# Coverage and Energy Efficiency Analysis for Two-Tier Heterogeneous Cellular Networks Based on Matérn Hard-Core Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model

_{k}, respectively. We abbreviate the network based on this model as MHCP-MHCP. Moreover, mobile user equipment (MUEs) are modelled as PPP in ${\mathbb{R}}^{2}$ with density λ

_{u}. According to Slivnyak’s theorem in random geometry, the MUE located at the represented origin is selected as the typical MUE [38]. The path loss and fast fading are taken into account when modelling the channel gain, the path loss exponent is denoted as α and the fast fading experienced by the tagged UE and all the BSs are assumed to be Rayleigh fading. Assuming that the serving BS is located at point ${x}_{k}\in {\mathsf{\Phi}}_{k}^{MHCP}\left(k=1,2\right)$, then the received power at the tagged MUE can be written as ${P}_{Tk}{h}_{{x}_{k}}{\Vert {x}_{k}\Vert}^{-\alpha}$, where α > 2 and ${h}_{{x}_{k}}~\mathrm{exp}(1)$ stand for the path-loss exponent and the Rayleigh fading, respectively.

^{PPP}with intensity λ

_{PPP}and give each point x (x ∈ Φ

^{PPP}) an independent mark m(x). m(x) is a random variable uniformly distributed among [0, 1]. We find all the points that have neighbours within the exclusion radius and have smaller m (x), then flag these points. Last, we remove all flagged points. Formally, $\mathsf{\Phi}\triangleq \left\{x\in {\mathsf{\Phi}}^{PPP}:m(x)<m(y)forally\in {\mathsf{\Phi}}^{PPP}\cap b(x,r)\backslash \left\{x\right\}\right\}$. Where b(x,r) denotes a ball centred at x with radius r, Φ represents the set of all removed points. Following the steps above, the remaining points form a MHCP Φ

^{MHCP}. The BSs distribution modelled by PPP with density 10

^{−4}m

^{−2}is shown in Figure 1a. The relevant BSs distribution modelled by MHCP with repulsion radius 50 m is shown in Figure 1b.

^{MHCP}can be calculated as [9]

_{MHCP}represents the density of Φ

^{MHCP}, r represents the repulsion radius, and λ

_{PPP}is the density of Φ

^{PPP}before removing points.

## 3. The ASAPPP Approach for Single-Tier MHCP Network

_{PPP}represents the MISR for PPP-modelled network, and MISR

_{MHCP}represents the MISR for MHCP-modelled network. For a network with BSs located at Φ and the serving BS located at x

_{0}, the MISR at the typical user is defined as [13,14,15,16]

_{MHCP}can be obtained by simulation and data fitting. Then, we can get the value of G

_{MHCP}.

## 4. Coverage Probability of MHCP-MHCP Network

_{k}is the SIR threshold for the k-th tier. It has been proved that, in the PPP-modelled network, if β

_{k}> 1 (0 dB), the tagged MUE can only access one base station at most [4]. On the basis of these existing conclusions, the coverage probability for tagged MUE in the MHCP-MHCP network in open access can be derived as follows:

_{k}is the SIR scaling factor of the k-th tier based on the MHCP model. However, due to the mutual exclusion between BSs in the MHCP modelled network, such processing will lead to increased interference and reduce the coverage of the results. Thus, the interference from ${\mathsf{\Phi}}_{k}^{MHCP}$ is upper bounded by the interference from a PPP with the same density. So, the coverage provided by step (a) will be slightly lower than the actual coverage. Step (b) follows from the assumption that ${\beta}_{k}/{G}_{k}>1\forall k$ [4]. Step (c) follows from the Campbell theorem [44]. The derivation of Step (d) is based on the Rayleigh fading channel. Here ${\mathcal{L}}_{{I}_{{x}_{k}}}(\cdot )$ is the Laplace transform of the cumulative interference from all the tiers when the tagged MUE access to a BS in k-th tier. Due to the stability of the point process, the interference is independent of the location x

_{k}. Therefore, we denote ${\mathcal{L}}_{{I}_{{x}_{k}}}(\cdot )$ by ${\mathcal{L}}_{I}(\cdot )$. In the PPP-PPP network, ${\mathcal{L}}_{I}(s)$ is derived as [4]

_{i}represents the density of the BSs in k-th tier in PPP-PPP network. Since we use the ASAPPP method, each tier of the MHCP network is approximated to the PPP network with the same density by SIR threshold scaling. λ

_{k}is numerically equal to λ

_{k-MHCP}. Substituting (8) into (7), the coverage probability of a tagged MUE in the MHCP-MHCP network can be derived as

## 5. Energy Efficiency of the MHCP-MHCP Network

#### 5.1. Downlink Channel Capacity

#### 5.2. Total Power Consumption

## 6. Data Fitting and Simulation Analysis

_{MHCP}. Then, a series of numerical simulations are carried out to verify the accuracy of our derived coverage and energy efficiency expressions. In order to keep the derivation tractable, we do not consider the MUEs’ QoS requirements and we assume that each BS equally allocates the frequency resource among its associated MUEs. Furthermore, the downlink channel capacity is calculated based on the preset target SIRs, and the target SIRs for MBSs and PBSs are set to be the same in the simulation. For the power consumption model, we refer to the statistics proposed in [48]. The simulation parameters are shown in Table 1.

