# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## Appendix B  
Because the event \(X_{M,in}<x\) is the event of \(R_{1}<r\) on the condition that the typical user is served by the MBS, the probability of \(X_{M,in}<x\) can be expressed as:  
\[\mathbb{P}\left(X_{M,in}<x\right)=\mathbb{P}\left(R_{1}<x\left|R_{1}<D\right. \right)=\left(1-\exp\left(-\pi\lambda_{1}x^{2}\right)\right)/\mathcal{Q}_{M,in},x\in\left(0,D\right].\]  
Then, the PDF of \(X_{M,in}\) can be obtained by differentiating the foregoing expression.  
Similarly, the cumulative distribution function (CDF) of \(X_{M,out}\) is also given by:  
\[\begin{split}\mathbb{P}\left(X_{M,out}<x\right)&= \mathbb{P}\left(R_{1}<x\left|R_{1}>D,P_{1}R_{1}^{-\alpha_{1}}/P_{2}R_{2}^{- \alpha_{2}}\geq\beta\right.\right)\\ &=\frac{\mathbb{P}\left(R_{1}<x,R_{2}\geq\left(\frac{\beta P_{2} }{P_{1}}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}\left|R_{1}>D \right.\right)\mathbb{P}\left(R_{1}>D\right)}{\mathcal{Q}_{M,out}}\end{split} \tag{10}\]  
The first term of the numerator in Equation (10) can be derived as:  
\[\begin{split}\mathbb{P}\left(R_{1}<x,R_{2}\geq\left(\frac{\beta P _{2}}{P_{1}}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}\left|R_{ 1}>D\right.\right)&=\int_{D}^{x}\mathbb{P}\left(R_{2}\geq\left( \frac{\beta P_{2}}{P_{1}}\right)^{1/\alpha_{2}}r^{\epsilon_{1}/\epsilon_{2}} \right)f_{R_{1}\left|R_{1}>D}\left(r\right)dr\\ &=\int_{D}^{x}\exp\left(-\pi\lambda_{2}\Big{(}\frac{\beta P_{2} }{P_{1}}\Big{)}^{2/\alpha_{2}}r^{2\epsilon_{1}/\epsilon_{2}}\right)\frac{2\pi \lambda_{1}r\exp\left(-\pi\lambda_{1}r^{2}\right)}{\exp\left(-\pi\lambda_{1}D^ {2}\right)}dr\end{split}\]  
Substituting back into Equation (10) and differentiating the resultant CDF, we have Equation (12).  
Then, the CDF of \(X_{F}\) is:  
\[\begin{split}\mathbb{P}\left(X_{F}<x\right)&= \mathbb{P}\left(R_{2}<x\left|R_{1}>D,P_{1}R_{1}^{-\alpha_{1}}/P_{2}R_{2}^{- \alpha_{2}}\leq 1\right.\right)\\ &=\frac{\mathbb{P}\left(R_{2}<x,R_{2}\leq\left(\frac{P_{2}}{P_{1 }}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}\left|R_{1}>D\right. \right)\mathbb{P}\left(R_{1}>D\right)}{\mathcal{Q}_{F}}\end{split} \tag{11}\]  
The first term of the numerator in Equation (11) can be derived as:  
\[\begin{split}\mathbb{P}\left(R_{2}<x,R_{2}\leq\left(\frac{P_{2}} {P_{1}}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}\left|R_{1}>D \right.\right)=\\ &\left\{\begin{array}{l}\mathbb{E}_{R_{1}}\left[\mathbb{P} \left(R_{2}<x\left|R_{1}>D\right.\right)\right],x\leq\left(\frac{P_{2}}{P_{1}} \right)^{1/\alpha_{2}}D^{\epsilon_{1}/\epsilon_{2}}\\ \mathbb{E}_{R_{1}}\left[\mathbb{P}\left(R_{2}<x,R_{2}\leq\left(\frac{P_{2}}{P_{1 }}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}\left|R_{1}>D\right. \right)\right],x>\left(\frac{P_{2}}{P_{1}}\right)^{1/\alpha_{2}}D^{\epsilon_{ 1}/\epsilon_{2}}\end{array}\right.\end{split}\]  
Using Equation (41) in [32], the PDF of \(X_{F}\) is given by Equation (13).  
Finally, the joint CDF of \(\mathbf{X}_{C}\) can be written as:  
\[\begin{split}\mathbb{P}\left(X_{1}<x_{1},X_{2}<x_{2}\right)& =\mathbb{P}\left(R_{1}<x_{1},R_{2}<x_{2}\left|\left(\frac{P_{2}}{P_{1 }}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}}<R_{2}<\left(\frac{ \beta P_{2}}{P_{1}}\right)^{1/\alpha_{2}}R_{1}^{\epsilon_{1}/\epsilon_{2}},R_{ 1}>D\right.\right)\\ &=\frac{\mathbb{P}\left(R_{1}<x_{1},