5G-NR

    5G New Radio
3GPP

    3rd Generation Partnership Project
ABS

    aerial base station
AC

    address coding
ACF

    autocorrelation function
ACR

    autocorrelation receiver
ADC

    analog-to-digital converter
AIC

    Analog-to-Information Converter
AIC

    Akaike information criterion
ARIC

    asymmetric restricted isometry constant
ARIP

    asymmetric restricted isometry property
ARQ

    Automatic Repeat Request
AUB

    asymptotic union bound
AWGN

    Additive White Gaussian Noise
AWGN

    additive white Gaussian noise
VHeNET

    Vertical Heterogeneous Network
LAP

    Low altitude platform
PSK

    asymmetric PSK
AWRICs

    asymmetric weak restricted isometry constants
AWRIP

    asymmetric weak restricted isometry property
BCH

    Bose, Chaudhuri, and Hocquenghem
BCHSC

    BCH based source coding
BEP

    bit error probability
BFC

    block fading channel
BG

    Bernoulli-Gaussian
BGG

    Bernoulli-Generalized Gaussian
BPAM

    binary pulse amplitude modulation
BPDN

    Basis Pursuit Denoising
BPPM

    binary pulse position modulation
BPSK

    Binary Phase Shift Keying
BPZF

    bandpass zonal filter
BSC

    binary symmetric channels
BU

    Bernoulli-uniform
BER

    bit error rate
BS

    base station
BW

    BandWidth
BLLL

     binary log-linear learning
CP

    Cyclic Prefix
CDF

    cumulative distribution function
CDF

    Cumulative Distribution Function
CDF

    cumulative distribution function
CCDF

    complementary cumulative distribution function
CCDF

    complementary CDF
CCDF

    complementary cumulative distribution function
CD

    cooperative diversity
CDMA

    Code Division Multiple Access
ch.f.

    characteristic function
CIR

    channel impulse response
CoSaMP

    compressive sampling matching pursuit
CR

    cognitive radio
CS

    compressed sensing
CS

    Compressed sensing
CS

    compressed sensing
CSI

    channel state information
CCSDS

    consultative committee for space data systems
CC

    convolutional coding
COVID-19

    Coronavirus disease
DAA

    detect and avoid
DAB

    digital audio broadcasting
DCT

    discrete cosine transform
DFT

    discrete Fourier transform
DR

    distortion-rate
DS

    direct sequence
DS-SS

    direct-sequence spread-spectrum
DTR

    differential transmitted-reference
DVB-H

    digital video broadcasting – handheld
DVB-T

    digital video broadcasting – terrestrial
DL

    DownLink
DSSS

    Direct Sequence Spread Spectrum
DFT-s-OFDM

    Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiplexing
DAS

    Distributed Antenna System
DNA

    DeoxyriboNucleic Acid
EC

    European Commission
EED

    exact eigenvalues distribution
EIRP

    Equivalent Isotropically Radiated Power
ELP

    equivalent low-pass
eMBB

    Enhanced Mobile Broadband
EMF

    ElectroMagnetic Field
EU

    European union
EI

    Exposure Index
eICIC

    enhanced Inter-Cell Interference Coordination
FC

    fusion center
FCC

    Federal Communications Commission
FEC

    forward error correction
FFT

    fast Fourier transform
FH

    frequency-hopping
FH-SS

    frequency-hopping spread-spectrum
FS

    Frame synchronization
FS

    frame synchronization
FDMA

    Frequency Division Multiple Access
GA

    Gaussian approximation
GF

    Galois field
GG

    Generalized-Gaussian
GIC

    generalized information criterion
GLRT

    generalized likelihood ratio test
GPS

    Global Positioning System
GMSK

    Gaussian Minimum Shift Keying
GSMA

    Global System for Mobile communications Association
GS

    ground station
GMG

     Grid-connected MicroGeneration
HAP

    high altitude platform
HetNet

    Heterogeneous network
IDR

    information distortion-rate
IFFT

    inverse fast Fourier transform
IHT

    iterative hard thresholding
i.i.d.

