## 3 Definition and Theory Analysis
### Theory Analysis
 \tag{13}\]where \(E(T_{slot})\) is the equivalent time slot length. It is presumed that there are \(n\) saturated sites, and each of them always has packets in the queue and is contending for the channel. The transmission probability of each site is \(\tau\). Then, the probability that the MAC layer is in an idle time slot is defined as:  
\[p_{i}=(1-\tau)^{n} \tag{14}\]  
We denote \(p_{s}\) as the successful transmission probability. That is to say, there is only one site transmitting data without collision from the other \(n-1\) sites in the current slot. It can be expressed as:  
\[p_{s}=C_{n}^{1}\tau(1-\tau)^{n-1}=n\tau(1-\tau)^{n-1} \tag{15}\]  
The collision probability \(p_{c}\) is:  
\[p_{c}=1-p_{i}-p_{s} \tag{16}\]  
In a time slot, the channel may be in an idle time slot, or in a collision, or transmitting the data successfully. Hence, there is:  
\[E(T_{slot})=p_{i}\sigma+p_{s}T_{SL}+p_{c}G \tag{17}\]  
where \(\sigma\) is the length of the backoff time slot, and it is a constant. From the above formulas, \(p_{s}\) and \(p_{i}\) can be replaced by \(\tau\) and \(n\), and \(T_{SL}\) and \(G\) can be used as constants when the packet size is fixed. Thus, \(E(T_{slot})\) is determined by \(\tau\), \(n\) and \(p_{c}\). \(P(N=n)\) obeys a geometric distribution [10]. That is:  
\[P(N=k)= \begin{cases}{p_{c}}^{k}\times(1-p_{c})&0\leq k<m\\ {p_{c}}^{k}&k=m\end{cases}  \tag{18}\]  
Combining Equations (10) and (18), we get:  
\[E(T_{FSD})= \begin{cases}\frac{\mathcal{C}W_{min}\times E(T_{slot})\times(1-2^ {M}{p_{c}}^{M+1}-...-2^{m-1}{p_{c}}^{m})}{2(1-2p_{c})}\\ +(T_{DIFS}+T_{SL})\times(1-{p_{c}}^{m})+\text{G}\times\frac{{p_{c}}-{p_{c}}^{m +1}}{1-p_{c}}&m<M\\ (T_{DIFS}+T_{SL})\times(1-{p_{c}}^{m})+\frac{\mathcal{C}W_{min}\times E(T_{slot })\times(1-2^{m}{p_{c}}^{m})}{2(1-2p_{c})}\\ +\text{G}\times\frac{{p_{c}}-{p_{c}}^{m+1}}{1-p_{c}}&m>M\end{cases}  \tag{19}\]  
As shown in Equation (19), \(E(T_{FSD})\) is determined by \(\tau\), \(n\) and \(p_{c}\), when the packet size is fixed, that is \(T_{F}=T_{CF}\). Hence, \(R_{FTE}\) can be:  
\[R_{FTE}=\frac{T_{SL}}{T_{FSD}}=\frac{E(T_{SL})}{E(T_{FSD})}=\frac{T_{SL}}{E(T _{FSD})} \tag{20}\]  
\(R_{FTE}\) is determined by \(\tau\), \(n\) and \(p_{c}\), if the size of the packet is fixed. In [24], the collision probability \(p_{c}\) is:  
\[p_{c}=1-(1-\tau)^{n-1} \tag{21}\]  
Thus, the frame transmission efficiency \(R_{FTE}\) is only related with each nodes' transmission probability \(\tau\), the number of neighbor nodes \(n\), data length \(T_{F}\) and collision data length \(T_{CF}\). For simplicity, the data length is fixed to be 1000 bytes. The correlation of frame transmission efficiency, transmission probability \(\tau\) and the number of neighbor nodes \(n\) is shown in Figure 3.  
It is indicated from Figure 3 that the frame transmission efficiency decreases with the increasing of transmission probability. Especially, when \(\tau\) is close to a certain threshold, if \(\tau\) increases, the value of \(R_{FTE}\) drops rapidly. It is also indicated from Figure 3 that increasing the number of neighbor nodes \(n\), the value of \(R_{FTE}\) drops when \(\tau\) is certain. The analysis above demonstrates that \(R_{FTE}\) is sensitive to the channel congestion at the link layer. This conclusion motivated us to investigate a novel approach by using \(R_{FTE}\) in congestion control in wireless ad hoc networks.