# A Channel Measurement-Based Listen-Before-Talk Algorithm for LTE-LAA and WLAN Coexistence
## 3 Proposed CM-LBT Scheme
### System Model and Transmission Probability of LTE-LAA eNB  
We adopt the LB-LBT procedure with a fixed contention window size for LTE-LAA eNB described in Section 2.3. The behavior of LTE-LAA eNB from applying our CM-LBT procedure is modeled as a non-saturated Markov chain [9]. From this non-saturated Markov chain, we can obtain the probability that the LTE-LAA eNB is attempting transmission  
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in the unlicensed band. This LTE-LAA transmission probability is used to estimate the throughput of LTE-LAA and WLAN systems in Section 3.3. Our model introduces two stages, \((k)_{e}\) and \((k)\) for \(k\in[0,W-1]\), where \(W\) is the chosen fixed contention window size of LTE-LAA eNB, as shown in Figure 2. Since a single LTE-LAA eNB shares the channel with multiple WLAN nodes, it employs two backoff stages, as shown in Figure 2, to ensure the transmission rate, even in the presence of collisions, unlike WLAN. The first and second stages are called post-backoff and backoff, respectively. The post-backoff stage, \((k)_{e}\), represents LTE-LAA eNB with the transmitted packet; however, it has no packets waiting. Conversely, LTE-LAA eNB must have a packet awaiting transmission in the backoff stage. The counter of the backoff stage, \(k\), is initially uniformly chosen between \([0,W-1]\) and is decremented while the medium is idle. We assume constant probability \(1-q\) that the LTE-LAA eNB buffer will have no packets awaiting transmission at the start of each counter decrement. This enables us to derive relationships between the following variables: \(q\), the probability of at least one packet awaiting transmission at the start of a counter decrement; \(p\), the probability of a collision given that LTE-LAA eNB is attempting transmission; \(P\), the Markov chain transition matrix, \(b\), the chain's stationary distribution, and \(\tau\), the stationary distribution's probability that the LTE-LAA eNB will transmit in a slot. Under the above assumptions, for \(0<k<W\), we have the following:  
\[ \begin{split} P[(k-1)|(k)]&=1,\\ P[(k-1)_{e}|(k)_{e}]&=1-q,\\ P[(k-1)|(k)_{e}]&=q.\end{split}  \tag{1}\]  
When the counter reaches \(0\) and a packet is queued, a transmission is initiated. If other WLAN nodes transmit at the same time, a collision occurs. In the case of a collision, we choose a random backoff counter again between \([0,W-1]\). In the case of a successful transmission, we return to the post-backoff stage, \((k)_{e}\), and the LTE-LAA eNB buffer is empty with probability \(1-q\). Accordingly, for \(k\geq 0\), we have the following:  
\[ \begin{split} P[(k)_{e}|(0)]&=\frac{(1-p)(1-q)}{W}, \\ P[(k)|(0)]&=\frac{(1-p)q+p}{W}.\end{split}  \tag{2}\]  
The final transitions are from the \((0)_{e}\) state, where post-backoff is complete, but the LTE-LAA eNB buffer is empty. In this case, this state is maintained if the station buffer remains empty. Then, a packet arrives. If the transmission succeeds, we return to the post-backoff stage. Otherwise, if a collision occurs or the medium is busy, we again begin  
Figure 2: Non\(-\)saturated Markov chain model for LTE-LAA eNB behavior. the backoff stage. With \(P_{idle}\) denoting the probability that the medium is sensed as idle during a typical slot, the transitions from the \((0)_{e}\) state are as follows:  
\[P[(0)_{e}|(0)_{e}]=1-q+\frac{qP_{idle}(1-p)}{W}, \tag{3}\] \[P[(k)_{e}|(0)_{e}]=\frac{qP_{idle}(1-p)}{W},\quad 0<k<W\] \[P[(k)|(0)_{e}]=\frac{qP_{idle}p+q(1-P_{idle})}{W},\;\;0\leq k<W.\]  
Given \(p\), \(P_{idle},q\), and \(W\), we can determine the stationary distribution of this Markov chain. This enables us to determine the probability, \(\tau\), that an LTE-LAA eNB is attempting transmission in a typical slot.