    independent, identically distributed
IoT

    Internet of Things
IR

    impulse radio
LRIC

    lower restricted isometry constant
LRICt

    lower restricted isometry constant threshold
ISI

    intersymbol interference
ITU

    International Telecommunication Union
ICNIRP

    International Commission on Non-Ionizing Radiation Protection
IEEE

    Institute of Electrical and Electronics Engineers
ICES

    IEEE international committee on electromagnetic safety
IEC

    International Electrotechnical Commission
IARC

    International Agency on Research on Cancer
IS-95

    Interim Standard 95
KPI

    Key Performance Indicator
LEO

    low Earth orbit
LF

    likelihood function
LLF

    log-likelihood function
LLR

    log-likelihood ratio
LLRT

    log-likelihood ratio test
LoS

    line-of-sight
LRT

    likelihood ratio test
LWRIC

    lower weak restricted isometry constant
LWRICt

    LWRIC threshold
LPWAN

    Low Power Wide Area Network
LoRaWAN

    Low power long Range Wide Area Network
NLoS

    Non-Line-of-Sight
Li-Fi

    light-fidelity
LED

    light emitting diode
LABS

    LoS transmission with each ABS
NLABS

    NLoS transmission with each ABS
MB

    multiband
MC

    macro cell
MDS

    mixed distributed source
MF

    matched filter
m.g.f.

    moment generating function
MI

    mutual information
MIMO

    Multiple-Input Multiple-Output
MISO

    multiple-input single-output
MJSO

    maximum joint support cardinality
ML

    maximum likelihood
MMSE

    minimum mean-square error
MMV

    multiple measurement vectors
MOS

    model order selection
${M}$-PSK

     $M$-ary phase shift keying
${M}$-PSK

     $M$-ary asymmetric PSK
MP

     multi-period
MINLP

    mixed integer non-linear programming
$M$-QAM

     $M$-ary quadrature amplitude modulation
MRC

    maximal ratio combiner
MSO

    maximum sparsity order
M2M

    Machine-to-Machine
MUI

    multi-user interference
mMTC

    massive Machine Type Communications
mm-Wave

    millimeter-wave
MP

    mobile phone
MPE

    maximum permissible exposure
MAC

    media access control
NB

    narrowband
NBI

    narrowband interference
NLA

    nonlinear sparse approximation
NLOS

    Non-Line of Sight
NTIA

    National Telecommunications and Information Administration
NTP

    National Toxicology Program
NHS

    National Health Service
LOS

    Line of Sight
OC

    optimum combining
OC

    optimum combining
ODE

    operational distortion-energy
ODR

    operational distortion-rate
OFDM

    Orthogonal Frequency-Division Multiplexing
OMP

    orthogonal matching pursuit
OSMP

    orthogonal subspace matching pursuit
OQAM

    offset quadrature amplitude modulation
OQPSK

    offset QPSK
OFDMA

    Orthogonal Frequency-division Multiple Access
OPEX

    Operating Expenditures
OQPSK/PM

    OQPSK with phase modulation
PAM

    pulse amplitude modulation
PAR

    peak-to-average ratio
PDF

    probability density function
PDF

    probability density function
PDF

    probability distribution function
PDP

    power dispersion profile
PMF

    probability mass function
PMF

    probability mass function
PN

    pseudo-noise
PPM

    pulse position modulation
PRake

    Partial Rake
PSD

    power spectral density
PSEP

    pairwise synchronization error probability
PSK

    phase shift keying
PD

    power density
$8$-PSK

     $8$-phase shift keying
PPP

    Poisson point process
PCP

    Poisson cluster process
FSK

    Frequency Shift Keying
QAM

    Quadrature Amplitude Modulation
QPSK

    Quadrature Phase Shift Keying
OQPSK/PM

    OQPSK with phase modulator
RD

    raw data
RDL

    ”random data limit”
RIC

    restricted isometry constant
RICt

    restricted isometry constant threshold
RIP

    restricted isometry property
ROC

    receiver operating characteristic
RQ

    Raleigh quotient
RS

    Reed-Solomon
RSSC

    RS based source coding
r.v.

