## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor
Due to the fact that there does not an exist effective algorithm for solving Equation (15), we compare the computational complexity of our proposed JBPDO algorithm with that of a traversal algorithm. As for solving Equation (15), the optimal values of pico CRE bias, power reduction factor and PBS density can be obtained by a traversal way, i.e., traversal pico CRE bias, power reduction factor and PBS density optimization (TBPDO) algorithm, which refers to traversing all possible values of these three parameters to maximize the objective function in Equation (15). Thus, the computational complexity of TBPDO algorithm will be \(O\left(n_{B_{p}}\times n_{\rho}\times n_{\lambda_{p}}\right)\), which signifies that the computational complexity of our proposed JBPDO algorithm is reduced effectively. As for solving the objective function in Equation (16), the optimal values of pico CRE bias and power reduction factor can also be obtained by a traversal way, i.e., traversal pico CRE bias and power reduction factor optimization (TBPO) algorithm, which refers to traversing all possible values of these two parameters to maximize the objective function in Equation (16). Indeed, TBPO algorithm consists of two nested ergodic sub-processes: (1) traversal pico CRE bias optimization (TBO) algorithm, which is executed by traversing all pico CRE bias to maximize the objective function in Equation (19) under fixed power reduction factor and PBS density; (2) traversal power reduction factor optimization (TPO) algorithm, which is executed by traversing all power reduction factor to maximize the objective function in Equation (20) under fixed pico CRE bias and PBS density. Therefore, the computational complexities of the TBO and TPO algorithms are \(O\left(n_{B_{p}}\right)\) and \(O\left(n_{\rho}\right)\), respectively. As a result, the computational complexity of the TBPO algorithm will be \(O\left(n_{B_{p}}\times n_{\rho}\right)\), which is obviously higher than that of our proposed JBPO algorithm. ## 6 Numerical Results and Analysis

In this section, we not only provide theoretical simulation, but also verify the effectiveness of proposed heuristic algorithms by Monte Carlo simulation. In the theoretical simulation, we considerws a network coverage area within a square region of \(1000\text{ m}\times 1000\text{ m}\). The deployments of PBS andMBS follow the PPP model and the typical UE is deployed in the origin. The simulation parameters used in this paper are summarized in \[0< B_{p}\leq 25\text{ dB}\]8 We took the average results from 30 times of network implementations as Monte Carlo simulations results. In each network implementation, the locations of MBSs, PBSs and UEs were modeled as spatial PPP, respectively. Then, the network EEs was calculated for all the different combination of pico CRE bias, power reduction factor, and PBS density values within their value ranges based on wireless channel quality. Finally, the maximum network EE and the optimal pico CRE bias, power reduction factor and PBS density can be obtained by comparing the calculated network EEs under all combinations. Indeed, the results of Monte Carlo simulation, including the maximal network EE, the optimal pico CRE bias, the optimal power reduction factor and the optimal PBS density, were obtained by a traversal way in each network implementation. Meanwhile, the performances of algorithms shown in the following simulation figures were just simulated based on theoretical derived Equation (14). Therefore, Monte Carlo simulation results had the best performance and can be referred to as a baseline for valuing the performances of those algorithms. At first, the network EE performances of the JBPO algorithm were compared with those of the TPO algorithm, TBO algorithm, theoretical simulation and Monte Carlo simulation, as shown in Figures 6 and 7, respectively. As shown in Figure 6, with the increase of PBS density, network EE increased accordingly. With PBS density increase, more UEs were offloaded into the coverage of low power PBSs.