# A Throughput Analysis Using a Non-Saturated Markov Chain Model for LTE-LAA and WLAN Coexistence
## 2 Analysis of LTE-LAA and WLAN System Throughputs
### Throughput of LTE-LAA and WLAN Systems  
In this subsection, we analyze the throughput of LTE-LAA and WLAN systems based on the random backoff operations of WLAN nodes and the random backoff operations of the LTE-LAA eNB, as explained in Section 2.2. The WLAN nodes employ random backoff operations, similar to those of LTE-LAA eNB described in Section 2.1, to control access to the shared wireless channel. The study in [15] analyzed the random backoff operations of WLAN nodes using a non-saturated Markov chain, with post-backoff and backoff stages denoted as \((0,k_{2})_{e}\) and \((k_{1},k_{2})\), respectively. Figure 6 illustrates the first two stages of the Markov chain model proposed in [15]. The following notations are used in this subsection:  
* \(k_{1}\): Backoff stage number. * \(k_{2}\): Value of the backoff counter. * \(W_{0}\): Minimum contention window size. * \(b(0,0)_{e}\): Stationary probabilities of being in the post-backoff states \((0,0)_{e}\). * \(p_{l}\): Collision probability of WLAN node \(l\). * \(q_{l}\): Probability of at least one packet awaiting transmission in the buffer of WLAN node \(l\) in a timeslot. * \(P_{idle}^{l}\): Probability that WLAN node \(l\) senses the medium as idle in a timeslot. * \(\tau_{l}\): Probability that WLAN node \(l\) is attempting transmission in a timeslot. * \(L(T)\): Set of WLAN nodes located within the CCA ranges of the nodes in the set \(T\), which does not include the nodes in the set \(T\). * \(L^{\prime}\): Set of WLAN nodes located in the CCA range of the LTE-LAA eNB. Then, referring to [15], the transmission probability \(\tau_{l}\) can be obtained as follows:  
\[\tau_{l}=b(0,0)_{e}\cdot\left(\frac{q_{l}^{2}W_{0}}{(1-p_{l})(1-q_{l})(1-(1-q_ {l})^{W_{0}})}-\frac{q_{l}^{2}P_{idle}^{l}}{1-q_{l}}\right) \tag{22}\]  
To simplify the analysis, we disregard the hidden-node problem; specifically, we assume that, for any transmitting and receiving pair of WLAN nodes, the receiver can only sense transmissions from WLAN nodes located within the CCA range of the transmitter. Thus,  
Figure 6: Markov chain model for the random backoff operations of a WLAN node. considering the transmissions of both the LTE-LAA eNB and WLAN nodes within the CCA range of transmitter \(l\), the collision probability \(p_{l}\) in Equation (22) is given by  
\[1-p_{l}=(1-\tau)\cdot\prod_{l^{\prime}\in L(\{l\})}(1-\tau_{l^{\prime}}) \tag{23}\]  
where the transmission probability of the LTE-LAA eNB, \(\tau\), can be obtained from Equation (15). Using Equation (23), \(P_{i\ddle}^{l}=1-p_{l}\). The probability that at least one WLAN node within the CCA range of the LTE-LAA eNB attempts transmission is given by  
\[\tau_{w}=1-\prod_{l^{\prime}\in L^{\prime}}(1-\tau_{l^{\prime}}) \tag{24}\]  
The channel usage state may be occupied due to either a successful transmission or a collision among transmissions; otherwise, the medium may be idle. Using Equations (15) and (24), the probability that the LTE-LAA eNB or at least one of WLAN nodes will attempt transmission is  
\[P_{tr}=1-(1-\tau)(1-\tau_{w}) \tag{25}\]  
The probabilities that the LTE-LAA eNB and the WLAN nodes within the CCA range transmit without collision are, respectively,  
\[ \begin{split}& P_{u,s}=\tau(1-\tau_{w})\\ & P_{w,s}=\tau_{w}(1-p_{w})(1-\tau)\end{split}  \tag{26}\]  
The probability that the transmission from the LTE-LAA eNB is interfered with by transmissions from WLAN nodes within the CCA range is given by  
\[P_{u,c}=\tau_{w}\cdot\tau \tag{27}\]  
The probability of a collision occurring among transmissions of WLAN nodes within the CCA range is  
\[P_{w,c}=\tau_{w}\cdot p_{w}\cdot(1-\tau) \tag{28}\]  
Suppose that \(n_{l}\) WLAN nodes, labeled \(l_{1},l_{2},\ldots,l_{n_{l}}\), are attempting transmission in a timeslot. For all these transmissions to succeed, the LTE-LAA eNB must remain idle, each WLAN node must be outside the CCA range of the others, and none of the WLAN nodes in the set \(L(l_{1},l_{2},\ldots,l_{n_{l}})\) should transmit.