    random variable
R.V.

    random vector
RMS

    root mean square
RFR

    radiofrequency radiation
RIS

    Reconfigurable Intelligent Surface
RNA

    RiboNucleic Acid
RRM

    Radio Resource Management
RUE

    reference user equipments
RAT

    radio access technology
RB

    resource block
SINR

    Signal-to-interference-plus-noise ratio
SG

    stochastic geometry
SA-Music

    subspace-augmented MUSIC with OSMP
SC

    small cell
SCBSES

    Source Compression Based Syndrome Encoding Scheme
SCM

    sample covariance matrix
SEP

    symbol error probability
SIMO

    single-input multiple-output
SINR

    signal-to-interference plus noise ratio
SIR

    signal-to-interference ratio
SISO

    Single-Input Single-Output
SMV

    single measurement vector
SNR

    signal-to-noise ratio
SP

    subspace pursuit
SS

    spread spectrum
SW

    sync word
SAR

    specific absorption rate
SSB

    synchronization signal block
SR

    shrink and realign
LOS

    line of sight
tUAV

    tethered Unmanned Aerial Vehicle
TBS

    terrestrial base station
uUAV

    untethered Unmanned Aerial Vehicle
PDF

    probability density functions
PL

    path-loss
TH

    time-hopping
ToA

    time-of-arrival
TR

    transmitted-reference
TW

    Tracy-Widom
TWDT

    TW Distribution Tail
TCM

    trellis coded modulation
TDD

    Time-Division Duplexing
TDMA

    Time Division Multiple Access
Tx

    average transmit
UAV

    unmanned aerial vehicle
URIC

    upper restricted isometry constant
URICt

    upper restricted isometry constant threshold
UWB

    ultrawide band
UWB

    Ultrawide band
URLLC

    Ultra Reliable Low Latency Communications
UWRIC

    upper weak restricted isometry constant
UWRICt

    UWRIC threshold
UE

    User Equipment
UL

    UpLink
WiM

    weigh-in-motion
WLAN

    wireless local area network
WM

    Wishart matrix
WMAN

    wireless metropolitan area network
WPAN

    wireless personal area network
WRIC

    weak restricted isometry constant
WRICt

    weak restricted isometry constant thresholds
WRIP

    weak restricted isometry property
WSN

    wireless sensor network
WSS

    Wide-Sense Stationary
WHO

    World Health Organization
Wi-Fi

    Wireless Fidelity
SpaSoSEnc

    sparse source syndrome encoding
VLC

    Visible Light Communication
VPN

    Virtual Private Network
RF

    Radio Frequency
FSO

    Free Space Optics
IoST

    Internet of Space Things
GSM

    Global System for Mobile Communications
2G

    Second-generation cellular network
3G

    Third-generation cellular network
4G

    Fourth-generation cellular network
5G

    Fifth-generation cellular network
gNB

    next-generation Node-B Base Station
NR

    New Radio
UMTS

    Universal Mobile Telecommunications Service
LTE

    Long Term Evolution
QoS

    Quality of Service

# System-Level Metrics for Non-Terrestrial Networks Under Stochastic Geometry
Framework

Qi Huang, Student Member, IEEE, Baha Eddine Youcef Belmekki, Ahmed M. Eltawil,
Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE The authors are
with King Abdullah University of Science and Technology (KAUST), CEMSE
division, Thuwal 23955-6900, Saudi Arabia (e-mail: {qi.huang,
bahaeddine.belmekki<EMAIL_ADDRESS>
###### Abstract

