## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor
10:endwhile
11:\(\hat{\rho}^{*}=\rho=0\), \(B_{p}=\hat{B}_{p}^{*}\). 12:while\(\rho<1\)do
13: Substituting \(\rho\) into Equation (14) with given \(\lambda_{u}\), \(\lambda_{m}\), \(\lambda_{p,m}\) and \(B_{p}\), the current network energy efficiency \(EE^{\prime}\) can be obtained as: \(EE^{\prime}=EE\left(\rho\right)|\lambda_{u},\lambda_{m},\lambda_{p,m},B_{p}\). 1

### Optimization of PBS Density

Similarly, suppose that \(B_{p}\), \(\rho\) and \(\lambda_{u}\) are known. Thus, the PBS density optimization problem can be formulated as follow:

\[0< B_{p}\leq 25\text{ dB}\]4 s.t. \[0< B_{p}\leq 25\text{ dB}\]5 \[0< B_{p}\leq 25\text{ dB}\]6 \[0< B_{p}\leq 25\text{ dB}\]7

Assume that the step length of \(\lambda_{p,m}\) is \(\Delta_{3}\), which is set to be 0.03. Thus, the optimal PBS density can be obtained by a linear search algorithm to maximize the network EE, which is described in Algorithm 2. Line 1 of Algorithm 2 indicates the network scenario and parameters initialization. From line 2 to line 8, the optimal PBS density is acquired by a linear search method to maximize the network \(EE\). ```
1:Initialization: Initialize the network scenario and the values \(\lambda_{u}\), \(\lambda_{m}\), \(B_{p}\) and \(\rho\), where \(B_{p}\in(0,25]\) and \(\rho\in[0,1)\). Set \(\Delta_{3}=0.03\), \(\lambda_{p,m}=0.3\), \(\lambda_{p}^{*}=\lambda_{p,m}\lambda_{m}\) and \(EE^{*}=0\). 2:while\(\lambda_{p,m}\leq 30\)do
3: Substituting \(\lambda_{p,m}\) into Equation (14) with given \(\lambda_{u}\), \(\lambda_{m}\), \(B_{p}\) and \(\rho\), the current network energy efficiency \(EE^{\prime}\) can be obtained as: \(EE^{\prime}=EE\left(\lambda_{p,m}\right)|\lambda_{u},\lambda_{m},B_{p},\rho\). 4:if\(EE^{\prime}>EE^{*}\)then
5:\(\lambda_{p}^{*}=\lambda_{p,m}\lambda_{m}\), \(EE^{*}=EE^{\prime}\). 6:endif
7:\(\lambda_{p,m}=\lambda_{p,m}+\Delta_{3}\). 8:endwhile
```

**Algorithm 2**Pico base stations (PBS) density optimization (PDO) algorithm. ### Joint Optimization of Pico CRE Bias, Power Reduction Factor and PBS Density

The FeICIC parameter optimization sub-problem and the PBS density optimization sub-problem are solved independently by the aforementioned optimization algorithms. Due to the fact that these variables are affected by each other, we further propose a heuristic pico CRE bias, power reduction factor and PBS density joint optimization algorithm to globally optimize network EE based on the joint pico CRE bias and power reduction factor optimization (JBPO) algorithm and PDO algorithm. The detailed procedure of our proposed heuristic algorithm is summarized in Algorithm 3. \(\varepsilon\) represents the positive tolerance value. \(N_{loop}\) is the iteration times of the algorithm. Line 1 of Algorithm 3 signifies the network scenario and parameters initialization. From line 3 to line 4, the JBPO algorithm is executed to obtain the current optimal pico CRE bias and power reduction factor at a given PBS density. From line 5 to line 6, the PDO algorithm is performed to acquire the current optimal PBS density based on the above obtained pico CRE bias and power reduction factor. From line 2 to line 9, the network EE is iteratively optimized until it cannot improve further within an arbitrary value \(\varepsilon\). As a result, the optimal pico CRE bias, power reduction factor and PBS density are obtained and the network EE is maximized. ### Computational Complexity

The computational complexity of the JBPO algorithm can be calculated as \(O\left(n_{B_{p}}+n_{\rho}\right)\), where \(n_{B_{p}}\) and \(n_{\rho}\) are the space sizes of the linear search for \(B_{p}\) and \(\rho\), respectively. The computational complexity of the PDO algorithm is \(O\left(n_{\lambda_{p}}\right)\), where \(n_{\lambda_{p}}\) is the space size of the linear search for \(\lambda_{p}\). Thus, the computational complexity of the JBPDO algorithm is \(O\left[\left(n_{B_{p}}+n_{\rho}+n_{\lambda_{p}}\right)\times N_{loop}\right]\).