## 3 Successful Transmission Probabilities for the Proposed Underwater Multi-DATA Train Protocol
If an RTS frame is transmitted from only one sensor node among \(n_{i}\) sensor nodes in the ring \(j\), an RTS frame is transmitted from only one sensor node among \(n_{j}\) sensor nodes in the ring \(j\), and no or more than one RTS frame is transmitted from underwater sensor nodes among \(N_{3}-n_{i}-n_{j}\) sensor nodes in the other ring, the two RTS frames are successfully and simultaneously transmitted from the sensor node in the ring \(i\) and the sensor node in the ring \(j\) to the underwater sink node with the following probability \(p_{\text{S},(i,j)|r=3}\) for \(\langle i,j\rangle=\langle 1,2\rangle,\ \ \langle 1,3\rangle,\ \ \langle 2,3\rangle\) as:  
\[p_{\text{B}|r=3}\cdot p_{\text{S},(1,2)|r=3} = n_{1}n_{2}\sum_{k=0,k\
eq 1}^{n_{3}}\left( \begin{array}{c}n_{3}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{2+k}(1-p_{\text{T}|r=3})^{N_{3}-2-k}, \tag{12}\] \[p_{\text{B}|r=3}\cdot p_{\text{S},(1,2)|r=3} = n_{1}n_{3}\sum_{k=0,k\
eq 1}^{n_{2}}\left( \begin{array}{c}n_{2}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{2+k}(1-p_{\text{T}|r=3})^{N_{3}-2-k},\] (13) \[p_{\text{B}|r=3}\cdot p_{\text{S},(2,3)|r=3} = n_{2}n_{3}\sum_{k=0,k\
eq 1}^{n_{1}}\left( \begin{array}{c}n_{1}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{2+k}(1-p_{\text{T}|r=3})^{N_{3}-2-k}. \tag{14}\]  
Let \(p_{\text{S},(i)|r=3}\) for \(i=1,\ 2,\ 3\) 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 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 in each of the other rings, only one RTS frame is successfully transmitted from the sensor node in the ring \(i\) to the underwater sink node with the following probability \(p_{\text{S},(i)|r=3}\) for \(i=1,\ 2,\ 3\) as:  
\[p_{\text{B}|r=3}\cdot p_{\text{S},(1)|r=3} = n_{1}\sum_{j=0,j\
eq 1}^{n_{2}}\sum_{k=0,k\
eq 1}^{n_{3}} \left( \begin{array}{c}n_{2}\\ j\end{array} \right)\left( \begin{array}{c}n_{3}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{1+j+k}(1-p_{\text{T}|r=3})^{N_{3}-1- j-k}, \tag{15}\] \[p_{\text{B}|r=3}\cdot p_{\text{S},(2)|r=3} = n_{2}\sum_{j=0,j\
eq 1}^{n_{1}}\sum_{k=0,k\
eq 1}^{n_{3}} \left( \begin{array}{c}n_{1}\\ j\end{array} \right)\left( \begin{array}{c}n_{3}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{1+j+k}(1-p_{\text{T}|r=3})^{N_{3}-1- j-k},\] (16) \[p_{\text{B}|r=3}\cdot p_{\text{S},(3)|r=3} = n_{3}\sum_{j=0,j\
eq 1}^{n_{1}}\sum_{k=0,k\
eq 1}^{n_{2}} \left( \begin{array}{c}n_{1}\\ j\end{array} \right)\left( \begin{array}{c}n_{2}\\ k\end{array} \right)p_{\text{T}|r=3}{}^{1+j+k}(1-p_{\text{T}|r=3})^{N_{3}-1- j-k}. \tag{17}\]  
The probability \(p_{\text{S}|r=3}\) that one or more RTS frames are successfully and simultaneously transmitted from underwater sensor nodes to an underwater sink node when \(r=3\) can be obtained as:  
\[p_{\text{S}|r=3}=\sum_{m=1}^{3}p_{\text{S},m|r=3}, \tag{18}\]  
where the probabilities \(p_{\text{S},m|r=3}\) for \(m=1,\ 2,\ 3\) can be derived as:  
\[p_{\text{S},3|r=3}=p_{\text{S},(1,2,3)|r=3}, \tag{19}\]  
\[p_{\text{S},2|r=3}=p_{\text{S},(1,2)|r=3}+p_{\text{S},(1,3)|r=3}+p_{\text{S}, (2,3)|r=3}, \tag{20}\]  
\[p_{\text{S},1|r=3}=p_{\text{S},(1)|r=3}+p_{\text{S},(2)|r=3}+p_{\text{S},(3)|r= 3}. \tag{21}\]