Non-terrestrial networks (NTNs) are considered one of the key enablers in
sixth-generation (6G) wireless networks; and with their rapid growth, system-
level metrics analysis adds crucial understanding into NTN system performance.
Applying stochastic geometry (SG) as a system-level analysis tool in the
context of NTN offers novel insights into the network tradeoffs. In this
paper, we study and highlight NTN common system-level metrics from three
perspectives: NTN platform types, typical communication issues, and
application scenarios. In addition to summarizing existing research, we study
the best-suited SG models for different platforms and system-level metrics
which have not been well studied in the literature. In addition, we showcase
NTN-dominated prospective application scenarios. Finally, we carry out a
performance analysis of system-level metrics for these applications based on
SG models.

##  I Introduction

During the last few years, the world has witnessed exponential growth in
network services, especially with new use cases of sixth-generation (6G)
wireless networks such as virtual reality and holographic communications. Non-
terrestrial networks (NTNs) have shown excellent promise as an enabling
technology for new wireless communication generations. Compared with the
densely deployed terrestrial networks, NTNs are still under development, with
many challenges that must be addressed and tackled such as the
interconnectivity between different types of airborne platforms and
terrestrial networks, and among themselves [1].

There are mainly three types of NTN platforms: Low altitude platform (LAP),
high altitude platform (HAP), and satellites. These platforms provide coverage
for areas that the terrestrial network cannot reach. They can also enhance
global connectivity with low latency and long-distance reach. Moreover, NTNs
benefit from a suitable communication environment where the signal is
propagated mostly through free space since it does not experience severe
shadowed fading or multi-path fading. NTN platforms will be part of the 6G
architecture networks. Therefore, system-level analysis, such as global
connectivity, becomes more significant than the performance analysis of a
single link. Unlike metrics that highlight the flying attitude and deployment
of the platform, system-level metrics focus on the performance of the entire
NTN system. The system-level metrics mentioned in this paper include coverage
probability, availability, channel capacity, propagation, latency, and energy
efficiency.

Network topology defined by the number, height, and distribution of platforms
is one of the most important factors influencing system-level metrics. As a
powerful mathematical tool, stochastic geometry (SG) assesses system-level
metrics in large-scale network topology. Moreover, SG is also one of a few
tools that can provide analytical results for system-level metrics while
incorporating co-channel interference into the analysis. Although the SG-based
analysis methods have powerful analytical tractability, it ignores the
correlation among the platforms. For instance, the binomial point process
(BPP), one of the most commonly-used SG models for satellites, assumed that
satellites are distributed on the spherical surface independently and
uniformly [2], which differs from reality where satellites are distributed on
fixed orbits. Fortunately, the SG-based system-level metrics analysis (such as
coverage probability) has been proven to approximate the lower bound of the
deterministic constellation, such as the Walker constellation, in the case of
a large-scale satellite constellation [3]. The contributions of the paper are
as follows:

  * •

While prior works focus on introducing and developing performance metrics of
NTNs, this work evaluates how these metrics influence the entire NTN system
performance, rather than an individual link.

  * •

To the best of the authors’ knowledge, this is the first paper to specifically
elucidate the system-level metrics in the NTNs, and analyze these metrics
based on special specific application scenarios, including remote and rural
coverage, post-disaster reconstructed networks, and military operations.
Finally, the system-level metrics of these scenarios are simulated according
to SG models.

  * •

This paper points out new research directions on the system-level metrics of
NTN platforms based on SG models, which will have a great improvement on the
overall performance evaluation of the NTN network combining LAPs, HAPs, and
satellites.

Figure 1: System framework diagram.

##  II NTN Platforms

NTN platforms have different characteristics and features, hence, different SG
models are used for each type of NTN. The radar diagram in the left bottom
part of Fig. 1 provides the quantitative results of metrics comparison.

