## 4 Analysis of Throughput and Delay for the Underwater Protocols
When the proposed underwater BACK protocol with \(r\) rings is utilized, \(\mathbb{E}\left[T_{\text{BACK},\text{SIF}}\right]\) can be obtained as:  
\[\mathbb{E}\left[T_{\text{BACK},\text{SIF}}\right]=\sum_{m=1}^{r}p_{\text{S},n \text{IF}}T_{\text{BACK},\text{S},m}, \tag{26}\]  
\[T_{\text{BACK},\text{S},m}=\left(T_{\text{RTS}}+\text{PROP}+\text{SIFS} \right)+\left(T_{\text{CTS}}+\text{PROP}+\text{SIFS}\right)+m\cdot\left(T_{ \text{PHY}}+T_{\text{MAC}}+T_{\text{P}}\right.\]  
\[\left.+\text{PROP}+\text{SIFS}\right)+T_{\text{BACK}}+\text{DIFS}, \tag{27}\]  
where \(T_{\text{BACK}}\) denotes the duration of BACK frame and PROP denotes the underwater acoustic propagation delay between a sink node and the farthest underwater sensor node to the sink node in an underwater cell. Since the underwater sink node is considered to be located at the center of an underwater cell, PROP is considered as \(\text{PROP}_{\text{max}}/2\). When the proposed underwater multi-DATA train protocol with \(r\) rings is utilized, \(\mathbb{E}\left[T_{\text{train},\text{SIF}}\right]\) can be obtained as:  
\[\mathbb{E}\left[T_{\text{train},\text{SIF}}\right]=\sum_{\forall\text{IF}}p_{ \text{S},\text{IF}}T_{\text{train},\text{S},\text{IF}}, \tag{28}\]  
where \(\text{IF}\) denotes an ordered list (tuple) of ring indexes when the number of rings is \(r\), \(p_{\text{S},\text{IF}}\) denotes the probability of the event that RTS frames are successfully and simultaneously transmitted from underwater sensor nodes in the rings whose indexes constitute the tuple \(\text{IF}\) to an underwater sink node, and \(T_{\text{train},\text{SIF}}\) denotes the duration of successful transmissions of RTS, CTS, DATA, and BACK frames when the event of successful and simultaneous transmissions of RTS frames in the case of the tuple \(l|r\) occurs with the probability \(p_{\text{S,Jr}}\) in the proposed underwater multi-DATA train protocol with \(r\) rings. When the number of rings \(r=2\), the duration \(T_{\text{train,S,Jr}}\) for the tuple \(l\in\{\langle 1\rangle,\langle 2\rangle,\langle 1,2\rangle\}\) can be expressed as:  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]0  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]1  
where \(\text{PROP}_{\text{Jr}}\) denotes the underwater propagation delay between a sink node and the farthest underwater sensor node in the ring \(i\) to the sink node in an underwater cell when the number of rings is \(r\), and \(\text{PROP}_{\text{Jr}}=\text{PROP}\). If \(|l_{2}-l_{1}|=|l_{1}-l_{0}|\) where the ring \(i\) has two radii \(l_{i-1}\) and \(l_{i}\), \(\text{PROP}_{\text{Jr}=2}=\text{PROP}/2\). When the number of rings \(r=3\), the duration \(T_{\text{train,S,Jr}}\) for the tuple \(l\in\{\langle 1\rangle,\langle 2\rangle,\langle 3\rangle,\langle 1,2\rangle, \langle 1,3\rangle,\langle 2,3\rangle,\langle 1,2,3\rangle\}\) can be expressed as:  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]2  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]3  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]4  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]5  
where \(\text{PROP}_{\text{Jr}=3}\) denotes \(\text{PROP}_{\text{Jr}}\) when the number of rings \(r=3\), and \(\text{PROP}_{\text{Jr}}=\text{PROP}\). If \(|l_{3}-l_{2}|=|l_{2}-l_{1}|=|l_{1}-l_{0}|\), \(\text{PROP}_{\text{Jr}=3}=i\cdot\text{PROP}_{\text{J}|r=3}\) for \(i=1,\;2,\;3\). The duration \(T_{\text{train,S,Jr}}\) is shorter than or equal to the duration \(T_{\text{BACK,S,m}}\) when \(m\) is equal to the number of elements of the tuple \(l|r\). It is because the underwater propagation delay between an underwater sink node and an underwater sensor node can be calculated and utilized in the proposed underwater multi-DATA train protocol. Thus, multiple DATA frames transmitted from underwater sensor nodes can be formed as a train at the underwater sink node in the proposed underwater multi-DATA train protocol.