## 4 Analytical Model for FASBA
If the transmission is still unsuccessful after \(m\) retry, the packet will drop and \(CW\) will reset. Hence, the channel contention can be described as a two-dimensional Markov chain model with \((s(t),b(t))\). In this model, every station is modelled by a pair of integers \((i,k)\), where \(i\) is the value for the backoff stage and \(k\) is the value of the backoff counter. The Markov model is presented in Figure 3. In wireless communication, a station's transmission can fail due to collision with other stations transmitting on the same channel at a given time. Therefore, the probability of collision, \(p_{coll}\), can be written as:  
\[p_{coll}=1-(1-\tau)^{n-1} \tag{1}\]  
where \(\tau\) is the transmission probability in a randomly chosen slot time and \(n\) represents the number of stations in the medium. In addition to collision, a frame may encounter transmission failure due to fading and/or noise. Thus, the frame failure transmission probability, \(p_{f}\), becomes:  
\[p_{f}=1-(1-p_{coll})(1-p_{err}) \tag{2}\]  
where \(p_{err}\) is the probability of frame error due to the channel condition. Let us consider that a station has a packet to transmit. Before initiating the transmission, the station will choose a random backoff timer from the window of \([0,CW_{min}]\). The station tried to transmit the frame for the last \((i-1)\) times but was unsuccessful; the station moves to any state on \((i,k)\) with probability \(p_{f}/CW_{i}\). If the transmission is successful at any stage, the station moves to backoff stage 0 with probability \((1-p_{f}/CW_{0})\). Hence, transition probabilities of the Markov chain can be written as:  
\[P(i,k|i,k+1)=1;k\in[0,CW_{i}-2],i\in[o,m]\]  
The backoff timer is decremented from \(k+1\) to \(k\) when a station has detected an idle time slot. \[P(0,k|i,0)=(1-p_{f})/CW_{0};k\in[0,CW_{0}-1],\] \[i\in[0,m]\]  
After successful transmission with probability \(1-p_{f}\), a station moves from backoff stage \(i\) to 0. \[P(i,k|i-1,0)=p_{f}/CW_{i};k\in[0,CW_{i}-1],\] \[i\in[1,m]\]At stage \(i-1\), the station attempts to transmit but fails and moves to backoff stage \(i\) with a randomly chosen backoff delay. Figure 2:0  
It is the same as number 3, except the station reaches maximum backoff stage m.  
Let \(b_{i,k}=lim_{t\rightarrow\infty}Ps(t)=i,b(t)=k,i\in(0,m),k\in(0,CW_{i}-1)\) be the stationary probability of the chain. Now, using the same reasoning [38], we can estimate the probability of transmission \(\tau\) at any random slot for both cases when retry limit \((m)\) is smaller than the maximum backoff stage \((m^{{}^{\prime}})\). Figure 2:1  
where \(X=(1-p_{f})CW(1-2p_{f}^{m+1})+(1-2p_{f})(1-p_{f}^{m+1})\) and \(Y=CW(1-2p_{f}^{m^{\prime}+1})(1-p_{f})+(1-2p_{f})(1-p_{f}^{m+1})+2m^{{}^{\prime }}CWp_{f}^{m^{\prime}+1}(1-2p_{f})(1-p_{f}^{m-m^{\prime}})\). When retry limit \((m)\) is greater or equal to the maximum backoff stage, the Equation (3) becomes:  
Figure 2:2  
Now, to solve \(p_{f}\), we first define the probability of frame error \(p_{err}\), which is a combination of the frame error of the MAC data frame \((FE_{data})\) or ACK frame \((FE_{ACK})\). Therefore, \(p_{err}\) can be expressed as follows:  
Figure 2:3 Figure 2:4  
Figure 2:5 Two—dimensional Markov chain model for FASBA-based 802.11n backoff. According to [39], frame error can be calculated based on maximum Doppler frequency \((f_{d})\) from station mobility and fading margin \((f_{m})\). Figure 2:6  
Now, the solution for the non-linear system represented by Equations (3)-(5) can be solved numerically with two unknown parameters \(\tau\) and \(p_{f}\). Knowing \(\tau\), we can calculate the probability \((P_{tr})\) that at least one station transmits a packet in a randomly selected time slot and the conditional probability \((\ P_{s})\) that the occurring packet is successfully transmitted as follows:  
Figure 2:7  
Figure 2:8