###  II-A Low-Altitude Platform

LAP refers to platforms that typically fly at 50 m – 4 km, and applies to
platforms flying under 10 km. The altitudes of most flying platforms, such as
unmanned aerial vehicles (UAVs), networked tethered flying platforms (NTFPs),
aerostats, and balloons, vary from tens of meters to hundreds of meters. LAPs
are widely used to form a temporary network to enhance coverage owing to their
flexible deployment. They are generally deployed in small regions at low
altitudes, forming a rapidly changing topology due to their frequent movement,
making the density of LAPs more important compared with their numbers.
Therefore, a two-dimensional Poisson point process (PPP) is suitable for
modeling a LAP network [4]. In typical NTN application scenarios, including
remote areas, post-disaster areas, and military communications (See Section
IV), LAPs are often carried on vehicles (such as trucks) and need frequent
charging, making it difficult to provide stable coverage and availability for
LAPs. Moreover, the coverage area of a single LAP is significantly smaller
than that of a HAP or a satellite due to its low altitude. In addition, a
small channel capacity suffices to support serving targets within LAPs
coverage. In comparison, LAPs possess high-energy efficiency and low
propagation latency due to their lower altitude.

###  II-B High-Altitude Platform

HAPs are flying platforms between 17 and 22 km in the stratosphere. It has not
been implemented until recent years with technological progress due to the
harsh environmental conditions such as low temperature, high wind speed, and
long exposure to radiation. HAPs are distributed on a thin hollow spherical
shell with 5 km thickness above the Earth. Therefore, HAPs are modeled
approximately by BPP over a sphere. BPP is more accurate than PPP for closed
surfaces such as spheres [3]. We note that BPP can also be used to model LAPs
distributed in a finite area.

The performances of HAPs are ranked between LAPs and satellites regarding
coverage area, propagation latency, channel capacity, and energy efficiency
due to their altitudes which are lower than satellites and higher than LAPs.
HAPs fly at high altitudes so they do not undergo frequency shadow fading
caused by objects such as buildings and trees. Thus, the links between HAPs
and service targets have a higher line-of-sight (LoS) probability than LAPs.
Consequently, HAPs have a stable signal-to-noise ratio (SNR) and negligible
fluctuations of system-level metric values. They also benefit from higher
availability (both transmitter and receiver are above the horizon) due to
their greater LoS probability if deployed strategically. Higher altitude leads
to better availability. Furthermore, HAPs can be powered by solar energy,
hence, they can stay aloft for several years in the stratosphere.

###  II-C Low Earth Orbit Satellite

Low Earth orbit (LEO) satellites orbit around the Earth with an altitude
between 160–2000 km above the Earth’s surface. Examples of LEO satellite
networks include Starlink and OneWeb. LEO satellites are not stationary
relative to the Earth while geosynchronous equatorial orbit (GEO) satellites
are geosynchronous at 35786 km. Since most GEO satellite constellations are
only composed of a few satellites, SG models are not accurate and suitable for
GEO satellites. For small-scale LEO satellite constellations composed of
dozens of satellites (e.g., Iridium), the orbit geometry model is the closest
one to approximate the actual distribution of satellites [5]. Unlike BPP and
PPP, the orbit geometry model is a semi-stochastic model, that is, the
positions of orbits are deterministic, while the distribution of satellites in
orbit is stochastic. As a stochastic model, details about orbits are ignored
in BPP. However, spherical BPP, as is described in subsection II-B, is well-
suited to approximate massive dense constellations. To account for the
circular shape and multiple altitudes of satellite orbits, the Cox point
process model was presented, where orbits are created based on PPP on a cuboid
and satellites are distributed conditionally on these orbits [6].

