## 5 Joint Optimization of FeICIC Parameters and Base-Station Density
### Joint Optimization of Pico CRE Bias and Power Reduction Factor  
In order to get the optimal values of \(B_{p}\) and \(\rho\) for network EE maximization, suppose that \(\lambda_{p,m}\) and \(\lambda_{u}\) are given. Thus, the pico CRE bias and power reduction factor joint optimization problem can be formulated as follow:  
\[\rho^{*},B_{p}^{*}=\underset{\rho,B_{p}}{\arg\max}\ EE|\lambda_{p,m}\] (16) s.t. \[0< B_{p}\leq 25\text{ dB}\] \[0\leq\rho<1\] \[\lambda_{p,m}\text{ is an arbitrary constant between 0 and 30,}\]  
where \(B_{p}^{*}\) and \(\rho^{*}\) are the optimal values of pico CRE bias and power reduction factor, respectively. To simplify the solving process, we assume that the overall SINR of PSF MUEs is identical to that of the USF MUEs. Hence, the result of resource allocation will have a direct influence on the network _EE_. In view of user fairness, the optimal network EE can be achieved when the relationship of association probabilities of PSF MUEs and USF MUEs obey the Equation (17). \[\theta=\frac{A_{pm}}{A_{pm}+A_{um}}. \tag{17}\]  
Combining with Equation (7), we can get the approximate relation of \(\rho\) and \(B_{p}\) as:  
\[\rho^{\prime} = \frac{B_{p}P_{p}}{P_{m}\left(k_{e}^{2}/\ A_{pp}-\lambda_{m}/ \lambda_{p}\right)^{2}}, \tag{18}\]  
where \(\rho^{\prime}\) denotes the approximate near-optimal value of \(\rho\). The relationships between \(\rho\) and \(B_{p}\) under different \(\lambda_{p}\) are shown in Figure 5. We can find that \(\rho^{\prime}\) is a strictly increasing function with respect to \(B_{p}\) at a given PBS density. Figure 5: The near-optimal power reduction factor \(\rho^{\prime}\) versus \(B_{p}\) with fixed \(\lambda_{u}=0.0018\). Substituting \(\rho^{\prime}\) into Equation (14), we can get the near-optimal value of \(B_{p}\), denoted as \(\hat{B}_{p}^{*}\) by solving the following univariate problem. \[\hat{B}_{p}^{*}=\underset{B_{p}}{\arg}\max EE|\lambda_{p,m},\rho= \rho^{\prime}\] (19) s.t. \[0< B_{p}\leq 25\text{ dB}\] \[0< B_{p}\leq 25\text{ dB}\]0  
Then, combining the value of \(\hat{B}_{p}^{*}\) with Equation (14), the near-optimal value of \(\rho\), denoted as \(\hat{\rho}^{*}\) can be obtained by following univariate problem. \[0< B_{p}\leq 25\text{ dB}\]1 (20) s.t. \[0< B_{p}\leq 25\text{ dB}\]2 \[0< B_{p}\leq 25\text{ dB}\]3  
Thus, \(\hat{B}_{p}^{*}\) and \(\hat{\rho}^{*}\) are obtained by an alternating algorithm, which is shown in Algorithm 1. \(\Delta_{1}\) and \(\Delta_{2}\) represent the step lengths of \(B_{p}\) and \(\rho\), which are set to be 0.1 and 0.05, respectively. \(EE^{*}\) denotes the optimal value of the network EE. The main idea of this altering algorithm is to obtain the near-optimal values of pico CRE bias and power reduction factor by a two-step linear search approach on the basis of the approximate relationship between them. In line 1 of Algorithm 1, the network scenario and relative parameters are initialized. From line 2 to line 10, the optimal pico CRE bias is obtained by a linear search way on the basis of approximate relationship between pico CRE bias and power reduction factor. From line 11 to line 18, the optimal power reduction factor is calculated based on Equation (14). ```
1:Initialize the network scenario and the values \(\lambda_{u}\), \(\lambda_{m}\) and \(\lambda_{p,m}\), where \(\lambda_{p,m}\in(0,30]\). Set \(\Delta_{1}=0.5\) dB, \(\Delta_{2}=0.05\), \(\hat{B}_{p}^{*}=B_{p}=0.5\) dB and \(EE^{*}=0\). 2:while\(B_{p}\leq 25\) dB do
3: Substituting \(B_{p}\) into Equation (18), the approximate near-optimal value of the power reduction factor \(\rho\) can be obtained as \(\rho^{\prime}\). 4:\(\rho=\rho^{\prime}\). 5: Substituting \(B_{p}\) and \(\rho\) into Equation (14) with given \(\lambda_{u}\), \(\lambda_{m}\) and \(\lambda_{p,m}\), the current network energy efficiency \(EE^{\prime}\) can be obtained as: \(EE^{\prime}=EE\left(B_{p},\rho\right)|\lambda_{u},\lambda_{m},\lambda_{p,m}\). 6:if\(EE^{\prime}>EE^{*}\)then
7:\(\hat{B}_{p}^{*}=B_{p},EE^{*}=EE^{\prime}\). 8:endif
9:\(B_{p}=B_{p}+\Delta_{1}\).