# DCO-MAC: A Hybrid MAC Protocol for Data Collection in Underwater Acoustic Sensor Networks
## 5 Approximate Performance Analysis
### RSV-MAC Protocol  
In the RSV-MAC protocol, a successful data transmission period, \(T_{\text{SUC}}\), is composed of a successful transmission of an RTR packet, several ATS packets, and an ORDER packet, followed by a data transmission period and the successful transmission of an ACK packet. The transmission period of CT-MAC protocol is illustrated in Figure 9.  
In the level-1 sub-network transmission phase, \(T_{\text{SUC}}\) can be calculated as follows:  
\[ \begin{split} T_{\text{SUC}}=T_{\text{RTS}}+T_{\text{ATS}}+T_{ \text{ORDER}}+T_{\text{DATA}}+T_{\text{ACK}}=\\ 5\tau_{\text{max}}+(N_{\text{level-1}}+3)\theta+N_{\text{level-1}} T_{\text{guard}}+\frac{K_{\text{level-1}}(T_{\text{guard}}+\delta)}{1-p_{\text{e}}},\end{split}  \tag{18}\]  
where \(N_{\text{level-1}}\) is the number of nodes in the level-1 sub-network, and \(K_{\text{level-1}}\) is the number of DATA packets transmitted in the level-1 sub-network in \(T_{\text{SUC}}\). In the level-1 sub-network, the level-1 nodes not only transmit the data packets generated by themselves to the sink, but also forward the gathered data packets from the corresponding level-2 nodes. Hence, the probability that no data packets are sent in the level-1 sub-network transmission phase is very small. To simplify the calculation, we assume that the level-1 nodes want to transmit data packets at the beginning of the level-1 sub-network transmission phase. Hence, the data transmission in the level-1 sub-network is assumed to be successful. So, the busy period can be given as follows:  
\[\overline{B}=T_{\text{SUC}} \tag{19}\]  
There are two types of data packets sent by the level-1 nodes, one is generated by the level-1 nodes during the last data transmission period, the other is gathered from the level-2 nodes during the last level-2 sub-network data transmission phase. Hence, \(K_{\text{level-1}}\) can be given as follows:  
\[K_{\text{level-1}}=N_{\text{level-1}}\lambda(\overline{B}+\overline{I})+\varphi _{\text{level-2}} \tag{20}\]  
where \(\varphi_{\text{level-2}}\), the expected number of data packets gathered from the level-2 sub-network by level-1 nodes, can be given as follows: \(\varphi_{\text{level-2}}=N_{\text{trans}}\mathcal{S}_{j=1}^{N_{\text{level-2}} }P_{\text{SUC},j}N_{j}\lambda(\tau_{\text{max}}+\theta)\). Combining Equations (18), (19), and (20), \(K_{\text{level-1}}\) can be rewritten as follows:  
\[ \begin{split} K_{\text{level-1}}=\frac{N_{\text{level-1}}\lambda \Big{[}5\tau_{\text{max}}+(N_{\text{level-1}}+3)\theta+N_{\text{level-1}}T_{ \text{guard}}+\overline{I}\Big{]}+\varphi_{\text{level-2}}}{1-\frac{N_{\text{ level-1}}\lambda(T_{\text{guard}}+\delta)}{1-p_{\text{e}}}}\end{split}  \tag{21}\]  
As \(K_{\text{level-1}}\) is always greater than 0, we have \(\frac{N_{\text{level-1}}\lambda(T_{\text{guard}}+\delta)}{1-p_{\text{e}}}<1\) in Equation (21), where \(\frac{T_{\text{guard}}+\delta}{1-p_{\text{e}}}\) is the expected receiving time of one DATA packet, and \(\frac{N_{\text{level-1}}\lambda(T_{\text{guard}}+\delta)}{1-p_{\text{e}}}\) denotes the number of data packets that are generated by the level-1 nodes during \(\frac{T_{\text{guard}}+\delta}{1-p_{\text{e}}}\). Hence, \(\frac{N_{\text{level-1}}\lambda(T_{\text{guard}}+\delta)}{1-p_{\text{e}}}<1\) indicates  
Figure 9: The transmission period of RSV-MAC protocol. that the data packet arrival rate of the level-1 nodes should be less than the data packet transmission rate. Otherwise, the network will be congested, and cannot operate normally. The expected time duration of the channel being used to transmit the DATA packets can be given as follows:  
\[\overline{U}=K_{\text{level-1}}\delta. \tag{22}\]  
The expected time duration of idle period equals the transmission time of the level-2 sub-networks.