# A Self-Scrutinized Backoff Mechanism for IEEE 802.11 ax in 5G Unlicensed Networks
## 4 Analytical Model
### Recursive Discrete-Time Markov Chain (R-DTMC) Model
On the other hand, Equation (7) describes how when a data frame transmission is successful at backoff stage \(b\), the backoff stage decreases to \(b-1\), and the new backoff value is uniformly chosen from the adaptively scaled-down contention window \(W_{b-1}\). In case the backoff stage reaches the value \(m\) (that is the maximum backoff value), it is not increased in the subsequent data frame transmission attempt. Figure 5: Recursive discrete-time Markov chain model (R-DTMC) for COSB mechanism. Let us assume that \(d_{b,\;u}=\lim\limits_{t\rightarrow\infty}P\{\pi(t)=b,\;\Omega(t)=u\},b\in(0,m),\;u \in(0,W_{b}-1)\) be the stationary distribution of the R-DTMC. From Figure 5, each state transition probability can be written as,  
\[d_{1,0}=\frac{p_{obs}}{1-p_{obs}}d_{0,0}. \tag{9}\]  
If \(\beta=\frac{p_{obs}}{1-p_{obs}}\), the above equation can be written as, \(d_{1,0}=\beta\times d_{0,0}\). Similarly, \(d_{b,0}=\beta\times d_{b-1,0}\) where \(d_{b-1,0}=\beta\times d_{b-2,0}\) till \(d_{1,0}=\beta\times d_{0,0}\). Therefore, we can write:  
\[d_{b,0}=\beta^{b}\times d_{0,0},\;0<b<m. \tag{10}\]  
Now for the backoff stage \(m\), the \(d_{m,0}\) can be written as,  
\[d_{m,0}=\beta^{m}\times d_{0,0} \tag{11}\]  
Owing to the Markov process-based chain regularities, for each \(u\in(1,W_{b}-1)\), the stationary distribution for \(\{\pi(t),\;\Omega(t)\}\) can be written as,  
\[d_{b,u}=\frac{W_{b}-u}{W_{b}}\times\left\{ \begin{array}{c}(1-p_{obs})\times d _{b+1,0}\;b=0\\ p_{obs}\times d_{b-1,0}+(1-p_{obs})\times d_{b+1,0}\;0<b<m\\ p_{obs}\times(d_{m-1,0}+d_{m,0})\;b=m\end{array} \right. \tag{12}\]  
The recursive characteristic of state transition probabilities can be combined as,  
\[\sum_{b=0}^{m-1}(b+1)\times d_{b,0}+m\times d_{m,0}\;=d_{0,0}\left(\frac{1- \beta^{m}-m\beta^{m}(1-\beta)}{(1-\beta)^{2}}\right) \tag{13}\]  
From Equations (9)-(11) and (13), Equation (12) can be re-written as,  
\[d_{b,u}=\frac{(W_{b}-u)(b+1)}{W_{b}}\;d_{b,0},\;b\in(0,m),u\in(0,W_{b}-1) \tag{14}\]  
From Equations (9)-(11) and (14), all the values \(d_{b,u}\) are expressed as a function of \(d_{0,0}\) and channel observation-based practical conditional collision probability \(p_{obs}\). \(d_{0,0}\) is finally determined by normalizing the R-DTMC states as follows,  
\[1=\sum_{b=0}^{m-1}(b+1)\times d_{b,0}\sum_{n=0}^{W_{b}-1}\frac{W_{b}-u}{W_{b}} +\frac{m\times(W_{m}-u)}{W_{m}}\;d_{m,0}, \tag{15}\]  
From \(W^{*}=W_{min}\times\omega^{p_{obs}}\) and few mathematical steps, the above normalization relation can be written as,  
\[1=\frac{d_{0,0}}{2}\left[W^{*}\left(\frac{1-\left(2\beta\right)^{m}-m\left(2 \beta\right)^{m}(1-2\beta)}{\left(1-2\beta\right)^{2}}\right)+\left(\frac{1- \beta^{m}-m\beta^{m}(1-\beta)}{\left(1-\beta\right)^{2}}\right)\right] \tag{16}\]  
Finally, we get \(d_{0,0}\) as follows,  
\[d_{0,0}=\frac{2(1-\beta)^{2}(1-\;\beta^{m})}{\left(1-\beta^{m}-m\beta^{m}(1- \beta)\right)}\times\frac{1}{\left((1-2\beta)(1-\;\beta^{m})(W^{*}+1)+\beta W ^{*}(1-\beta)\big{(}1-\;\left(2\beta\right)^{m}\big{)}\right)}. \tag{17}\]Since, a transmission occurs only when the backoff counter of the WE reaches zero regardless of the backoff stage, transmission probability \(\gamma\) can be expressed as follows,  
\[\gamma=\sum_{b=0}^{m-1}(b+1)d_{b,0}+md_{m,0}=d_{0,0}\Bigg{(}\frac{1-\beta^{m}-m \beta^{m}(1-\beta)}{\left(1-\beta\right)^{2}}\Bigg{)}. \tag{18}\]  
Furthermore, after performing a few mathematical steps to Equation (18) using the value of \(d_{0,0}\) from Equation (17) we get,  
\[\gamma=\frac{2}{\left(W^{*}+\beta W^{*}\bigg{(}\frac{\sum_{b=0}^{m-1}\left(2 \beta\right)^{b}}{\sum_{b=0}^{m-1}\left(\beta\right)^{b}}\bigg{)}+1\right)}. \tag{19}\]  
However, in general \(\gamma\) depends on the practical collision probability \(p_{obs}\), which is always unknown until the channel is observed for the busy slots. A transmitted data frame encounters the collision if at least one of the \(n-1\) remaining WEs transmit.