## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor
Finally, the network EE performances of the JBPDO algorithm were compared with those of the PDO algorithm, JBPO algorithm with fixed \(\lambda_{p}=10\lambda_{m}\), TBPDO algorithm, and Monte Carlo simulation in Figure 10. In the TBPDO algorithm, pico CRE bias, power reduction factor and PBS density are

Figure 8: The network EE versus \(\lambda_{p}\) with different \(\lambda_{u}\). Figure 9: The network EE versus \(\lambda_{u}\) with different \(\lambda_{p}\). jointly optimized to maximize network EE by an exhaustive traversal algorithm based on Equation (14). The Monte Carlo simulation results show the maximum network EE within the value range of pico CRE bias, power reduction factor and PBS density at different network load. As shown in Figure 10, the proposed JBPDO algorithm can obtain better network EE than that of the JBPO algorithm since the JBPDO algorithm further optimizes the PBS density on the basis of the JBPO algorithm. Meanwhile, the accuracy and effectiveness of our proposed JBPDO algorithm are once again verified by Monte Carlo simulation results. In addition, although the network EE of the TBPDO algorithm outperforms our proposed JBPDO algorithm, the computational complexity of TBPDO algorithm is \(O\left(n_{B_{p}}\times n_{\rho}\times n_{\lambda_{p}}\right)\), which is far higher than that of the JBPDO algorithm. The convergence of JBPDO algorithm is provided in Figure 11. From Figure 11, we can find that JBPDO algorithm can converge after three iterations. It is proved that the computational complexity of the JBPDO algorithm is much lower than that of the TBPDO algorithm and more suitable for the real-time network. Figure 11: The network EE versus \(N_{loop}\) for JBPDO algorithm. Figure 8.0 The network EE versus \(\lambda_{u}\) with different optimization algorithms. ## 7 Conclusions

In this paper, pico CRE bias, PSF power reduction factor and PBS density are jointly optimized to maximize network EE for a two-tier HetNets with FeICIC. First, we derive the closed-form expression of network EE based on stochastic geometry theory. Then, the near-optimal values of pico CRE bias and power reduction factor are obtained by an alternating algorithm based on the equivalence relation between them at a given PBS density deployment. With fixed pico CRE bias and power reduction factor, the PBS density is optimized by a linear search method. Finally, a heuristic algorithm is proposed to optimize the pico CRE bias, power reduction factor, and PBS density jointly for network EE maximization. Extensive simulation results show the accuracy of network EE theoretical deduction and the effectiveness of our proposed low-complexity heuristic algorithm for network EE improvement. [PERSON] conceived of the study and proposed the idea; [PERSON] designed the simulations and wrote the paper; [PERSON] and [PERSON] proposed analysis model and corrected the manuscript; [PERSON], [PERSON] and [PERSON] participated in the coordination of the research, corrected the manuscript and searched for funding to support the work. This work was supported by research grants from the National Natural Science Foundation of P.R. China (No. 61701293, 61671011, 61673253, 61420106011) and The National Key R&D Program of China (No. 2017 YFE0121400). The authors declare no conflict of interest. ## Appendix A Proof of Lemma 1

Considering a typical UE, it is at a distance \(r_{k}\) away from its nearest BS in the \(k\) tier, that is to say, there is no BS closer than \(r_{k}\) in the \(k\) tier. Due to the locations of MBSs and PBSs follow two iid. PPPs, the cumulative distribution function (CDF) of two-point distance \(r_{k}\) is

\[ \begin{split} F_{r_{k}}(r)&=1-\text{Prob}\left(r_{k }>r\right)\\ &=1-\text{Prob}\left(no\ BS\ closer\ than\ r_{k}\right)\\ &=1-\exp\left(-\pi r^{2}\lambda_{k}\right)\,,\end{split}  \tag{1}\]

where \(\lambda_{k}\) is the BS density of \(k\) tier.