## 4 Analysis of Throughput and Delay for the Underwater Protocols  
In this section, in order to compare the performances of our proposed protocols in the case of \(r\) rings with the performances of other protocols, we analyze the throughput performance \(S_{protocol}\) and the delay performance \(D_{protocol}\) of our proposed underwater multi-DATA train protocol (\(S_{\text{train}|r}\) and \(D_{\text{train}|r}\)), our proposed underwater BACK protocol (\(S_{\text{BACK}|r}\) and \(D_{\text{BACK}|r}\)), the underwater multi-RTS reception protocol (\(S_{\text{M}\text{M}\text{T}\text{S}|r}\) and \(D_{\text{M}\text{R}\text{T}\text{S}|r}\)) [17], and the conventional CSMA/CA protocol (\(S_{\text{CONV}|r}\) and \(D_{\text{CONV}|r}\)) based on [6; 14]. The throughput performance \(S_{protocol}\) is the normalized throughput defined as the time fraction of successful transmissions of DATA payloads over an underwater acoustic channel when the number of rings is \(r\)[6; 17]. The normalized throughput \(S_{protocol}\) for \(protocol\in\{\text{CONV},\text{MRTS},\text{BACK},\text{train}\}\) can be obtained as:  
\[S_{protocol}=\frac{p_{\text{BIF}}\cdot\sum_{r=1}^{r}\left(m\cdot p_{ \text{SIF}}\right)\cdot T_{\text{P}}}{\left(1-p_{\text{BIF}}\right)\cdot T_{1} +p_{\text{BIF}}\left(1-p_{\text{BIF}}\right)\cdot T_{\text{C}}+p_{\text{BIF}} \cdot\mathbb{E}\left[T_{protocol,S_{\text{P}}}\right]}, \tag{22}\]  
for \(protocol\in\{\text{MRTS},\text{BACK},\text{train}\}\),  
\[S_{\text{CONV}_{r}}=S_{\text{MRTS}\text{IF}=1}, \tag{23}\]  
where \(T_{\text{P}}\) denotes the duration of DATA payload, \(T_{\text{I}}\) denotes the duration of an idle time slot, \(\mathbb{E}\left[T_{protocol,S_{\text{P}}}\right]\) for \(protocol\in\{\text{MRTS},\text{BACK},\text{train}\}\) denotes the expected duration of successful transmissions of RTS, CTS, DATA, and ACK or BACK frames when the number of rings is \(r\), and \(T_{\text{C}}\) denotes the duration of colliding RTS transmissions. \(T_{\text{C}}\) is considered as \(T_{\text{RTS}}+\text{PROP}_{\text{max}}+\text{EIFS}\) where \(\text{PROP}_{\text{max}}\) denotes the maximum underwater acoustic propagation delay between the farthest underwater sensor nodes in an underwater cell and EIFS denotes an extended inter-frame space [5; 6]. The normalized throughput of the conventional CSMA/CA protocol \(S_{\text{CONV}_{r}}\) can be obtained as that of the underwater multi-RTS reception protocol \(S_{\text{MRTS}\text{IF}}\) in the case of \(r=1\). The probabilities \(p_{\text{BIF}}\), \(p_{\text{SIF}}\), and \(p_{\text{S},n\text{IF}}\) in Equation (22) can be obtained from Equations (2)-(21). The expected duration \(\mathbb{E}\left[T_{protocol,S_{\text{P}}}\right]\) for \(protocol\in\{\text{MRTS},\text{BACK},\text{train}\}\) in Equation (22) can be obtained from the following Equations (24)-(34). When the underwater multi-RTS reception protocol with \(r\) rings is utilized, \(\mathbb{E}\left[T_{\text{MRTS},\text{SIF}}\right]\) can be obtained as [17]:  
\[\mathbb{E}\left[T_{\text{MRTS},\text{SIF}}\right]=\sum_{m=1}^{r}p_{\text{S},n \text{IF}}T_{\text{MRTS},\text{S},m}, \tag{24}\]  
\[T_{\text{MRTS},\text{S},m}=\left(T_{\text{RTS}}+\text{PROP}_{\text{max}}+ \text{SIFS}\right)+\left(T_{\text{CTS}}+\text{PROP}_{\text{max}}+\text{SIFS} \right)+m\]  
\[\cdot\left(T_{\text{PHY}}+T_{\text{MAC}}+T_{\text{P}}+\text{PROP}_{\text{max} }+\text{SIFS}\right)+\left(m-1\right)\]  
\[\cdot\left(T_{\text{ACK}}+\text{PROP}_{\text{max}}+\text{SIFS}\right)+T_{ \text{ACK}}+\text{DIFS},\]  
where \(T_{\text{CTS}}\) denotes the duration of CTS frame, \(T_{\text{PHY}}\) denotes the duration of PHY header, \(T_{\text{MAC}}\) denotes the duration of MAC header, \(T_{\text{ACK}}\) denotes the duration of ACK frame, and DIFS denotes a distributed inter-frame space [5; 6].