## 3 Successful Transmission Probabilities for the Proposed Underwater Multi-DATA Train Protocol  
In this section, we calculate successful transmission probabilities for our proposed underwater multi-DATA train protocol. We consider that an underwater sink node is located at the center of an underwater cell and multiple underwater sensor nodes are distributed in the underwater cell. The underwater cell can be divided into \(r\) rings [17]. Each ring \(i=1,\ldots,r\) has the boundary \([l_{i-1},l_{i})\) where \(l_{0}=0\) and the ring \(i\) has two radii \(l_{i-1}\) and \(l_{i}\). Let \(d_{x,a}\) denote the distance between the underwater sensor node \(x\) in the ring \(a\) and the underwater sink node. Let \(d_{y,b}\) denote the distance between the underwater sensor node \(y\) in the ring \(b\) and the underwater sink node. If the underwater sensor node \(x\) in the ring \(a\) and the underwater sensor node \(y\) in the ring \(b\) send RTS frames simultaneously to the underwater sink node and the following condition (1) is satisfied, the RTS frames sent by the underwater sensor nodes \(x\) and \(y\) to the underwater sink node may not overlap each other at the underwater sink node in the time domain as:  
\[\left|d_{x,a}-d_{y,b}\right|\geq v_{\text{UWA}}\cdot(T_{\text{RTS}}+\text{ SIFS}), \tag{1}\]  
where \(v_{\text{UWA}}\) denotes the speed of sound via the underwater acoustic channel, \(T_{\text{RTS}}\) denotes the duration of RTS frame, and SIFS denotes a short inter-frame space [5; 6]. When \(a=b\), as the probability that RTS frames sent simultaneously from the same ring to a sink node overlap each other at the sink node is high, the RTS frames are assumed to be overlapped at the sink node. When \(a\
eq b\), as the probability that RTS frames sent simultaneously from different rings to a sink node overlap each other at the sink node is low, the RTS frames are assumed to be not overlapped at the sink node. Let \(N_{r}\) denote the number of underwater sensor nodes in an underwater cell when the number of rings is \(r\). Let \(n_{i}\) denote the number of underwater sensor nodes located in the ring \(i=1,\ldots,r\). The number of underwater sensor nodes \(N_{r}\) can be expressed as:  
\[N_{r}=\sum_{i=1}^{r}n_{i}. \tag{2}\]  
If at least one of underwater sensor nodes among \(N_{r}\) sensor nodes sends an RTS frame, the underwater acoustic channel becomes busy. The channel busy probability in the case of \(r\) rings \(p_{\text{BIF}}\) can be obtained as:  
\[p_{\text{BIF}}=1-(1-p_{\text{TIF}})^{N_{r}}, \tag{3}\]  
where \(p_{\text{TIF}}\) is the probability that an underwater sensor node sends an RTS frame to an underwater sink node when the number of rings is \(r\). When the number of rings \(r=1\), let \(p_{\text{SIF}=1}\) denote the probability that an RTS frame is successfully transmitted from an underwater sensor node to an underwater sink node. If the RTS frame is transmitted from only one underwater sensor node among \(N_{1}\) sensor nodes, the RTS frame is successfully transmitted to the underwater sink node as:  
\[p_{\text{BIF}=1}\cdot p_{\text{SIF}=1}=N_{1}p_{\text{TIF}=1}(1-p_{\text{TIF}=1 })^{N_{1}-1}. \tag{4}\]  
When the number of rings \(r=2\), let \(p_{\text{S}(1,2)\text{IF}=2}\) denote the probability that two RTS frames are successfully and simultaneously transmitted from two underwater sensor nodes in two rings to an underwater sink node.