## 4 Joint Parameters Optimization
### _Initialization_
that JBPDO Algorithm can realize pico CRE bias and PBS density joint optimization, the performance of JBPDO Algorithm can always outperform CBO Algorithm and PDO Algorithm obviously with different user densities. The network EE performance gap is also given in Figure 7 between our proposed JBPDO Algorithm and Traverse Algorithm, which can also realize pico CRE bias and PBS density joint optimization with higher computational complexity. Simulation results show that the network EE performance of our proposed JBPDO Algorithm is just slightly worse than that of Traverse Algorithm. Referring to Figures 6 and 7 together, the performance of Traverse Algorithm outperforms JBPDO Algorithm slightly. However, the computational complexity of JBPDO Algorithm is much lower than that of Traverse Algorithm. The computational complexity of Traverse Algorithm will be \(O\left({{n_{p,cre}}\times{n_{p,density}}}\right)\) and that of JBPDO Algorithm will be \(O\left[{{\left({{n_{p,cre}}+{n_{p,density}}}\right)\times{N_{loop}}}}\right]\). The convergence of JBPDO Algorithm is simulated in Figure 8, which is averaged with 30 different UE densities. It is seen from Figure 8 that JBPDO Algorithm can converge after two iterations. Hence, although the network EE performance of JBPDO Algorithm is slightly smaller than that of Traverse Algorithm, the computational complexity of JBPDO is



## 6 Conclusions

In this paper, we analyze the network EE for a two-tier HetNets consisting of MBSs and PBSs by means of stochastic geometry theory, where CRE implemented on PBS and ABS based on an eICIC scheme is adopted by MBS for downlink interference mitigation to PBS CRE UEs. We first derive the closed-form expression of the network EE. Then, a linear search algorithm is adopted to optimize the pico CRE bias and PBS density, respectively. Finally, a heuristic based algorithm is proposed to optimize the pico CRE bias and PBS density jointly to achieve the network EE maximization. Extensive simulation results show the accuracy of our theory deduction and the effectiveness of our proposed optimization algorithms for the network EE optimization with reduced complexity. **Acknowledgments:** The authors would like to thank the National Natural Science Foundation of China under Grant Nos. 61501289, 61701293, 61420106011, 61671011, and 61673253 and the Shanghai Science and Technology Development Funds under Grant No. 17010500400. **Author Contributions:** [PERSON] conceived of the study and proposed the idea; [PERSON] designed the simulations and wrote the paper; [PERSON] and [PERSON] proposed the analysis model and corrected the manuscript; [PERSON] and [PERSON] participated in the coordination of the research, corrected the manuscript and searched for funding to support the work. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A

**Proof of Lemma 1. ** Considering a typical user, it is at a distance \(r\) away from its closest BS in the \(k\)th tier. Then, there is no BS closer than \(r\) in the \(k\)th tier. Due to the fact that the void probability of 2D PPP in an area of \(\Psi\) is \(e^{-\lambda_{k}\Psi}\), we can obtain

\[P\left(r_{k}>r\right)=P\left(no\ BS\ closer\ than\ r_{k}\right)= e^{-\lambda_{k}\pi r^{2}}, \tag{1}\]

where \(r_{k}\) denotes the distance of this typical user to the closest BS in the \(k\)th tier, and \(\lambda_{k}\) is the BS density of \(k\)th tier. Then, we can get the cumulative distribution function (CDF) of \(r_{k}\) as follows:

\[F_{r_{k}}\left(r\right)=1-\exp\left(-\lambda_{k}\pi r^{2}\right). \tag{2}\]

Hence, the probability density function (PDF) of the distance \(r_{k}\) is

\[f_{r_{k}}\left(r\right)=2\pi\lambda_{k}r\exp\left(-\pi\lambda_{k}r^{2}\right). \tag{3}\]

Figure 8: The network EE versus iteration times \(N_{l}oop\) of JBPDO Algorithm. Referring to the user type formular expressions in Equation (2), we deduce the probability of a similar expression \(Ar_{2}<r_{1}<Br_{2}\) as:

\[