## 3 Definition and Theory Analysis
### Theory Analysis  
We derive frame transmission efficiency:  
\[R_{FTE}=\frac{T_{SL}}{T_{FSD}}=\frac{E(T_{SL})}{E(T_{FSD})}, \tag{3}\]  
where \(E(T_{SL})\) is the mathematic expectation of \(T_{SL}\). \(E(T_{FSD})\) is the mathematic expectation of \(T_{FSD}\). 1. The numerical value of \(\mathbf{T_{SL}}\): As we can see in Figures 1 and 2, the time that a frame is successfully transmitted with the RTS/CTS mechanism is: \[T_{SL}=T_{RTS}+T_{CTS}+T_{F}+T_{ACK}+3T_{SIFS},\] (4) where \(T_{F}\) is the time of the data frame transmission. The time that a frame is successfully transmitted while using the basic access mechanism is: \[T_{SL}=T_{F}+T_{SIFS}+T_{ACK}.\] (5) According to the system parameters, \(T_{SIFS}\) is 10 \(\upmu\)s. The lengths of RTS, CTS and ACK are 160 bits, 112 bits and 112 bits, respectively. Thus, \(T_{SL}\) is determined by the size of the data frame and is independent of the number of collisions, that is \(E(T_{SL})=T_{SL}\). If the size of the data frame is determined, \(T_{SL}\) will be a constant value. (2) The numerical value of \(\mathbf{T_{FSD}}\): The numerical value of \(T_{FSD}\) is: \[T_{FSD}=T_{CL}+T_{SL},\] (6)  
Figure 1: The delay of RTS/CTS (Request To Send/Clear To Send) mechanism. Figure 2: The delay of the basic access mechanism. where:  
\[T_{\text{CL}}=T_{\text{DIFS}}+\sum_{j=0}^{N}(T_{C_{j}}+T_{\text{CW}_{j}})+T_{ \text{CW}_{N+1}}, \tag{7}\]  
where \(T_{\text{CL}}\) is the total time of congestion. \(T_{C_{j}}\) is the time of each congestion. \(T_{\text{CW}_{j}}\) is the time of congestion waiting. \(N\) is the number of collisions, and the range of \(N\) is period [1, \(m\)], where \(m\) is the maximum number of collisions. Next, we will calculate the mathematic expectation of \(T_{FSD}\). \[E(T_{FSD}|N=k)=E(T_{\text{CL}})+E(T_{\text{SL}})=T_{\text{DIFS}}+E(\sum_{j=0}^{ k}(T_{C_{j}}+T_{\text{CW}_{j}}))+E(T_{\text{CW}_{k+1}})+E(T_{\text{SL}}), \tag{8}\]  
that is  
\[E(T_{FSD}|N=k)=\sum_{k=0}^{m}(T_{DIFS}+E(\sum_{j=0}^{k}(T_{C_{j}}+T_{\text{CW}_ {j}}))+E(T_{\text{CW}_{k+1}})+E(T_{\text{SL}}))\times P(N=k) \tag{9}\]  
where \(P(N=k)\) is the function of the collision numbers \(k\). We have \(E(T_{\text{SL}})\)=\(T_{\text{SL}}\); thus:  
\[E(T_{FSD}|N=k)=\sum_{k=0}^{m}(T_{DIFS}+E(\sum_{j=0}^{k}(T_{C_{j}}+T_{\text{CW} _{j}}))+T_{\text{CW}_{k+1}}+T_{\text{SL}})\times P(N=k) \tag{10}\]  
With the RTS/CTS mechanism, there is:  
\[T_{C_{j}}= \begin{cases}0&j=0\\ T_{\text{RTS}}+T_{\text{SIFS}}+T_{\text{CTS}}&0<j\leq m\end{cases} . \tag{11}\]  
With the basic access mechanism, there is:  
\[T_{C_{j}}= \begin{cases}0&j=0\\ T_{\text{CF}}+T_{\text{SIFS}}+T_{\text{ACK}}&0<j\leq m\end{cases} , \tag{12}\]  
where \(T_{\text{CF}}\) is the time of collision data frame transmission. According to the default value of the system parameters, \(T_{SIFS}\) is 10 \(\mu\)s. The lengths of RTS, CTS and ACK are 160, 112 and 112 bits, respectively. Thus, \(T_{C_{j}}\) is determined by the size of the collision data frame. If the packet size is fixed, there is \(T_{F}=T_{\text{CF}}\). Thus, \(\sum_{j=0}^{k}T_{C_{j}}\) is a constant, named \(G\). According to the IEEE 802.11 standard, the node uses the binary backoff algorithm to reserve the channel and send data. In the binary backoff algorithm, an integer is selected as the backoff time slot number \(W_{j}\) randomly from the uniformly-distributed period \([0,CW]\). The backoff window \(CW\) will be exponential increased with the number of retransmissions \(j\), that is \(CW_{j}=2^{j}\times CW_{min},j\in[0,M]\), where \(CW_{min}\) is the constant and \(M\) is the maximum backoff stage. When the node performs the backoff algorithm in the first time, \(CW\) will be \(CW_{min}\). The following backoff time slot will be \(CW_{j}=2^{j-1}\times CW_{min}\). When the number of collisions exceeds \(M\), there is \(W_{j}=2^{M-1}\times CW_{min}\). Hence the average backoff time is:  
\[E(T_{CW_{j+1}})= \begin{cases}2^{j-1}CW_{min}*E(T_{slot})&0<j\leq m\\ 2^{M-1}CW_{min}*E(T_{slot})&j>M\end{cases}