# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. Array Gain Model

#### 2.2. Signal Model and Service Probability

**Remark**

**1.**

## 3. CF of the Aggregated Interference in LOS

#### 3.1. Uniform Cylindrical Arrays

**Remark**

**2.**

**Remark**

**3.**

#### 3.2. Specific Cases

#### 3.3. Analysis of AP Height

#### 3.4. Numerical Validation on Aggregated Interference

**Remark**

**4.**

**Remark**

**5.**

## 4. Statistical Characterization of the Aggregated Interference Power

## 5. Outage Analysis in the Presence of NLOS, Noise and Blockage

#### 5.1. NLOS Paths

#### 5.2. NLOS Paths

#### 5.3. Noise

#### 5.4. Blockage

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Singularity Point

## Appendix B. Derivative of Mean Aggregated Interference

## Appendix C. Proof of MRC

**h**is ${h}_{j}={\beta}_{j}/{r}_{j}^{b}$ for distance ${r}_{j}$, and the CF of the interference ${I}_{j}={\left|{\iota}_{j}\right|}^{2}$ is known ${\mathsf{\Psi}}_{\ell}\left(\omega \right)$. The MRC are designed to maximize the SIR $\mathsf{{\rm Y}}$, and thus the service probability ${P}_{MRC}\left(\mathbf{c}\right)$ where the instantaneous SIR is

## Appendix D. Multipath Interference

## References

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**Figure 1.**Configuration of a ${N}_{c}\times {N}_{v}$ uniform cylindrical array (UcylA) with line-of-sight (LOS) and non-line-of-sight (NLOS) links: (${\varphi}_{i,\ell}$,${\theta}_{i,\ell}$) are the azimuth and elevation angels; h is the height of the array; and the pointing directions are toward all the LOS ($\ell =1$) and NLOS ($\ell =2,3,\cdots $) links arriving from the user equipment (UE) over a propagation path ${R}_{i,\ell}$.

**Figure 2.**Service probability vs. radius ${R}_{o}$ for varying $\lambda =\{{10}^{-3},{10}^{-2}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, where the access point (AP) employs a UCA with ${N}_{c}=128$ or a UcylA with ${N}_{c}\times {N}_{v}=32\times 4$. Parameters: ${R}_{max}={R}_{max}^{\left(num\right)}=300$ m, unless mentioned otherwise; SIR threshold ${\left[T\right]}_{dB}=0$; normalized amplitude $|{\beta}_{o}|=1$, $2b=2.6$, $h=10$ m.

**Figure 3.**Service probability vs. radius ${R}_{o}$ for varying $\lambda =\{{10}^{-2},{10}^{-3}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, where the AP employs a UCA with ${N}_{c}=128$. Parameters: ${R}_{max}=200$ m, SIR threshold ${\left[T\right]}_{dB}=0$; normalized amplitude $|{\beta}_{o}|=1$; $2b=2.6$ and $2b=3.6$, $h=10$ m.

**Figure 4.**Received power and mean aggregated interference power vs. AP height h, for a target UE located at ${R}_{0}=20$ m equipped with a uniform circular array (UCA) with ${N}_{c}=128$ that has average power gain $\overline{{G}_{c}^{2}}=0.012$. Parameters: SIR threshold ${\left[T\right]}_{dB}=\{0,1,2,3,4,5\}$, interferer density $\lambda =5\times {10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, path-loss exponent $2b=2.4$, central frequency ${F}_{c}=28$ GHz, and transmit power ${\left[{P}_{tx}\right]}_{dBm}=20$, $|{\beta}_{o}|=\sqrt{{P}_{tx}}\phantom{\rule{0.166667em}{0ex}}4\pi {F}_{c}/c$, ${R}_{max}\to \infty $.

**Figure 5.**SIR vs. AP height h, for a target UE located at ${R}_{0}=\{20,30,40\}$ m equipped with a uniform circular array (UCA) with ${N}_{c}=128$ that has average power gain $\overline{{G}_{c}^{2}}=0.012$, for two sets of interfere density $\lambda =\{{10}^{-4},{10}^{-3}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$. Parameters: path-loss exponent $2b=2.4$, normalized amplitude $|{\beta}_{o}|=1$, ${R}_{max}\to \infty $.

