# Full-Duplex MAC Protocol for CSMA/CA-Based Single-Hop Wireless Networks
## 3 Proposed FD MAC Protocol
### Modeling and Analyzing the Performance of the Proposed Protocol
\right)\pi_{T1}(1-\pi_{T1})^{n-2}\) term represents the probability that only one other node (excluding the current node) in the network \"actively\" sends a frame, and the \(\frac{1}{n-1}\) term represents the probability that the destination address of the frame is the current node. In the FD_3 mode, the solution of this probability (denoted by \(\beta_{2}\)) needs to be discussed according to the communication mode of the two nodes. Basically, we can obtain the probability that they simultaneously send the frames, as follows:  
\[\beta_{2(basic)}\ =\ \left( \begin{array}{c}n-1\\ 2\end{array} \right)\pi_{T1}^{2}(1-\pi_{T1})^{n-3}. \tag{8}\]  
First, in the mode as shown in Figure 1a, the two nodes initiate the transmission actively. We have \(\beta_{2(1)}=0\). Second, in the mode as shown in Figure 1b, we can obtain \(\beta_{2(2)}\) as follows:  
\[\beta_{2(2)}\ =\ \beta_{2(basic)}\cdot 2\Big{(}\frac{n-2}{n-1}\frac{1}{n-1} \Big{)}\cdot\frac{1}{2}\Big{(}\frac{1}{n-2}\Big{)}, \tag{9}\]  
where the \(2\Big{(}\frac{n-2}{n-1}\frac{1}{n-1}\Big{)}\) term represents the probability that the two nodes communicate in this mode, and the \(\frac{1}{2}\Big{(}\frac{1}{n-2}\Big{)}\) term represents the probability that one of the two nodes obtains the transmission opportunity and selects the current node as the destination. Third, in the mode as shown in Figure 1c, we can obtain \(\beta_{2(3)}\) as follows:  
Then, based on the value of \(\pi_{T1}\), we can solve the system normalized throughput (denoted by \(Th_{FD}\)). At a time slot, 0, 1, 2, or more nodes may send frames. For different cases, the behaviors of the nodes and channel access times they occupy are different. To solve the average length of the time interval between the two consecutive back-off counts of the node, we need to analyze these cases respectively. First, the probability that no node \"actively\" sends the frame (denoted by \(P_{idle}\)) is as follows:  
\[P_{idle}\ =\ (1-\pi_{T1})^{n}. \tag{14}\]  
The channel access time occupied by this case is exactly the length of a time slot (\(\alpha\)). Second, the probability that only one node \"actively\" sends the frame (denoted by \(P_{spl}\); here \"sgl\" is short for \"single\") is as follows:  
\[P_{sgl}\ =\ \left( \begin{array}{c}n\\ 1\end{array} \right)\pi_{T1}(1-\pi_{T1})^{n-1}. \tag{15}\]  
In this case, the destination node will send a \"reverse\" frame after receiving the frame's header. As shown in Figure 2, the corresponding channel access time is as follows (we ignored the propagation time of the signal, similarly hereinafter):  
\[T_{ggl}=\text{DIFS}+T_{Hdr}+T_{Hdr}+T_{Pyl}+\text{SIFS}+T_{Ack}. \tag{16}\]  
Third, the probability that two nodes \"actively\" send the frames simultaneously (denoted by \(P_{dbl}\); here \"dbl\" is short for \"double\") is as follows:  
\[P_{dbl}\ =\ \left( \begin{array}{c}n\\ 2\end{array} \right)\pi_{T1}^{2}(1-\pi_{T1})^{n-2}, \tag{17}\]  
where the probability that the two are mutual destinations (denoted by \(P_{bi}\), here \"bi\" is short for \"bidirectional\") is as follows:  
\[P_{bi}\ =\ P_{dbl}\left(\frac{1}{n-1}\right)^{2}. \tag{18}\]  
As shown in Figure 3, the channel access time occupied by this case (denoted by \(T_{bi}\)) is as follows:  
\[T_{bi}=\text{DIFS}+T_{Hdr}+T_{Pyl}+\text{SIFS}+T_{Ack}. \tag{19}\]  
Moreover, the probability that the nodes communicate in the FD_3 mode is as follows:  
\[P_{non-bi}=P_{dbl}-P_{bi}. \tag{20}\]  
The channel access time occupied by this case (denoted by \(T_{non-bi}\)) is as follows:  
\[T_{non-bi}=T_{ggl}+\text{SIFS}+T_{Hdr}. \tag{21}\]  
When more than two nodes \"actively\" send the frames simultaneously, the collisions will occur.