## 3 Successful Transmission Probabilities for the Proposed Underwater Multi-DATA Train Protocol
If each of two RTS frames is transmitted from only one sensor node among sensor nodes in the ring 1 and only one sensor node among \(n_{2}\) sensor nodes in the ring 2, respectively, the two RTS frames are successfully and simultaneously transmitted to the underwater sink node as:  
\[p_{\text{BJ}r=2}\cdot p_{\text{S},(1,2)|r=2}=n_{1}n_{2}p_{\text{TJ}r=2}{}^{2}(1- p_{\text{TJ}r=2})^{N_{2}-2}. \tag{5}\]  
Let \(p_{\text{S},(i)|r=2}\) for \(i=1,\ 2\) denote the probability that an RTS frame is successfully transmitted from an underwater sensor node in the ring \(i\) to an underwater sink node and RTS frames are not successfully transmitted from underwater sensor nodes in the other ring between two rings to the underwater sink node. If an RTS frame is transmitted from only one underwater sensor node among \(n_{i}\) sensor nodes in the ring \(i\) and no or more than one RTS frame is transmitted from underwater sensor nodes among \(N_{2}-n_{i}\) sensor nodes in the other ring, only one RTS frame is successfully transmitted from the sensor node in the ring \(i\) to the underwater sink node with the following probability \(p_{\text{S},(i)|r=2}\) for \(i=1,\ 2\) as:  
\[p_{\text{BJ}|r=2}\cdot p_{\text{S},(1)|r=2}=n_{1}\sum_{j=0,j\
eq 1}^{n_{2}} \left( \begin{array}{c}n_{2}\\ j\end{array} \right)p_{\text{TJ}r=2}{}^{1+j}(1-p_{\text{TJ}r=2})^{N_{2}-1-j}. \tag{6}\]  
\[p_{\text{BJ}|r=2}\cdot p_{\text{S},(2)|r=2}=n_{2}\sum_{j=0,j\
eq 1}^{n_{1}} \left( \begin{array}{c}n_{1}\\ j\end{array} \right)p_{\text{TJ}r=2}{}^{1+j}(1-p_{\text{TJ}r=2})^{N_{2}-1-j}. \tag{7}\]  
Let \(p_{\text{S},m|r}\) denote the probability that the number of RTS frames transmitted successfully and simultaneously from underwater sensor nodes to an underwater sink node is \(m\) when the number of rings is \(r\). The probability \(p_{\text{S}|r=2}\) that one or more RTS frames are successfully and simultaneously transmitted from underwater sensor nodes to an underwater sink node when \(r=2\) can be obtained as:  
\[p_{\text{S}|r=2}=\sum_{m=1}^{2}p_{\text{S},m|r=2}, \tag{8}\]  
where the probabilities \(p_{\text{S},2|r=2}\) and \(p_{\text{S},1|r=2}\) can be derived as:  
\[p_{\text{S},2|r=2}=p_{\text{S},(1,2)|r=2}, \tag{9}\]  
\[p_{\text{S},1|r=2}=p_{\text{S},(1)|r=2}+p_{\text{S},(2)|r=2}. \tag{10}\]  
When the number of rings \(r=3\), let \(p_{\text{S},(1,2,3)|r=3}\) denote the probability that three RTS frames are successfully and simultaneously transmitted from three underwater sensor nodes in three rings to an underwater sink node. If each of three RTS frames is transmitted from only one sensor node among \(n_{1}\) sensor nodes in the ring 1, only one sensor node among \(n_{2}\) sensor nodes in the ring 2, and only one sensor node among \(n_{3}\) sensor nodes in the ring 3, respectively, the three RTS frames are successfully and simultaneously transmitted to the underwater sink node as:  
\[p_{\text{BJ}r=3}\cdot p_{\text{S},(1,2,3)|r=3}=n_{1}n_{2}n_{3}p_{\text{TJ}r= 3}{}^{3}(1-p_{\text{TJ}r=3})^{N_{3}-3}, \tag{11}\]  
Let \(p_{\text{S},(i,j)|r=3}\) for \(\langle i,j\rangle=\langle 1,2\rangle,\ \ \langle 1,3\rangle,\ \ \langle 2,3\rangle\) denote the probability that two RTS frames are successfully and simultaneously transmitted from one sensor node in the ring \(i\) and one sensor node in the ring \(j\) to an underwater sink node and RTS frames are not successfully transmitted from underwater sensor nodes in the other ring to the underwater sink node.