**Figure 6.**Average number of users served vs. antenna ratio ${log}_{2}({N}_{c}/{N}_{v})$, within $\overline{R}=100\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ while keeping the total number of antennas $({N}_{c}{N}_{v}=256)$ constant for different SIR thresholds ${\left[T\right]}_{dB}=5$, AP height $h=5$ m, ${R}_{max}=400$ m, $|{\beta}_{0}{|}^{2}=1$, $\lambda =\{{10}^{-3},5\times {10}^{-2}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$.

**Figure 7.**Average number of users served vs. antenna ratio ${log}_{2}({N}_{c}/{N}_{v})$, within $\overline{R}=50$ m, while keeping constant the total number of antennas $({N}_{c}{N}_{v}=256)$ for different SIR thresholds ${\left[T\right]}_{dB}=\{0,5,10\}$, AP height $h=5$ m, ${R}_{max}=400$ m, normalized amplitude $|{\beta}_{0}|=1$, $\lambda =5\times {10}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$: solid lines correspond to $2b=2$ and dashed lines corresponding to $2b=3.6$.

**Figure 8.**Average number of the users served vs. antenna ratio ${log}_{2}({N}_{c}/{N}_{v})$, within $\overline{R}=100$ m while keeping constant the total number of antennas $({N}_{c}{N}_{v}=256)$ for different array height $h=\{5,10,15,20\}$ m, ${R}_{max}=400$ m, SIR threshold ${\left[T\right]}_{dB}=5$, normalized amplitude $|{\beta}_{0}|=1$, $\lambda =5\times {10}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$: Solid lines correspond to $2b=2$ and dashed lines corresponding to $2b=3.6$.

**Figure 9.**Average number of the users served vs. antenna ratio ${log}_{2}({N}_{c}/{N}_{v})$, within $\overline{R}=100$ m while keeping constant the total number of antennas $({N}_{c}{N}_{v}=256)$ for different interferer density $\lambda =\{1,5,10,20,50\}\times {10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, SIR threshold ${\left[T\right]}_{dB}=5$, AP height $h=5$ m, ${R}_{max}=400$ m, normalized amplitude $|{\beta}_{0}|=1$: Solid lines correspond to $2b=2$ and dashed lines corresponding to $2b=3.6$.

**Figure 10.**Real part of the ${\Xi (\alpha ,\omega )}^{\prime}$ vs. $\omega $ for $2b=2.6$ and $\lambda =1\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-2}$, with a single isotropic antenna for different heights of the array. The two guidelines are parallel to ${\omega}^{\alpha}$ and ${\omega}^{2}$. For large heights, ${\Xi}^{\prime}(\alpha ,\omega )$ vs. $\omega $ behaves as ${\omega}^{2}$, and for very small heights it behaves as ${\omega}^{\alpha}$, while for medium heights, it has two different slopes based on $\omega $.

**Figure 11.**Real part of the ${\Xi (\alpha ,\omega )}^{\prime}$ vs. $\omega $ for $2b=2.6$ and $\lambda =1\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, comparing the UCA with single antenna. The two guidelines are parallel with ${\omega}^{\alpha}$ and ${\omega}^{2}$: Solid lines correspond to a point antenna while dashed lines correspond to a UCA with ${N}_{c}=16$ isotropic antennas on a ring.

**Figure 12.**Comparison of the CDF of the aggregated interference power, within ${R}_{max}=400$ m using a UCA of size ${N}_{c}=16$ positioned at AP height $h=2$ m, for different interferer density $\lambda =\{{10}^{-3},{10}^{-2},{10}^{-1}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, path-loss exponent $2b=2.6$, transmit power ${\left[{P}_{tx}\right]}_{dBm}=20$ dBm, central frequency ${F}_{c}=28$ GHz and $|{\beta}_{o}|=\sqrt{{P}_{tx}}\phantom{\rule{0.166667em}{0ex}}4\pi {F}_{c}/c$, for three cases: (a) true CF model described in (19); (b) approximate mixture model described in (40); and (c) $\alpha $-stable distribution.

**Figure 13.**Comparison of the CDF of the aggregated interference power, within ${R}_{max}=400$ m using a UCA of size ${N}_{c}=16$ positioned at AP height $h=30$ m, for different interferer density $\lambda =\{1,5,10\}\times {10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, path-loss exponent $2b=2.6$, transmit power ${\left[{P}_{tx}\right]}_{dBm}=20$ dBm, central frequency ${F}_{c}=28$ GHz and $|{\beta}_{o}|=\sqrt{{P}_{tx}}\phantom{\rule{0.166667em}{0ex}}4\pi {F}_{c}/c$, for three cases: (a) true CF model described in (19); (b) approximate mixture model described in (40); and (c) $\alpha $-stable distribution.

