# 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
Thus, the probability of all these transmissions succeeding is  
\[P_{succ}(T)=(1-\tau)\cdot\mathcal{A}(T)\cdot\prod_{l^{\prime}\in L(T)}(1-\tau_ {l^{\prime}}) \tag{29}\]  
where \(T=\{l_{1},l_{2},\cdots,l_{n_{l}}\}\) and \(\mathcal{A}(T)\) is an indicator function. If each node in \(T\) is outside the CCA range of every other node in \(T\), then \(\mathcal{A}(T)=1\); otherwise, \(\mathcal{A}(T)=0\). Using Equation (29), the probability that all transmissions succeed when some WLAN nodes located within the CCA range of the LTE-LAA eNB transmit in a timeslot is  
\[P_{w,s}=\prod_{T\subset L^{\prime}}\left(P_{succ}(T)\cdot\prod_{l^{\prime} \in T}\tau_{l^{\prime}}\right) \tag{30}\]Using the equation above, the probability of a collision occurring among transmissions from WLAN nodes located within the CCA range of the LTE-LAA eNB is  
\[P_{\text{tr},c}=(1-\tau)\cdot(\tau_{\text{tr}}-P_{\text{tr},s}) \tag{31}\]  
Let \(T_{laa}\) denote the expected time for a transmission by the LTE-LAA eNB, and let \(T_{\text{tr},s}\) and \(T_{\text{tr},c}\) denote the expected times for a successful transmission by a WLAN node and a collision experienced by the WLAN node, respectively. Using \(P_{tr}\), \(P_{\text{tr},s}\), \(P_{\text{tr},c}\), \(P_{\text{tr},s}\), and \(P_{\text{tr},c}\) in Equations (25)-(28), (30), and (31), respectively, the expected time spent per channel state is  
\[E_{s}=(1-P_{tr})\sigma+P_{\text{tr},s}T_{laa}+P_{\text{tr},s}T_{\text{tr},s}+P_ {\text{tr},c}\cdot\max\{T_{laa},T_{\text{tr},c}\}+P_{\text{tr},c}T_{\text{tr},c} \tag{32}\]  
where \(\sigma\) is the timeslot size. Let \(R_{W}\) denote the data rate of the WLAN system, as defined in the 802.11 standard [20]. The throughput of the WLAN system can then be obtained as follows:  
\[S_{w}=\frac{P_{\text{tr},s}T_{\text{tr},s}^{data}R_{W}}{E_{s}} \tag{33}\]  
Here, \(T_{\text{tr},s}^{data}\) represents the portion of \(T_{\text{tr},s}\) during which data is transmitted. Unlike for the WLAN system, we account for throughput degradation due to collisions to estimate the throughput of the LTE-LAA system. Let \(P_{I,in}\) represent the average interference power caused by WLAN nodes within the CCA range. Then, similarly to Equation (20), we can calculate \(P_{I,in}\); however, in this case, \(y=\alpha\cdot R_{CCA}\) and \(dy=R_{CCA}\cdot d\alpha\). Thus, \(P_{I,in}\) is given by  
\[P_{I,in}\simeq 2\lambda_{n,in}P_{n}K\cdot\int_{0}^{\pi}\int_{0}^{1}v(\alpha, \theta)^{-\eta}\cdot R_{CCA}d\alpha d\theta \tag{34}\]  
where \(\lambda_{n,in}\) represents the average number of WLAN nodes transmitting simultaneously within the CCA range. Let \(\gamma_{in}\) denote the SINR measured during a collision. Then, referring to Equation (16), it is given by \(\gamma_{in}=\frac{P_{\text{tr},s}}{P_{I}+P_{\text{tr},s}+T_{N}}\). Thus, using Equation (21), the throughput of the LTE-LAA system is  
\[S_{laa}=\frac{T_{laa}^{data}\cdot\{P_{\text{tr},s}R_{laa}(\gamma)+P_{\text{ tr},c}R_{laa}(\gamma_{in})\}}{E_{s}} \tag{35}\]  
Here, \(T_{laa}^{data}\) represents the portion of \(T_{laa}\) during which data is transmitted, and \(\gamma\) denotes the SINR measured when there is no interference from WLAN nodes within the CCA range, which can be calculated using Equation (16). If multiple LTE-LAA devices are present, \(\gamma\) and \(\gamma_{in}\) are computed as the average SINRs across all LTE-LAA devices.