  
Mobility of CN is envisioned in such a way that \(CN_{1}\) is in phase with \(CN_{3}\) and \(CN_{2}\) is in phase with \(CN_{4}\). Initially \(CN_{1}\) and \(CN_{3}\) are at bottom end of their respective elliptical paths and \(CN_{2}\) and \(CN_{4}\) are at the top end of their corresponding elliptical paths. Every node moves in clockwise direction and reaches to next destination after time \(T_{i}/4\). In this way a node in its one revolution has four points to stop each after phase angle of \(\theta/2\). Their motion is synchronized in such a way that when \(CN_{1}\) is about the top. Each node moves upward to their predefined destination, transfer data to next CN, and move back to initial stage. This motion pattern is synchronized in such a way that when a certain courier node is about to reach its destination next forwarder has just started its tour, first node find next CN to take its data to next CN.