# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## Appendix E  
The CDF of \(X_{NC_{k}}\) is:  
\[\begin{split}\mathbb{P}\left(X_{NC_{k}}<x\right)&= \mathbb{P}\left(R_{k}<x\left|R_{1}>D,P_{k}R_{k}^{-a_{k}}\ \geqslant\ \max_{i,j\
eq k}\beta_{i}P_{i}R_{i}^{-a_{i}}\right.\right)\\ &=\mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\ \geqslant\ \max_{i,j\
eq k}\beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right) \mathbb{P}\left(R_{1}>D\right)\bigg{/}\mathcal{Q}_{NC_{k}}\end{split} \tag{10}\]  
The first term of the numerator in Equation (10) can be derived as:  
\[\begin{split}&\mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\geq \max_{i,j\
eq k}\beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right)\\ &=\prod\limits_{i=1,i\
eq k}^{K}\mathbb{P}\left(R_{k}<x,P_{k}R_{ k}^{-a_{k}}\geq\beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right)\\ &=\left\{\begin{array}{l}\mathbb{P}\left(R_{k}<x\left|R_{1}>D \right.\right)\prod\limits_{i>1,i\
eq k}^{K}\mathbb{P}\left(R_{k}<x,P_{k}R_{ k}^{-a_{k}}\geq\beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right),x\leq \left(\frac{P_{k}}{\beta_{1}P_{1}}\right)^{1/a_{k}}\!\!D^{a_{1}/a_{k}}\\ &\prod\limits_{i=1,i\
eq k}^{K}\mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\geq \beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right),x>\left(\frac{P_{k}}{ \beta_{1}P_{1}}\right)^{1/a_{k}}\!\!D^{a_{1}/a_{k}}.\end{array}\right.\end{split}\]  
Substituting back into Equation (10), the numerator in Equation (10) can be derived as:  
\[\begin{split}\mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\geq \max_{i,j\
eq k}\beta_{i}P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right)\mathbb{ P}\left(R_{1}>D\right)=\\ &\left\{\begin{array}{l}\prod\limits_{i=1,i\
eq k}^{K}\varphi_{i} \left(x\right),x\leq\left(\frac{P_{k}}{P_{1}P_{1}}\right)^{1/a_{k}}\!\!D^{a_{1 }/a_{k}}\\ \prod\limits_{i=1,i\
eq k}^{K}\varphi_{i}\left(x\right),x>\left(\frac{P_{k}}{ \beta_{1}P_{1}}\right)^{1/a_{k}}\!\!D^{a_{1}/a_{k}}\end{array}\right.\end{split}\]  
where  
\[\varphi_{i}\left(x\right)=\left\{\begin{array}{l}\mathbb{P}\left(R_{k}<x \left|R_{1}>D\right.\right)\mathbb{P}\left(R_{1}>D\right.\!,i=1\\ \mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\geq\beta_{i}P_{i}R_{i}^{-a_{i}}\left| R_{1}>D\right.\right)\mathbb{P}\left(R_{1}>D\right.\right),i\
eq 1\end{array}\right.\]  
and \(\phi_{i}\left(x\right)=\mathbb{P}\left(R_{k}<x,P_{k}R_{k}^{-a_{k}}\geq\beta_{i }P_{i}R_{i}^{-a_{i}}\left|R_{1}>D\right.\right)\mathbb{P}\left(R_{1}>D\right)\).  
Similarly to the derivation in Lemma 2, \(\varphi_{i}\left(x\right)\) and \(\phi_{i}\left(x\right)\) are further given by:  
\[\varphi_{i}\left(x\right)=\left\{\begin{array}{l}2\pi\lambda_{1}\int_{D}^{ \infty}r\exp\left(-\pi\lambda_{1}r^{2}\right)\left[1-\exp\left(-\pi\lambda_{k} x^{2}\right)\right]dr,i=1\\ 2\pi\lambda_{i}\left[\int_{0}^{\left(\frac{\beta_{i}P_{i}}{\lambda_{k}}\right)^{ 1/a_{i}}x^{\alpha_{k}/a_{i}}}r\exp\left(-\pi\lambda_{i}r^{2}\right)\left[1- \exp\left(-\pi\lambda_{k}\left(\frac{P_{k}}{P_{i}P_{i}}\right)^{2/a_{k}}x^{2a_ {i}/a_{k}}\right)\right]dr+\\ \int_{\left(\frac{\beta_{i}P_{i}}{\lambda_{k}}\right)^{1/a_{i}}x^{ \alpha_{k}/a_{i}}}^{\infty}r\exp\left(-\pi\lambda_{i}r^{2}\right)\left[1-\exp \left(-\pi\lambda_{k}x^{2}\right)\right]dr\right],i\
eq 1\end{array}\right.\]  
\[\begin{split}\phi_{i}\left(x\right)=2\pi\lambda_{i}\left[\int_{d \left(i\right)}^{\left(\frac{\beta_{i}P_{i}}{\lambda_{k}}\right)^{1/a_{i}}x^{ \alpha_{k}/a_{i}}}r\exp\left(-\pi\lambda_{i}r^{2}\right)\left[1-\exp\left(-\pi \lambda_{k}\left(\frac{P_{k}}{P_{i}P_{i}}\right)^{2/a_{k}}x^{2a_{i}/a_{k}} \right)\right]dr+\\ &\int_{\left(\frac{\beta_{i}P_{i}}{P_{k}}\right)^{1/a_{i}}x^{ \alpha_{k}/a_{i}}}^{\infty}r\exp\left(-\pi\lambda_{i}r^{2}\right)\left[1-\exp \left(-\pi\lambda_{k}x^{2}\right)\right]dr\right]\end{split}\]  
where \(d\left(i\right)=\left\{\begin{array}{l}D,i=1\\ 0,i\
eq 1\end{array}\right..\) Using the product rule for derivatives, the PDFs of \(X_{NC_{k}}\) and \(\mathbf{X}_{C_{kj}}\) can be obtained by differentiating the foregoing expression.