# Ocean-Current-Motion-Model-Based Routing Protocol for Void-Avoided UASNs
## 4 OCMR Protocol
### Protection Radius-Based Construction of the Candidate Forwarding Set  
The traditional WOA still has some disadvantages, namely, a low convergence accuracy and rate. Therefore, we need to infer the radius of protection as the selection area of the candidate forwarding set to improve the convergence rate of the traditional WOA.  
Data transmission will be affected in a certain area regardless of whether there are too few or too many nodes. If the nodes in the area are few in a sparse network, the point-to-point distance will exceed the maximum communication distance, which will lead to ineffective data transmission. If the nodes in the area are dense, the load of the network will increase, resulting in congestion and data interruption. In order to find an applicable set of forwarding candidates and maintain the nodes of the network in an equilibrium state, we construct the candidate forwarding set with the optimal outage probability as the decision threshold. A suitable WOA search coverage with the optimal outage probability is derived and an appropriate relay node is found within the search coverage. The range is a circle with radius R, namely, the protection radius, in which it can not only select an appropriate relay node to overcome the routing void in the network, but also improve the efficiency of the search.  
As shown in Figure 4, a circle is constructed with \(H_{00}\) as the center and \(R\) as the radius. \(H_{i0}\) is the \(i\)th node, distributed randomly in the circle. \(X_{i}\) is the distance between \(H_{00}\) and \(H_{i0}\). We can calculate the communication outage probability among all nodes within the circle as follows:  
\[\begin{array}{ll}q(\lambda)&=\Pr(\frac{H_{00}R^{-\alpha}}{\sum\limits_{i\in \Pi}H_{i0}X_{i}^{-\alpha}}<\beta)\\ &=\Pr(H_{00}<\beta R^{\alpha}I_{\prod interf})\end{array} \tag{11}\]  
where \(\alpha\) is the attenuation coefficient, \(\beta\) is the SNR threshold, and \(I_{\prod interf}\) is the total interference signal calculated as  
\[I_{\prod interf}=\sum\limits_{i\in\Pi}H_{i0}X_{i}^{-\alpha} \tag{12}\]  
Figure 4: Protection radius diagram.  
As \(H_{ij}\) follows an exponential distribution, we obtain  
\[f_{H}(x)=\tau\exp(-\tau x),x\geq 0 \tag{13}\]  
Therefore, Equation (11) is converted into  
\[q(\lambda)=E[\int_{0}^{\beta R^{\kappa}I_{\Pi}}\tau\exp(-\tau x)dx]=1-\mathcal{L }_{I_{\Pi interior}}(\tau\beta R^{\alpha}) \tag{14}\]  
where \(\mathcal{L}_{I_{\Pi interior}}(x)\) is the Laplace transform of \(I_{\Pi interior}\), which is expressed as  
\[\mathcal{L}_{I_{\Pi interior}}(s)=\exp(-\alpha\pi\lambda\int_{d}^{\infty} \frac{sx^{-\alpha+1}}{sx^{-\alpha}+u}dx) \tag{15}\]  
Then, we take the iteration from \(d\) to \(\alpha\). By substituting (15) into (11), Equation (11) can be converted into  
\[\begin{array}{ll}q(\lambda)&=1-\exp(-2\pi\lambda\int_{d}^{\infty}\frac{ \tau\beta R^{\kappa}x^{-\alpha+1}}{\tau\beta R^{\kappa}x^{-\alpha+1}+\tau}dx) \\ &=1-\exp(-2\pi\lambda\int_{d}^{\infty}\frac{x^{-\alpha+1}}{x^{-\alpha}+\beta ^{-1}R^{-\alpha}}dx)\end{array} \tag{16}\]  
We set \(\alpha=4\) due to the Rayleigh fading, and we obtain  
\[\begin{array}{ll}q_{\rm a=4}(\lambda)&=1-\exp\left(-2\pi\lambda\int_{d}^{ \infty}\frac{x^{-3}}{x^{-4}+\beta^{-1}R^{-4}}dx\right)\\ &=1-\exp[-\pi\lambda\beta^{0.5}R^{2}arctan(\beta^{0.5}R^{2}d^{-2})]\end{array} \tag{17}\]  
Let \(q(\lambda)=\varepsilon\), and \(q(\lambda)\) can be rewritten as  
\[q(\lambda)=1-\exp\{-\lambda\pi\beta^{\frac{2}{4}}R^{2}[\frac{1}{\alpha-2}+\ln( \frac{2}{1+d^{2}R^{-2}\beta^{-\frac{2}{4}}})]\} \tag{18}\]  
According to the above equation with the optimal outage probability, the optimal protection radius could be calculated as  
\[d_{\varepsilon}=\sqrt{\frac{2}{\frac{\alpha}{4}\beta^{\frac{2}{4}}R^{2}\pi \csc(\frac{2}{\alpha}\pi)}{1-e\ln(1-\varepsilon)}} \tag{19}\]  
It can be inferred from the above derivation that the nodes in this circle with radius \(d_{\varepsilon}\) are the alternative nodes that form the candidate forwarding set.