For well-designed constellations, the unavailability of satellites on the
horizon hardly occurs owing to the non-temporary nature of satellite networks.
The coverage area and capacity of a single LEO or GEO satellite are much
larger than those of HAPs [7]. A single LEO satellite system provides seamless
coverage for the whole Earth, while global coverage can be achieved with only
three GEO satellites. The channel capacity of LEO satellite systems suffices
to support communications in remote areas. The tens to hundreds of
milliseconds propagation latency of LEO satellites can support most real-time
communication scenarios. Satellites, including LEO and GEO, have a low energy
efficiency due to the satellite-ground distance. Even with a large
transmission power, the received SNR may still not exceed the decoding
threshold. Due to severe attenuation in the ground clutter layer, satellite
communication often requires HAPs or ground gateways for relaying.

##  III Typical Communication Analyses

Depending on the number of hops, there are three communication analyses:
coverage, relay, and routing. The coverage analysis using coverage probability
as a metric has been the most studied system-level metric in SG literature.
However, the system-level metrics in relay communications (namely availability
and channel capacity) and routing (namely latency and energy efficiency) still
need further investigations.

###  III-A Coverage-Based Analysis: Single-Hop Communication

Coverage probability is a key metric in coverage-based analysis, which is the
probability of received signal-to-interference plus noise ratio (SINR) being
larger than a predefined threshold. Factors that affect coverage probability
include [7]: 1) the interference power; 2) the distance to the serving
platform; 3) and small-scale fading. An important feature of SG-based analysis
is that the aforementioned three factors are assumed to be random variables
subject to certain distributions.

Interference power is the sum of the received powers of all the transmitting
nodes in the network excluding the serving platform power. The authors in [3]
show that, by taking the Laplace transform of the interference power, the
specific distribution of the interference will not affect the analytical
expression of the coverage probability. Second, in [2], the distance to the
serving platform in the SG model (contact distance distribution), is the
distribution of the distance between the user and the serving platform with
the strongest average received power. Finally, small-scale fading in the SG-
based channel model also contributes to the randomness of received power. The
air-to-ground link passing through the ground clutter layer induces a severe
multi-path effect, leading to small-scale fading. Shadowed-Ricean fading is
the most accurate small-scale fading model satellite to ground user link or
HAP to ground [8] while that from LAP to ground is best captured by
Nakagami-$m$ fading [9].

###  III-B Relay-Based: Dual-Hop Communication

Relay communication can be viewed as the combination of an uplink and a
downlink coverage, or a basic unit of routing. Compared with coverage and
routing, availability and channel capacity are more suitable for relay-based
analysis. The results of these two system-level metrics obtained in relay-
based analysis provide better insight than a coverage-based one and can be
easily extended to results in multi-hop routing.

We start with HAP availability between two ground users. Due to blockage
caused by the Earth, platforms below the horizon are out of the ground user
LoS range and cannot provide a stable service. Therefore, the HAP has to be at
the overlapping region composed of two spherical caps above the horizons. The
area of this overlapping region is significantly smaller than a single-user
LoS spherical cap when considering single-hop communications. Therefore, it is
more relevant to discuss HAP availability in relay-based analysis. When HAPs
are modeled as BPP, SG analysis can provide concise analytical results for HAP
availability using the null probability. The SG analysis can also provide the
probability of HAPs availability in the overlapping region.

TABLE I: System framework diagram

Platform | Altitude | SG Model | Advantages | Challenges | Fading model
---|---|---|---|---|---
LAP* | UAV | 150 m–200 m |  $\bullet$ PPP $\bullet$ BPP for a finite area |  $\bullet$ High energy efficiency $\bullet$ Low propagation latency |  $\bullet$ Limited coverage area, availability, and channel capacity |  Shadowed-Racian fading
Balloon | 150 m–700 m
Blimps | 100 m–5 km
Helikites | 100 m-1.5 km
HAP | Aerostat |  17 km–22 km | $\bullet$ BPP |  $\bullet$ More LoS probability than LAP; $\bullet$ Longer lifetime; Stable system metrics values |  $\bullet$ High deployment expense and difficulty; $\bullet$ Low mobility |  Shadowed-Racian fading
Airship
| Balloon
---
Satellite | LEO | 160 km–2000 km |  $\bullet$ BPP $\bullet$ Semi-stochastic model $\bullet$ Cox point process |  $\bullet$ Large coverage area and availability |  $\bullet$ Long propagation latency; $\bullet$ Low energy efficiency; $\bullet$ Severe attenuation |  Nakagami-m fading
| GEO
---
35786 km

* All LAP systems can be tethered to the ground [1].

