## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor
## Appendix C Proof of Lemma 3

Due to \(W_{l}\) and \(N_{l}\) can be seen as constant and \(f_{l}(r)\) can be obtained based on Equation (6), we can deduce the closed-form expression of the average achievable downlink rate of UE \(l\) as:

\[ \begin{split} R_{l}&=\frac{W_{l}}{N_{l}}E\left[\log_{ 2}\left(1+\gamma_{l}\right)\right]\\ &=\frac{W_{l}}{N_{l}}\int_{0}^{\infty}E\left[\log_{2}\left(1+ \frac{\rho_{l}P_{l}n_{l}^{-1}}{\rho_{l}^{\prime}I_{lm}+I_{p}+\rho^{2}}\right) \right]\!f_{l}\left(r\right)dr,\end{split} \] (A8)

where \(P_{l}\) is full transmission power of serving BS of user type \(l\), \(r_{l}\) represents the distance between UE \(l\) and its serving BS. \(\rho_{l}\) is determined as \(\rho_{l}=\rho\) when \(l\in\{pm\}\) and \(\rho_{l}^{\prime}\) is determined as \(\rho_{l}^{\prime}=\rho\) when \(l\in\{pm,pp\}\). Otherwise \(\rho_{l}={\rho_{l}}^{\prime}=1\). Considering the complexity of Equation (10), we can deduce \(E\left[\log_{2}\left(1+\gamma_{l}\right)\right]\) first as Equation (11). \[ \begin{split} E\left[\log_{2}\left(1+\frac{\rho_{l}P_{l}h_{l}^{- \alpha}}{{\rho_{l}^{\prime}}^{\prime}l_{m}+l_{p}+\sigma^{2}}\right)\right]& \stackrel{{(a)}}{{=}}\int_{0}^{\infty}\text{Prob} \left[\log_{2}\left(1+\frac{\rho_{l}P_{l}h_{l}^{-\alpha}}{{\rho_{l}^{\prime}} ^{\prime}l_{m}+l_{p}+\sigma^{2}}\right)>t\right]dt\\ &\stackrel{{(b)}}{{=}}\int_{0}^{\infty}\text{Prob} \left[h>\left(2^{t}-1\right)\left({\rho_{l}^{\prime}}^{\prime}l_{m}+l_{p}+ \sigma^{2}\right)\tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right]dt\\ &\stackrel{{(b)}}{{=}}\int_{0}^{\infty}\exp\left[- \left(2^{t}-1\right){\rho_{l}^{\prime}}^{\prime}l_{m}{\rho_{l}^{\prime}}^{ \alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right]\exp\left[-\left(2^{t}-1\right)I_{p} \tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right]\\ &\stackrel{{(c)}}{{=}}\int_{0}^{\infty}\mathcal{L}_{ l_{m}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{\prime}}{\rho_{l}^{-1}}P_{l}^{-1} \right)\mathcal{L}_{I_{p}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1 }\right)\\ &\stackrel{{(d)}}{{=}}\int_{0}^{\infty}\mathcal{L}_{ l_{m}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{\prime}}{\rho_{l}^{-1}}P_{l}^{-1} \right)\mathcal{L}_{I_{p}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1 }\right)\\ &\stackrel{{(e)}}{{=}}\int_{0}^{\infty}\left(\tau \tau_{l}^{\alpha}{\rho_{l}^{\prime}}^{\prime}{\rho_{l}^{-1}}P_{l}^{-1}\right) \mathcal{L}_{I_{p}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right) \\ &\stackrel{{(f)}}{{\cdot}}\exp\left[-\tau\sigma^{2}r_{l }^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right]dt,\end{split}  \tag{11}\]

where (a) is derived according to \(E\left(X\right)=\int_{0}^{\infty}P\left[X>x\right]dx\), (b) is obtained referring to \(h\sim\exp\left(1\right)\), (c) is obtained through the Laplace transform of \(I_{m}\) and \(I_{p}\) by setting \(\tau=2^{t}-1\). Referring to [6], the Laplace transform formulations can be expressed as:

\[\mathcal{L}_{I_{m}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}}^{\prime}{\rho_{l}^{-1 }}P_{l}^{-1}\right)=\exp\left\{-\pi\lambda_{m}\bigg{(}\frac{{\rho_{l}}^{\prime }P_{m}}{{\rho_{l}P_{l}}}\bigg{)}^{2/\alpha}r^{2}\tau^{2/\alpha}\int_{\left[B_{ m}/\left({\rho_{l}}^{\prime}{\rho_{l}^{-1}}B_{l}\right)\right]}^{2/\alpha}\frac{1}{1+x^{ \alpha/2}}dx\right\} \tag{12}\]

\[\mathcal{L}_{I_{p}}\left(\tau\tau_{l}^{\alpha}{\rho_{l}^{-1}}P_{l}^{-1}\right) =\exp\left\{-\pi\lambda_{p}\bigg{(}\frac{P_{p}}{{\rho_{l}P_{l}}} \bigg{)}^{2/\alpha}r^{2}\tau^{2/\alpha}\int_{\left[B_{p,l}/\left(B_{l}\right) \right]}^{\infty}\frac{1}{1+x^{\alpha/2}}dx\right\}, \tag{13}\]

where \(B_{l}\) denotes the association bias of the typical UE \(l\) to select its serving BS. When \(l=pp\), \(B_{l}=B_{p}\), otherwise \(B_{l}=1\). In order to simplify the analysis, we can define variable \(\beta=B_{k,l}/B_{l},k\in\{m,p\}\), where \(B_{m,l}\)=\(B_{m}/\left({\rho_{l}}^{\prime}{\rho_{l}^{-1}}\right)\) and \(B_{p,l}\)= \(\left\{ \begin{array}{l}1,\text{if }l=up\\ B_{p},\text{ otherwise }\end{array}