# 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  
According to the above protocol description, each node actually has a finite number of states. Moreover, each state is only related to the previous one. Hence, referring to the method in [2], we used a discrete Markov chain to model the states of each node and their transitions. After solving the steady states of the Markov chain, we further modeled the analytical performance of the protocol. As a comparison, we first provide the state transition diagram of the HD node in the network [7], as follows:  
In Figure 6, \(S_{1}\)-\(S_{W-1}\) denote the back-off states, and \(T\) (also can be regarded as \(S_{0}\)) denotes the transmission state. According to the CSMA mechanism, if the node senses that the channel is idle at the \(t\)-1 time slot, then it will reduce the back-off count at the \(t\) time slot. Otherwise, the back-off count remains unchanged. The actual time interval between the two consecutive back-off counts may be the length of a slot time (denoted by \(\sigma\)) or the length of the time required for a frame transmission. Let \(t\)\(\rightarrow\)\(\infty\), [7] solves the steady probability of the node in the \(T\) state (we used \(\pi_{T}\) to denote its probability, similarly hereinafter). That is, at any time slot, the transmission probability of the node is as follows:  
\[\pi_{T}\ =\ \frac{2}{W+1}. \tag{1}\]  
In the HD networks, when two or more nodes in the \(T\) state are present at a time slot, the collisions will occur. \(P_{\omega(HD)}\) denotes the collision probability. We also denote this using \(T_{\omega(HD)}\), the length of the time wasted by the collision. According to the value of \(\pi_{T}\), without using or using the control frames, [2] solves the system normalized throughputs in both cases, denoted by \(Th_{HD(basic)}\) and \(Th_{HD(RTS)}\), respectively. In the physical sense, the throughput can be obtained from dividing the total number of the transmitted bits in a time interval (denoted by \(L_{tol}\); here \"tol\" is short for \"total\") by the average length of the time interval (denoted by \(L_{ave}\); here \"ave\" is short for \"average\"). Then, we give the state transition diagram of the FD node in the network, as shown in Figure 7:  
Figure 6: State transition diagram of the HD node in the network. Figure 7: State transition diagram of the FD node in the network. Note that each node has two transmission states (\"active\" and \"passive\"), which are denoted by \(T_{1}\) and \(T_{2}\), respectively. Here, the node can jump directly from the back-off states to the transmission state with probability \(\beta\). Let \(t\)\(\rightarrow\)\(\infty\), because \(\alpha+\beta=1\), the steady probability of each state of the Markov chain in Figure 7 satisfies Equations (2) and (3). \[\pi_{S_{i}}\ =\ \alpha\pi_{S_{i+1}}+\frac{1}{W}(\pi_{T1}+\pi_{T2}),\ \ \ \ \ i\ =\ 0,\ldots,W-2\ \, \tag{2}\]  
where \(\pi_{S0}\) = \(\pi_{T1}\). For the last back-off state, we have:  
\[\pi_{S_{i}}\ =\ \frac{1}{W}(\pi_{T1}+\pi_{T2}),\ \ \ \ \ i\ =\ W-1\ . \tag{3}\]  
For \(\pi_{T2}\), we have:  
\[\pi_{T2}\ =\ \beta\sum_{i\ =\ 1}^{W-1}\pi_{S_{i}}. \tag{4}\]  
By Equations (2) and (3), the iterative expressions among each \(\pi_{Si}\)(\(i=1,\,\ldots,\,W-1\)) can be obtained as follows:  
\[\pi_{S_{i}}\ =\ \frac{1}{1+\alpha}\pi_{S_{i-1}},\ \ \ \ \ i\ =\ W-1\ \, \tag{5}\]  
\[\pi_{S_{i}}\ =\ \left(1-\frac{\alpha^{W-i}}{\sum\limits_{ \begin{subarray}{c }k\ =\ 0\end{subarray} }^{W-i}\alpha^{k}}\pi_{S_{i-1}},\ \ \ \ \ i\ =\ 1,\ldots,W-2\ \.\right. \tag{6}\]  
According to the protocol description in Section 3.2, if only two nodes communicate in the FD_1 or FD_3 mode, then one of them can jump from the back-off states to the \(T_{2}\) state. In the FD_1 mode, the probability (denoted by \(\beta_{1}\)) can be obtained as follows:  
\[\beta_{1}\ =\ \left( \begin{array}{c}n-1\\ 1\end{array} \right)\pi_{T1}(1-\pi_{T1})^{n-2}\frac{1}{n-1}, \tag{7}\]  
where the \(\left( \begin{array}{c}n-1\\ 1\end{array}