# DCO-MAC: A Hybrid MAC Protocol for Data Collection in Underwater Acoustic Sensor Networks
## 5 Approximate Performance Analysis
### CT-MAC Protocol  
In the CT-MAC protocol, a successful data collection period, \(T_{\text{SUC}}\), is composed of a successful transmission of the RTS packet, INVITE packet, several NOTICE packets, and ORDER packet, followed by a data transmission period and a successful transmission of the ACK packet. The transmission period of the CT-MAC protocol is illustrated in Figure 8.  
Figure 8: The transmission period of CT-MAC protocol. If the bit-error-rate of the underwater acoustic channel is high, the retransmission of the DATA packets is required when a packet error occurs. Therefore, in the level-2 sub-network, \(j\), the transmission time of hte DATA packet can be calculated as follows:  
\[T_{\text{DATA}}=\tau_{\text{max}}+\sum_{i=0}^{\infty}iK_{\text{level-2}}(\delta+ T_{\text{guard}})p_{\text{e}}^{i-1}(1-p_{\text{e}})=\tau_{\text{max}}+\frac{K_{ \text{level-2}}(\delta+T_{\text{guard}})}{1-p_{\text{e}}} \tag{10}\]  
Hence, \(T_{\text{SUC}}\) is given as follows:  
\[ \begin{split} T_{\text{SUC}}=T_{\text{RTS}}+T_{\text{INVITE}}+T_{ \text{NOTICE}}+T_{\text{ORDER}}+T_{\text{DATA}}+T_{\text{ACK}}=\\ 6\tau_{\text{max}}+(N_{j}+4)\theta+N_{j}T_{\text{guard}}+\frac{ K_{\text{level-2}}(\delta+T_{\text{guard}})}{1-p_{\text{e}}},\end{split}  \tag{11}\]  
where \(K_{\text{level-2}}\), the number of DATA packets transmitted in a level-2 sub-network during \(T_{\text{SUC}}\), can be calculated as \(K_{\text{level-2}}=N_{j}\lambda(\tau_{\text{max}}+\theta)\); if the Poisson arrival is assumed, \(N_{j}\) is the number of level-2 nodes in the considered level-2 sub-network; \(\lambda\) is the packet arrival rate at each level-2 node; and \(p_{\text{e}}\) is the packet error rate. For a successful channel reservation with the RTS packet, only one level-2 node is allowed to send an RTS packet during \(T_{\text{RTS}}\), and the RTS packet transmission of other level-2 nodes and interference nodes should be avoided. Therefore, the probability of a successful channel reservation is as follows:  
\[ \begin{split} P_{\text{SUC}}=\Pr(\text{only one RTS arrival in $T_{\text{RTS}}$}|\text{RTS arrivals in $T_{\text{RTS}}$ period})=\\ \frac{N_{j}\lambda(\tau_{\text{max}}+\theta)e^{-\lambda(N_{j}+N_{ \text{int},j})(\tau_{\text{max}}+\theta)}}{1-e^{-N_{j}\lambda(\tau_{\text{max }}+\theta)}},\end{split}  \tag{12}\]  
where \(N_{\text{int},j}\) is the number of interference nodes in level-2 sub-network, \(j\). A failed transmission period, \(T_{\text{FAIL}}\), is the length of an RTS transmission period. That is, as follows:  
\[T_{\text{FAIL}}=T_{\text{RTS}}=\tau_{\text{max}}+\theta. \tag{13}\]  
A busy period can be a period of a successful or a failed data transmission. Hence, \(\overline{B}\) is given as follows:  
\[\overline{B}=T_{\text{SUC}}P_{\text{SUC}}+T_{\text{FAIL}}(1-P_{\text{SUC}}) \tag{14}\]  
The expected time duration of the channel being used to transmit the DATA packets is given as follows:  
\[\overline{U}=K_{\text{level-2}}\delta P_{\text{SUC}}. \tag{15}\]  
According to the definition of the Poisson distribution, the probability that no RTS packets are generated in \(T_{\text{RTS}}\) is \(P_{0}=e^{-N_{j}\lambda(\tau_{\text{max}}+\theta)}\). The expected time duration of the idle period equals the expected time duration without the generated RTS packets.