_{MHCP}and path loss. We use the fitting toolbox cftool in MATLAB to obtain the mathematical model of MISR about path loss as

_{MHCP}to calculate G

_{MHCP}according to Equation (4), and then substitute G

_{MHCP}into Equation (9) to obtain the approximate coverage of the two-tier MHCP-MHCP network. It should be noted that the fitting results in Table 2 are related to the repulsion radius.

^{6}to 1.3 × 10

^{6}bits/Joule higher than that of the PPP-PPP network under the same parameters, which means that the EE of the MHCP-MHCP network is much higher than that of the PPP-PPP network. This is due to the relative regularity of base station deployment in the MHCP-MHCP network, which greatly reduces the mutual interference between BSs. Compared with the PPP-PPP network, the MHCP-MHCP network can improve downlink minimum achievable data rates with the same power consumption. It can also be seen that EE shows a trend of rising first and then declining. This is because, with the increase of the transmit power of the PBS, more users choose to access PBS, so EE is on the rise. However, with the further increase of the transmit power of PBS, the interference between the BSs is also increased, which leads to the decline of the MUE’s downlink rate, so the EE is gradually reduced. Therefore, EE is a convex function of the transmit power of the PBS and there exists an optimal transmit power of PBS which can maximize the EE. Furthermore, it has been pointed out that EE can be improved by optimizing BS density [29]. We also compare EE of the MHCP-MHCP network with that of the PPP-PPP network which uses the optimization algorithms proposed in [29]. It can be seen from Figure 4 that even if the fixed BS density is adopted, the EE of the MHCP-MHCP network is still higher than that of the PPP-PPP network adopting the optimal BS density.

^{6}to 1.3 × 10

^{6}bits/Joule higher than that of the PPP-PPP network under the same parameters. Similar to Figure 4, the EE shows a trend of rising first and then declining, so it is a convex function of the transmit power of the MBS. Therefore, there exists an optimal transmit power of MBS which can maximize the EE.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Base stations distribution modelled by the Poisson point process (PPP) and Matérn hard-core process (MHCP).

Parameters Name | Values |
---|---|

Bandwidth | 10^{7} Hz |

MHCP-MHCP network area | 1000 × 1000 m^{2} |

density of MUs, λu | 0.15 m^{−2} |

density of MBSs, λ_{1-MHCP} | 2 * 10^{−5} m^{−2} |

density of PBSs, λ_{2-MHCP} | 2 * 10^{−4} m^{−2} |

repulsion radius of MBSs | 100 m |

repulsion radius of PBSs | 50 m |

path-loss exponent, α | [3, 5] |

circuit power consumption of MBS, P_{C1} | 130 W |

transmit power of MBS, P_{T1} | 20 W |

number of transmit antenna elements of MBS, N_{TR1} | 6 |

slope of power consumption depends on load of MBS, θ_{1} | 4.7 |

circuit power consumption of PBS, P_{C2} | 6.8 W |

transmit power of PBS, P_{T2} | 0.13 W |

number of transmit antenna elements of PBS, N_{TR2} | 2 |

slope of power consumption depends on load of PBS, θ_{2} | 4.0 |

Coefficients Name | a | b | c |
---|---|---|---|

1-th tier network (MBSs) | 5.917 (5.55, 6.285) | −1.551 (−1.641, −1.461) | 0.001449 (−0.04122, 0.04412) |

2-th tier network (PBSs) | 18.09 (16.82, 19.36) | −2.602 (−2.678, −2.527) | 0.1721 (0.1558, 0.1883) |

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**MDPI and ACS Style**

Yang, J.; Pan, Z.; Guo, L.
Coverage and Energy Efficiency Analysis for Two-Tier Heterogeneous Cellular Networks Based on Matérn Hard-Core Process. *Future Internet* **2020**, *12*, 1.
https://doi.org/10.3390/fi12010001

**AMA Style**

Yang J, Pan Z, Guo L.
Coverage and Energy Efficiency Analysis for Two-Tier Heterogeneous Cellular Networks Based on Matérn Hard-Core Process. *Future Internet*. 2020; 12(1):1.
https://doi.org/10.3390/fi12010001

**Chicago/Turabian Style**

Yang, Jie, Ziyu Pan, and Lihong Guo.
2020. "Coverage and Energy Efficiency Analysis for Two-Tier Heterogeneous Cellular Networks Based on Matérn Hard-Core Process" *Future Internet* 12, no. 1: 1.
https://doi.org/10.3390/fi12010001