# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## 3 Analysis of the Coverage Probability and Average Rate
### The Extension to the Multi-Tier Network  
The performance analysis for our proposed scheme in the previous sections can be generalized to a network with more than two tiers. In the \(K\)-tier (\(K>2\)) setup, the small cells within the inner region of each MBS will all be deactivated. Similarly to Equations (1) and (2), a user \(u\in U_{k}\) can be divided into two disjoint sets for their serving tier \(k>1\):  
\[u\in\left\{ \begin{array}{l}U_{NC_{k}}\,\ \mathrm{if}\ R_{1}>D\ \&\ R_{R}R_{k}^{-\alpha_{k}}/P_{i}R_{i}^{- \alpha_{i}}\ \geqslant\ \beta_{i},\ \forall i\
eq k\\ U_{C_{k}}\,\ \mathrm{if}\ R_{1}>D\ \&\ 1<\beta_{i}P_{i}R_{i}^{-\alpha_{i}}/P_{k}R_{k}^{ -\alpha_{k}}\ <\ \beta_{k},\forall i\
eq k\end{array} \right.\]  
Hence, the users in tier \(k\) can be classified into the non-CoMP users \(U_{NC_{k}}\) and the CoMP users \(U_{C_{k}}\). The set \(U_{k}\triangleq U_{NC_{k}}\cup U_{C_{k}}\) is the set of users in tier \(k\), where \(U_{C_{k}}=\bigcup_{j=1,j\
eq k}^{K}U_{C_{kj}}\). The non-CoMP users are directly served by the nearest BS in tier \(k\), while the CoMP users are jointly served by the satisfactory BSs from two tiers. Together with the nearest BS in tier \(k\), the offloaded BS from tier \(j\) (\(j\
eq k\)) can cooperate to transmit the data for the offloaded users, in order to tackle the dominant interference from their original tier. More precisely, the cooperative BS set \(\mathcal{C}_{k}=\bigcup_{j=1,j\
eq k}^{K}\mathcal{C}_{kj}\) is defined as:  
\[\mathcal{C}_{kj}=\left\{BS_{k,0}\cup BS_{j,0}\left|R_{1}>D\&\beta_{i}P_{i}R_{i} ^{-\alpha_{i}}/P_{k}R_{k}^{-\alpha_{k}}<\beta_{k},\forall i\
eq k\&1<\beta_{j} P_{j}R_{j}^{-\alpha_{j}}/P_{k}R_{k}^{-\alpha_{k}},j\
eq k\right.\right\}\]  
**Lemma 3**. : _Let \(\mathcal{Q}_{NC_{k}}\) and \(\mathcal{Q}_{C_{kj}}\) denote the probabilities that a typical user belonging to tier \(k\) operates in the non-CoMP mode and in the CoMP mode, respectively. Then,_  
\[\mathcal{Q}_{NC_{k}}=\left[\exp\left(-\pi\lambda_{1}D^{2}\right)-2\pi\lambda _{1}\int_{D}^{\infty}r\exp\left(-\pi\left(\lambda_{1}r^{2}+\lambda_{k}\left( \frac{\mathbb{P}_{k}}{\beta_{1}P_{1}}\right)^{2/\alpha_{k}}r^{2\alpha_{1}/ \alpha_{k}}\right)\right)dr\right]\ \times \tag{22}\]  
\[\mathcal{Q}_{C_{kj}}=\left\{ \begin{array}{l}\left[2\pi\lambda_{1}\int_{D}^{ \infty}r\exp\left(-\pi\lambda_{1}r^{2}\right)\left[\exp\left(-\pi\lambda_{k} \left(\frac{\mathbb{P}_{k}}{\beta_{1}P_{1}}\right)^{2/\alpha_{k}}r^{2\alpha_{ 1}/\alpha_{k}}\right)-\exp\left(-\pi\lambda_{k}\left(\frac{\mathbb{P}_{k}P_{ 1}}{\beta_{1}P_{1}}\right)^{2/\alpha_{k}}r^{2\alpha_{1}/\alpha_{k}}\right) \right]dr\right]\times\\ \left[2\pi\lambda_{1}\int_{0}^{\infty}r\exp\left(-\pi\lambda_{1}r^{2}\right) \left[1-\exp\left(-\pi\lambda_{k}\left(\frac{\mathbb{P}_{k}P_{1}}{\beta_{1}P_{ 1}}\right)^{2/\alpha_{k}}r^{2\alpha_{1}/\alpha_{k}}\right)\right]dr\right] \times\\ \left[2\pi\lambda_{j}\int_{0}^{\infty}r\exp\left(-\pi\lambda_{1}r^{2}\right) \left[\exp\left(-\pi\lambda_{k}\left(\frac{\mathbb{P}_{k}P_{1}}{\beta_{1}P_{ 1}}\right)^{2/\alpha_{k}}r^{2\alpha_{1}/\alpha_{k}}\right)-\exp\left(-\pi \lambda_{k}\left(\frac{\mathbb{P}_{k}P_{1}}{\beta_{1}P_{1}}\right)^{2/\alpha_{ k}}r^{2\alpha_{1}/\alpha_{k}}\right)\right]dr\right]\times\\ \left[\underset{i>1,j\
eq k}{\mathbb{P}_{j}}\int_{0}^{\infty}2\pi\lambda_{i}r \exp\left(-\pi\lambda_{i}r^{2}\right)\left[1-\exp\left(-\pi\lambda_{k}\left( \frac{\mathbb{P}_{i}P_{1}}{\beta_{1}P_{1}}\right)^{2/\alpha_{k}}r^{2\alpha_{1}/ \alpha_{k}}\right)\right]dr\right]forj\
eq 1\end{array} \right. \tag{23}\]  
Proof. : See Appendix D.  
**Lemma 4**. : _Let \(X_{NC_{k}}\) and \(\mathbf{X}_{C_{kj}}=\left(X_{j},X_{k}\right)\) denote the distances between the typical user and their serving BSs. Then, the PDFs of \(X_{NC_{k}}\) and \(\mathbf{X}_{C_{kj}}\) are:_  
\[f_{X_{NC_{k}}}\left(x\right)\approx\left\{