**Figure 14.**NLOS model: every user has a few NLOS links in addition to an LOS link. LOS links are shown with thick red lines, and NLOS links are shown with dashed lines reflected back from the perimeter of a circle around the UE. Here, the $U{E}_{o}$ is the user of interest and $U{E}_{i}$ is an interferer.

**Figure 15.**Average users served ${M}_{s}$ within square area $200\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\times 200$ m vs. UCA height h, and solid lines are the no NLOS ($L=1$), while dashed lines with a marker are the with NLOS for $L=2,3,4$, with maximal ratio combining (MRC) and selection combining (SC) receivers for ${N}_{c}=500,\lambda ={10}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, $2b=2.6$, SIR threshold ${\left[T\right]}_{dB}=0$, ${R}_{max}=400$ m, the normalized amplitude $|{\beta}_{\ell}|=1$ for every path ℓ.

**Figure 16.**Average users served ${M}_{s}$ within square area $200\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\times 200$ m vs. UCA height h, solid lines is the no NLOS ($L=1$), while dashed lines with marker are with NLOS for $L=2,3,4$, with MRC and SC receivers for ${N}_{c}=500,\lambda ={10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$; $2b=2.6$, SIR threshold ${\left[T\right]}_{dB}=0$, ${R}_{max}=400$ m, the normalized amplitude $|{\beta}_{\ell}|=1$ for every path ℓ.

**Figure 17.**Average service probability vs. the probability or blockage of all the links, for $\lambda =\{5\times {10}^{-3},5\times {10}^{-2}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, and number of paths $L=\{1,4\}$, where $L=1$ means that only LOS link exists. Parameters: ${\left[{P}_{tx}\right]}_{dB}=20$ dB, $NF=7$ dB, $BW=400$ MHz, ${F}_{c}=28$ GHz, $|{\beta}_{\ell}|=\sqrt{{P}_{tx}}\phantom{\rule{0.166667em}{0ex}}4\pi {F}_{c}/c$ for every path ℓ, service area = $100\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\times 100$ m square, AP height $h=10$ m, signal-to-noise-and-interference (SINR) threshold ${\left[T\right]}_{dB}=0$, ${R}_{max}=400$ m.

**Figure 18.**Average service probability vs. the probability or blockage of all the links, for $\lambda =\{5\times {10}^{-3},5\times {10}^{-2}\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$, and number of paths $L=\{1,4\}$, where $L=1$ means that only LOS link exists. Parameters: ${\left[{P}_{tx}\right]}_{dB}=20$ dB, $NF=7$ dB, $BW=400$ MHz, ${F}_{c}=28$ GHz, $|{\beta}_{\ell}|=\sqrt{{P}_{tx}}\phantom{\rule{0.166667em}{0ex}}4\pi {F}_{c}/c$ for every path ℓ, service area = $100\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\times 100$ m square, AP height $h=10$ m, SINR threshold ${\left[T\right]}_{dB}=0$, ${R}_{max}=400$ m.

**Table 1.**Array configurations and reference to the characteristic function (CF) of aggregated interference $\mathsf{\Psi}\left(\omega \right)=\mathbb{E}\left[{e}^{j\omega I}\right]$.

UcylA | UCA $\mathit{h}>0$ | UCA $\mathit{h}=0$ | Isotropic $\mathit{h}>0$ | Isotropic $\mathit{h}=0$ | |
---|---|---|---|---|---|

${N}_{c}$ | >1 | >1 | >1 | 1 | 1 |

${N}_{v}$ | >1 | 1 | 1 | 1 | 1 |

h | ≥0 | ≥0 | 0 | ≥0 | 0 |

Section 3.1 CF: (15) | Section 3.2 CF: (19) | Section 3.2 CF: (23) | Section 3.2 CF: (25) | Ref. [11] |

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**MDPI and ACS Style**

Aghazadeh Ayoubi, R.; Spagnolini, U. Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry. *Mathematics* **2022**, *10*, 1156.
https://doi.org/10.3390/math10071156

**AMA Style**

Aghazadeh Ayoubi R, Spagnolini U. Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry. *Mathematics*. 2022; 10(7):1156.
https://doi.org/10.3390/math10071156

**Chicago/Turabian Style**

Aghazadeh Ayoubi, Reza, and Umberto Spagnolini. 2022. "Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry" *Mathematics* 10, no. 7: 1156.
https://doi.org/10.3390/math10071156