# Energy Efficient Pico Cell Range Expansion and Density Joint Optimization for Heterogeneous Networks with eICIC

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Works

## 2. Network Model

## 3. Analytical Model

#### 3.1. User Type Probability

**Lemma**

**1.**

#### 3.2. Distribution of Serving BS Distance

**Lemma**

**2.**

#### 3.3. The Ratio of Almost Blank Subframe

#### 3.4. Average Ergodic Rate

**Lemma**

**3.**

**Corollary**

**1.**

#### 3.5. BS Power Consumption

#### 3.6. Network Energy Efficiency

## 4. Joint Parameters Optimization

#### 4.1. Optimization of Pico CRE Bias

Algorithm 1: CRE Bias Optimization (CBO) Algorithm. | |

1. Initialization:(1) Initialize the network scenario and the values ${\lambda}_{u}$, ${\lambda}_{m}$ and ${\rho}_{p,m}$, where ${\rho}_{p,m}\in \left(0,30\right]$. (2) Set the initial value of ${B}_{p}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.1$. (3) Denote $\kappa $ as the variable step length of ${B}_{p}$. Denote $E{E}^{*}$ as the optimal value of the network EE. Denote ${B}_{p}^{*}$ as the optimized CRE bias. Let ${B}_{p}^{*}={B}_{p}$, $E{E}^{*}=EE\left({B}_{p}\right)\left|{}_{{\lambda}_{p},{\rho}_{p,m}}\right.$ and $\kappa =0.1$. 2. Calculate the optimal pico CRE biaswhile ${B}_{p}\le 25$ do${B}_{p}={B}_{p}+\kappa $ $E{E}^{\prime}=EE\left({B}_{p}\right)\left|{}_{{\lambda}_{p},{\rho}_{p,m}}\right.$ according to Equation (14) if $E{E}^{\prime}>E{E}^{*}$ then${B}_{p}^{*}={B}_{p}$, $E{E}^{*}=E{E}^{\prime}$ end ifend while |

#### 4.2. Optimization of PBS Density

Algorithm 2: PBS Density Optimization (PDO) Algorithm. | |

1. Initialization:(1) Initialize the network scenario and the values ${\lambda}_{u}$, ${\lambda}_{m}$ and ${B}_{p}$, where ${B}_{p}\in \left(0,25\right]$. (2) Set the initial value of ${\rho}_{p,m}=0.03$. (3) Denote $\sigma $ as the variable step length of ${\rho}_{p,m}$. Denote $E{E}^{*}$ as the optimal value of the network EE. Denote ${\rho}_{p,m}^{*}$ as the optimized ratio between PBS density MBS density. Let ${\rho}_{p,m}^{*}={\rho}_{p,m}$, $E{E}^{*}=EE\left({\rho}_{p,m}\right)\left|{}_{{\lambda}_{u},{B}_{p}}\right.$ and $\sigma =0.03$. 2. Calculate the optimal ratio of PBS density to MBS densitywhile ${\rho}_{p,m}\le 30$ do${\rho}_{p,m}={\rho}_{p,m}+\sigma $ $E{E}^{\prime}=EE\left({\rho}_{p,m}\right)\left|{}_{{\lambda}_{u},{B}_{p}}\right.$ according to Equation (14) if $E{E}^{\prime}>E{E}^{*}$ then${\rho}_{p,m}^{*}={\rho}_{p,m}$, $E{E}^{*}=E{E}^{\prime}$ end ifend while3. Obtain the optimal PBS density${\lambda}_{p}^{*}={\lambda}_{m}{\rho}_{p,m}^{*}$ |

#### 4.3. Joint Optimization of Pico CRE Bias and PBS Density

Algorithm 3: Joint Pico CRE Bias and PBS Density Optimization (JBPDO) Algorithm. | |

