# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## Appendix A  
Following the PPP's void probability [43], the probability a typical user \(u\) is located in the inner region \(\mathcal{A}_{in}\) can be calculated by:  
\[\mathcal{Q}_{M,in}=\mathbb{P}\left(u\in\mathcal{A}_{in}\right)=\mathbb{E}_{R_ {1}}\left[\mathbb{P}(R_{1}\leq D)\right]=1-\exp\left(-\pi\lambda_{1}D^{2}\right)\]  
When the typical user located in \(\mathcal{A}_{out}\) is also served by a MBS, the probability \(\mathcal{Q}_{M,out}\) is hence derived as:  
\[\mathcal{Q}_{M,out} =\mathbb{E}_{R_{1}}\left[\mathbb{P}\left(R_{1}>D,P_{1}R_{1}^{- \alpha_{1}}/P_{2}R_{2}^{-\alpha_{2}}\geq\beta\right)\right]\] \[\overset{(a)}{=}\int_{D}^{\infty}\mathbb{P}\left(R_{2}\geq\left( \frac{\beta P_{2}}{P_{1}}\right)^{1/\alpha_{2}}r^{\alpha_{1}/\alpha_{2}}\right) f_{R_{1}|R_{1}>D}\left(r\right)dr\times\mathbb{P}\left(R_{1}>D\right)\] \[\overset{(b)}{\approx}2\pi\lambda_{1}\int_{D}^{\infty}r\exp \left(-\pi\left(\lambda_{1}r^{2}+\lambda_{2}\bigg{(}\frac{\beta P_{2}}{P_{1}} \bigg{)}^{2/\alpha_{2}}r^{2\alpha_{1}/\alpha_{2}}\right)\right)dr\]  
which: (a) follows the expectation \(g(x)\) of a measurable function of the random variable \(X\), given by \(\mathbb{E}\left[g(X)\right]=\int_{-\infty}^{\infty}g(x)f(x)dx\), given that \(X\) has a PDF of \(f(x)\), \(f_{R_{1}|R_{1}>D}\left(r\right)=\frac{f_{R_{1}}(r)}{\mathbb{P}(R_{1}>D)}=\frac {2\pi\lambda_{1}r\exp\left(-\pi\lambda_{1}D^{2}\right)}{\exp\left(-\pi\lambda_ {1}D^{2}\right)}\) and \(\mathbb{P}\left(R_{1}>D\right)=\exp\left(-\pi\lambda_{1}D^{2}\right)\), which can be both obtained from [32], and (b) is obtained by approximating the density of FBSs in the vicinity of the outer region as \(\lambda_{2}\)[31; 32; 33]. In fact, the locations of FBSs originally drawn from a HPPP form a Poisson hole process (PHP) by the creation of exclusion zones around MBSs [43; 44], which makes the characterization for the interference field and for the contact distance distribution of the PHP challenging. To maintain analytical tractability, it makes sense to ignore the effect of holes and to approximate the PHP by its original HPPP [31; 32; 33; 44]. This is because of the fact that it is desirable to derive simple but tight approximations for obtaining useful system design insights. This approach leads to the lower but tight bound on the Laplace transform of the interference power [44].  
Accordingly, the probability \(\mathcal{Q}_{F}\) is similarly derived as:  
\[\mathcal{Q}_{F} =\mathbb{E}_{R_{1}}\left[\mathbb{P}\left(R_{1}>D,P_{1}R_{1}^{- \alpha_{1}}/P_{2}R_{2}^{-\alpha_{2}}\leq 1\right)\right]\] \[=\int_{D}^{\infty}\mathbb{P}\left(R_{2}\leq\left(\frac{P_{2}}{P_ {1}}\right)^{1/\alpha_{2}}r^{\alpha_{1}/\alpha_{2}}\right)f_{R_{1}|R_{1}>D} \left(r\right)dr\times\mathbb{P}\left(R_{1}>D\right)\] \[\approx\exp\left(-\pi\lambda_{1}D^{2}\right)-2\pi\lambda_{1}\int_ {D}^{\infty}r\exp\left(-\pi\left(\lambda_{1}r^{2}+\lambda_{2}\bigg{(}\frac{P_ {2}}{P_{1}}\bigg{)}^{2/\alpha_{2}}r^{2\alpha_{1}/\alpha_{2}}\right)\right)dr\]  
Following from the law of total probability, the probability \(\mathcal{Q}_{C}\) is eventually given by Equation (6).