## 4 Joint Parameters Optimization
### _Initialization_
\begin{array}{l}P\left(Ar_{2}<r_{1}<Br_{2}\right)=P\left(Ar_{2}<r_{1}<Br_{2},r_ {2}\right)\\ =\int_{0}^{\infty}P\left(Ar_{2}<r_{1}<Br_{2}\right)f_{r_{2}}\left(r\right)dr\\ =\int_{0}^{\infty}\left[\int_{Ar}^{Br}2\pi\lambda_{1}r\exp\left(-\pi\lambda_{1 }r^{2}\right)dr\right]f_{r_{2}}\left(r\right)dr\\ =\int_{0}^{\infty}\left[\exp\left(-\lambda_{1}\tau A^{2}r^{2}\right)-\exp \left(-\lambda_{1}\pi B^{2}r^{2}\right)\right]\\ \quad\quad\cdot\ 2\pi\lambda_{2}r\exp\left(-\pi\lambda_{2}r^{2}\right)dr\\ =\frac{\lambda_{2}}{\lambda_{2}+A^{2}\lambda_{1}}-\frac{\lambda_{2}}{\lambda_ {2}+B^{2}\lambda_{1}},\end{array}  \tag{10}\]

where \(A\) and \(B\) denote the arbitrary value coefficients and the subscript 1 and 2 denote the tier 1 and tier 2, respectively. In the same way, we can get the expressions of \(A_{m_{u}}\), \(A_{p_{c}}\) and \(A_{p_{e}}\) combining Equations (2) and (10). ## Appendix B

Proof of Lemma 2.: To get the PDF of the distance between a typical user and its serving BS, the user type of this typical UE has to be identified firstly. Then, this issue can be converted to solving a conditional probability problem, which can be solved by Bayes rule. Assuming that the distance between the typical user and its serving BS is \(r_{l}\), then the PDF of it according to its user type can be expressed as:

\[ \begin{array}{l}f_{m_{u}}\left(r_{l}\right)=f_{m_{u}}\left(r_{m}\left|P_{m} hr_{m}^{-\alpha}>B_{p}P_{p}hr_{p}^{-\alpha}\right.\right),\\ f_{p_{c}}\left(r_{l}\right)=f_{p_{c}}\left(r_{p}\left|P_{p}hr_{p}^{-\alpha}>P_ {m}hr_{m}^{-\alpha}\right.\right),\\ f_{p_{e}}\left(r_{l}\right)=f_{p_{e}}\left(r_{p}\left|P_{p}hr_{p}^{-\alpha}<P_ {m}hr_{m}^{-\alpha}<B_{p}P_{p}hr_{p}^{-\alpha}\right.\right),\end{array}  \tag{11}\]

where \(f_{m_{u}}\left(r_{l}\right)\), \(f_{p_{c}}\left(r_{l}\right)\) and \(f_{p_{e}}\left(r_{l}\right)\) have similar expressions. If we can deduce the results of \(f_{p_{e}}\left(r_{l}\right)\), then \(f_{m_{u}}\left(r_{l}\right)\) and \(f_{p_{c}}\left(r_{l}\right)\) can be obtained in the same way. Referring to the expression of \(f_{p_{e}}\left(r_{l}\right)\), we first calculate the following similar conditional probability:

\[P\left[r_{2}\left|Ar_{2}<r_{1}<Br_{2}\right.\right]=\frac{P\left[r_{2}<r,Ar_{2} <r_{1}<Br_{2}\right]}{P\left[Ar_{2}<r_{1}<Br_{2}\right],} \tag{12}\]

where the subscripts 1 and 2 denote tier 1 and tier 2 of HetNets, respectively, and \(A\) and \(B\) represent the arbitrary value coefficients. \(P\left[Ar_{2}<r_{1}<Br_{2}\right]\) has been obtained in Equation (10) and is independent of \(r\). Therefore, the PDF of distance \(r_{2}\) can be obtained by the derivation of the CDF function without considering \(P\left[Ar_{2}<r_{1}<Br_{2}\right]\). Substituting \(C\) for \(P\left[Ar_{2}<r_{1}<Br_{2}\right]\), the PDF of distance \(r_{2}\) can be expressed as:

\[ \begin{array}{l}f_{r_{2}}\left(r\right)=\frac{d}{dr}P\left[r_{2}\left|Ar_{2}< r_{1}<Br_{2}\right.\right]\\ =\frac{1}{C}\frac{d}{dr}P\left[r_{2}<r,Ar_{2}<r_{1}<Br_{2}\right]\\ =\frac{2\pi\lambda_{2}}{C}\left\{\exp\left[-\pi r^{2}\left(\lambda_{2}+A \lambda_{1}\right)\right]\right.\\ \quad\quad-\exp\left[-\pi r^{2}\left(\lambda_{2}+B\lambda_{1}\right)\right] \right\}.\end{array}  \tag{13}\]

Similarly, we can also get the expression of \(f_{m_{u}}\left(r_{l}\right)\), \(f_{p_{c}}\left(r_{l}\right)\) and \(f_{p_{e}}\left(r_{l}\right)\). ## Appendix C

Proof of Lemma 3.: On the basis of the expression of the SINR in Equation (1), \(W_{l}\) and \(N_{l}\) can be seen as constants. \(f_{l}\left(r\right)\) can be obtained according to Equation (A7). Thus, we can deduce the expression of the average user rate of user type \(l\) as:

\[ \begin{array}{l}R_{l}=\frac{W_{l}}{N_{l}}E\left[\log_{2}\left(1+SINR_{l} \right)\right]\\ =\frac{W_{l}}{N_{l}}\int_{0}^{\infty}E\left[\log_{2}\left(1+\frac{p_{l}n_{l} ^{-a}}{\sum_{k=1}^{2}l_{l,k}+\sigma^{2}}\right)\right]f_{l}\left(r\right)dr \end{array} \big{]},\] (A8)

where

\[