## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor
We can obtain the PDF of the distance by the differential of Equation (1) over \(r\) as follows:

\[f_{r_{k}}(r)=2\pi r\lambda_{k}\exp\left(-\pi r^{2}\lambda_{k}\right). \tag{2}\]

Hence, given an arbitrary coefficient value of \(n\), the probability of \(r_{p}>nr_{m}\) can be given as:

\[ \begin{split}\text{Prob}\left(r_{p}>nr_{m}\right)& =\text{Prob}\left(no\ BS\ closer\ than\ nr_{m}|r_{m}\right)\\ &=\text{Prob}\left(r_{p}>nr_{m},r_{m}\right)\\ &=\int_{0}^{\infty}\text{Prob}\left(r_{p}>nr_{m},r_{m}\right)f_{ r_{m}}\left(r\right)dr\\ &=\int_{0}^{\infty}\left[\int_{nr}^{\infty}f_{r_{p}}\left(r \right)dr\right]f_{r_{m}}\left(r\right)dr\\ &=\int_{0}^{\infty}\left[\int_{nr}^{\infty}2\pi r\lambda_{p} \exp\left(-\pi r^{2}\lambda_{p}\right)dr\right]f_{r_{m}}\left(r\right)dr\\ &=\int_{0}^{\infty}\left[-\exp\left(-\pi r^{2}n^{2}\lambda_{p} \right)\right]2\pi r\lambda_{m}\exp\left(-\pi r^{2}\lambda_{m}\right)dr\\ &=\frac{\lambda_{m}}{\lambda_{m}+n^{2}\lambda_{p}}.\end{split}  \tag{3}\]

Further, given two arbitrary coefficient values of \(n_{a}\) and \(n_{b}\), the probability of \(n_{a}r_{m}<r_{p}<n_{b}r_{m}\) can be expressed as:

\[ \begin{split}\text{Prob}\left(n_{a}r_{m}<r_{p}<n_{b}r_{m}\right)& =\text{Prob}\left(r_{p}>n_{a}r_{m}\right)-\text{Prob}\left(r_{p}>n_{b}r_{m} \right)\\ &=\frac{\lambda_{m}}{\lambda_{m}+n^{2}\lambda_{p}}-\frac{\lambda_ {m}}{\lambda_{m}+n^{2}\lambda_{p}}.\end{split}  \tag{4}\]Combing Equation (A4) with Equation (2), the association probabilities of the typical UE can be obtained as Equation (5). ## Appendix B Proof of Lemma 2

According to the Bayes rule, we can obtain the conditional probability as

\[\text{Prob}\left(r_{m}|r_{p}>nr_{m}\right)=\frac{\text{Prob}\left(r_{m}<r,r_{p}>nr _{m}\right)}{\text{Prob}\left(r_{p}>nr_{m}\right)},\] (A5)

where \(\text{Prob}\left(r_{p}>nr_{m}\right)\) is given in Equation (A3). After taking partial derivation with respect to the variable \(r\), the PDF of distance \(r\) between a typical UE to its serving BS under the condition of \(r_{p}>nr_{m}\) is

\[ \begin{split} f_{r_{m}|r_{p}>nr_{m}}\left(r\right)& =\frac{d}{dr}\text{Prob}\left(r_{m}|r_{p}>nr_{m}\right)\\ &=\frac{1}{\text{Prob}\left(r_{p}>nr_{m}\right)}\frac{d}{dr} \text{Prob}\left(r_{m}<r,r_{p}>nr_{m}\right)\\ &=\frac{2\pi\lambda_{m}}{\text{Prob}\left(r_{p}>nr_{m}\right)} \frac{d}{dr}\int_{0}^{r}\text{Prob}\left(r_{p}>nr_{m}|r_{m}=r\right)f_{r_{m}} \left(r\right)dr\\ &=\frac{2\pi\lambda_{m}}{\text{Prob}\left(r_{p}>nr_{m}\right)} \frac{d}{dr}\int_{0}^{r}r\exp\left(-\pi r^{2}n^{2}\lambda_{p}\right)\exp\left( \pi r^{2}\lambda_{m}\right)dr\\ &=\frac{\lambda_{m}+n^{2}\lambda_{p}}{\text{Prob}\left(r_{p}>nr _{m}\right)}\frac{d}{dr}\left\{1-\exp\left[-\pi r^{2}\left(\lambda_{m}+n^{2} \lambda_{p}\right)\right]\right\}\\ &=\frac{2\pi r\lambda_{m}}{\text{Prob}\left(r_{p}>nr_{m}\right)} \exp\left[-\pi r^{2}\left(\lambda_{m}+n^{2}\lambda_{p}\right)\right].\end{split} \] (A6)

Further, we can obtain as:

\[ \begin{split} f_{r_{m}|n_{a}r_{m}<r_{p}<nr_{m}}\left(r\right)& =\frac{d}{dr}\text{Prob}\left(r_{m}|n_{a}r_{m}<r_{p}<n_{b}r_{m} \right)\\ &=\frac{d}{dr}\frac{\text{Prob}\left(r_{m}<r,n_{a}r_{m}<r_{p}<n_{ b}r_{m}\right)}{\text{Prob}\left(n_{a}r_{m}<r_{p}<n_{b}r_{m}\right)}\\ &=\frac{1}{\text{Prob}\left(n_{a}r_{m}<r_{p}<n_{b}r_{m}\right)} \frac{d}{dr}\left[\text{Prob}\left(r_{m}<r,r_{p}>n_{a}r_{m}\right)\right.\\ &\left.-\text{Prob}\left(r_{m}<r,r_{p}>n_{b}r_{m}\right)\right] \\ &=\frac{2\pi r\lambda_{m}}{\text{Prob}\left(n_{a}r_{m}<r_{p}<n_{ b}r_{m}\right)}\left\{\exp\left[-\pi r^{2}\left(\lambda_{m}+n_{a}^{2} \lambda_{p}\right)\right]\right.\\ &\left.-\exp\left[-\pi r^{2}\left(\lambda_{m}+n_{b}^{2}\lambda_{ p}\right)\right]\right\}.\end{split} \] (A7)

According to Equations (5) and (A7), we can get the PDFs of distance \(r\) between a typical UE \(l\) to its serving BS in Equation (6).