1. Initialization:(1) Initialize the network scenario and the values ${\lambda}_{u}$, ${\lambda}_{m}$, ${B}_{p}$ and ${\rho}_{p,m}$, where ${B}_{p}\in \left(0,25\right]$ and ${\rho}_{p,m}\in \left(0,30\right]$. (2) Let $E{E}^{*}=0$ represent the initial optimal value of the network EE. Initialize algorithm iteration number ${N}_{loop}=0$. Given a tolerance $\epsilon >0$. (3) Denote ${B}_{p}^{*}$ as the optimized PBS CRE bias. Denote ${\rho}_{p,m}^{*}$ as the optimized ratio of PBS density to MBS density. Denote ${\lambda}_{p}^{*}$ as the optimized PBS density. 2. Calculate the suboptimal CRE bias according to CBO Algorithm${B}_{p}^{\mathit{\text{sub\_opt}}}=\underset{{B}_{p}}{argmaxEE}\left({B}_{p}\right)\left|{}_{\lambda u},{\rho}_{p,m}\right.$ ${B}_{p}={B}_{p}^{\mathit{\text{sub\_opt}}}$ 3. Calculate the suboptimal ratio of PBS density to MBS density according to PDO Algorithm$\underset{{\rho}_{p,m}}{{\rho}_{p,m}^{\mathit{\text{sub\_opt}}}=argmaxEE}\left({\rho}_{p,m}\right)\left|{}_{{\lambda}_{u},{B}_{p}^{\mathit{\text{sub\_opt}}}}\right.$ ${\rho}_{p,m}={\rho}_{p,m}^{sub\_opt}$ $E{E}^{\mathit{\text{sub\_opt}}}=EE({B}_{p}^{sub\_opt},{\rho}_{p,m}^{sub\_opt})$ 4. Termination of the loopif $\left|E{E}^{\mathit{\text{sub\_opt}}}-E{E}^{*}\right|>\epsilon $ then$E{E}^{*}=E{E}^{\mathit{\text{sub\_opt}}}$, ${N}_{loop}={N}_{loop}+1$, go to step 2 else${B}_{p}^{*}={B}_{p}$, ${\rho}_{p,m}^{*}={\rho}_{p,m}$ ${\lambda}_{p}^{*}={\lambda}_{m}{\rho}_{p,m}^{*}$, $E{E}^{*}=E{E}^{\mathit{\text{sub\_opt}}}$ end if |

## 5. Numerical Results and Analysis

#### 5.1. Performance Analysis for Pico CRE Bias Optimization

#### 5.2. Performance Analysis for PBS Density Optimization

#### 5.3. Performance Analysis for Joint Optimization of CRE Bias and PBS Density

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

## Appendix B

**Proof**

**of**

**Lemma**

**2.**

## Appendix C

**Proof**

**of**

**Lemma**

**3.**

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**Figure 5.**The network EE versus ${\lambda}_{u}$ with ${B}_{p}=5\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$.

Parameters | Value |
---|---|

Carrier frequency f | 2 GHz |

Path loss exponent $\alpha $ | 4 |

Path Loss L | $L=10log\left({L}_{0}\right)+\alpha 10log\left({r}_{l}\right)$, where ${L}_{0}={\left(4\pi f/\phantom{4\pi fc}\phantom{\rule{0.0pt}{0ex}}c\right)}^{2}$, $c=3\times {10}^{8}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$ |

MBS transmit power ${P}_{m}$ or ${P}_{m,t}$ | 43 dBm or 20 W |

PBS transmit power ${P}_{p}$ or ${P}_{p,t}$ | 30 dBm or 1 W |

Bandwidth W | 10 MHz |

MBS static power ${P}_{m,s}$ | 800 W |

PBS static power ${P}_{p,s}$ | 130 W |

MBS density ${\lambda}_{m}$ | $0.00003$ |

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## Share and Cite

**MDPI and ACS Style**

Sun, Y.; Xia, W.; Zhang, S.; Wu, Y.; Wang, T.; Fang, Y. Energy Efficient Pico Cell Range Expansion and Density Joint Optimization for Heterogeneous Networks with eICIC. *Sensors* **2018**, *18*, 762.
https://doi.org/10.3390/s18030762

**AMA Style**

Sun Y, Xia W, Zhang S, Wu Y, Wang T, Fang Y. Energy Efficient Pico Cell Range Expansion and Density Joint Optimization for Heterogeneous Networks with eICIC. *Sensors*. 2018; 18(3):762.
https://doi.org/10.3390/s18030762

**Chicago/Turabian Style**

Sun, Yanzan, Wenqing Xia, Shunqing Zhang, Yating Wu, Tao Wang, and Yong Fang. 2018. "Energy Efficient Pico Cell Range Expansion and Density Joint Optimization for Heterogeneous Networks with eICIC" *Sensors* 18, no. 3: 762.
https://doi.org/10.3390/s18030762