# 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  
Consider there are \(n\) number of WEs competing for the channel in a WLAN. In the saturated condition, each WE has immediately a data frame available for transmission after each successful transmission. Thus, due to the consecutive data frame transmission, each data frame needs to wait for a random backoff time before transmitting. Let \(b\) be the backoff stage counter for a tagged WE and \(m\) be the maximum number of backoff stages \(b\) can experience for a data frame, that is \(b\in(0,m)\), such that \(W_{b}=2^{b}\times W_{min}\times\omega^{p_{obs}}\) for \(b^{\text{th}}\) backoff stage and \(W_{max}=2^{m}\times W_{min}\times\omega^{p_{obs}}\) for the \(m^{th}\) backoff stage contention window, where \(W_{b}\) is the contention window size at \(b^{\text{th}}\) backoff stage and \(p_{obs}\) is the observed channel collision probability. Let us adopt the notation \(W_{b+1}=2^{b+1}\times W_{min}\times\omega^{p_{obs}}\), for the adaptively scaled-up contention window for \(b+1\) backoff stage, when transmission is failed at the \(b^{\text{th}}\) backoff stage. Similarly, let \(W_{b-1}=2^{b-1}\times W_{min}\times\omega^{p_{obs}}\) be the adaptively scaled-down contention window for the \(b-1\) backoff stage, when successfully transmitted at the \(b^{\text{th}}\) backoff stage. Assume \(\Omega(t)\) is the function for the stochastic process representing the backoff counter \(u\) for a tagged WE, where \(u\in(0,W_{cur}-1)\). Since time is discretized as an integer time scale, \(t\) and \(t+1\) correspond to the beginning of two consecutive transmission time slots, and the backoff time counter of each WE decreases at the beginning of each slot time. Figure 3 illustrates that the backoff time decreases when the communication channel is sensed as idle (\(\sigma\)), and it stops when the channel is sensed as busy, which may be due to a successful or unsuccessful transmission of any other WE. Therefore, the time interval between two consecutive slot time beginnings may be much longer and different from the idle slot time size, i.e., \(\sigma\). Let \(\pi(t)\) be the stochastic process representing the backoff stages \((0,\;1,\;2,\;\ldots,\;m)\) of the tagged WE at time \(t\). The key articulation in our R-DTMC model is that, at each data frame transmission attempt regardless of the number of retransmission attempts, each data frame collides with a practically observed and independent collision probability \(p_{obs}\). With these assumptions, COSB can be modeled as the two dimensional process \(\{\pi(t),\;\Omega(t)\}\) with the R-DTMC as depicted in Figure 5. In this R-DTMC, the transition probabilities are described as follows. The tagged WE remains at the first backoff stage after a successful transmission on the first backoff stage with the probability,  
\[P\{(0,u)|(0,0)\}=(1-p_{obs})/W_{0},\;u\in(0,W_{0}-1) \tag{4}\]  
The backoff counter decreases when the channel is sensed as idle with the probability,  
\[P\{(b,u)|(b,u+1)\}=1,\;b\in(0,m),u\in(0,W_{b}-2) \tag{5}\]The tagged WE scales-up the current contention window and moves to the next stage \(b\) if a data frame transmission failed on backoff stage \(b-1\) with the probability,  
\[P[(b,u)|(b-1,0)\}=p_{obs}/W_{b},\ b\in(0,m),u\in(0,W_{b}-1) \tag{6}\]  
The tagged WE scales-down the current contention window and decreases its backoff stage for the next transmission attempt to \(b-1\) after a successful transmission on backoff stage \(b\) with the probability,  
\[P[(b-1,u)|(b,0)\}=(1-p_{obs})/W_{b-1},\ b\in(0,m),u\in(0,W_{b}-1) \tag{7}\]  
The tagged WE remains at the \(m^{th}\) backoff stage after an unsuccessful transmission with the probability,  
\[P[(m,u)|(m,0)\}=p_{obs}/W_{m},\ u\in(0,W_{m}-1) \tag{8}\]  
In particular, to the above transition probabilities, as considered in Equation (6), when a data frame transmission is collided at backoff stage \(b-1\), the backoff stage increases to \(b\), and the new backoff value is uniformly chosen from the adaptively scaled-up contention window \(W_{b}\).