In terms of satellite system availability, we focus on LEO satellites that are
not geosynchronous. LEO satellite availability is called pass duration. Pass
duration is the continuous time interval when the LEO satellite is serving a
given user. Denoting the arc drawn by the satellite during this interval as
the session arc, the duration can be given by a fraction of the orbital
period, which is equal to the ratio of the session arc length to the orbit
length [10]. The SG-based framework provides tools to derive analytical
expressions of the session arc length and pass duration. Finally, similar to
the concept of LEO satellite pass duration, LAP availability is defined as the
ratio of serving time to the whole period, since LAPs need to be recharged
frequently. The whole period is the summation of serving time, back-haul
duration, and charging duration [9].

Channel capacity is defined as the ergodic capacity given by the Shannon-
Hartley theorem over a fading channel [3]. Generally, a large packet is
decomposed into many small packets. The channel capacity in a multi-hop link
is determined by the hop with the lowest capacity. In addition, since the
channel capacity in uplink and downlink might be different, it is more
relevant to investigate channel capacity in relay-based analysis. Under
limited transmitting power, the SG-based method maximizes channel capacity by
selecting platforms at optimal locations. To make the analysis of channel
capacity tractable, the SG-based method often selects a suboptimal relay
strategy [11]. Such suboptimal relay strategy selects the platform with the
maximum receiver capacity as the relay among a subset of platforms that has a
communication capacity exceeding a given threshold.

###  III-C Routing-Based Analysis: Multi-Hop Communication

Path planning and relay selections are the main focus when conducting a
routing-based analysis [12]. Unlike relay selection in channel capacity
analysis, path planning and relay selection in routing is based on all links
of the entire network. When selecting a relay platform, the best relay
locations for the next few hops have to be considered in advance.

Because each hop has an equal effect on the total latency, propagation latency
is a typical system-level metric considered in routing. Propagation latency is
defined as the time spent by the signal to propagate through the transmission
medium. Propagation latency in densely distributed LAP networks has been
comprehensively studied using SG framework [13]. As for the propagation
latency as a system-level metric in HAP and satellite networks, SG is the only
mathematical tool providing analytical results for spherical routing. Through
contact distance distribution, the authors in [12] provide accurate lower and
upper bounds for propagation latency in multi-hop routing.

Energy efficiency is the average number of bits that can be transmitted per
unit of energy consumed, which can be measured as the ratio of channel
capacity to total transmission power. Generally, there are two main strategies
of path and relay selection: minimum propagation latency routing and maximum-
efficiency routing. The minimum propagation latency routing uses the least
number of hops. Smaller number of hops leads to lower latency, but with the
same transmission distance, it has a lower energy efficiency. In comparison,
larger number of hops leads to larger energy efficiency since the transmission
of each bit requires less energy, but the latency is larger when the
transmission distance is the same.

##  IV Application Scenarios

In this section, we first describe three relevant application scenarios of
NTNs under the SG framework. Then, we select the most relevant system-level
metrics for each application according to its specific requirement. Finally,
we put forward the prospect of future research directions under the SG
framework.

###  IV-A Remote and Rural Coverage

The cost of laying terrestrial infrastructures, such as optical fibers, is
high. Remote areas, such as mountain areas, have sparse populations and
usually low income compared to urban populations. Consequently, these areas
are lacking economic incentives for mobile network operators (MNOs) to deploy
terrestrial infrastructure.

The most crucial communication aspects in remote areas are coverage and access
equity. Therefore, investigating these two aspects is of utmost importance
when designing an NTN system in these areas. Covering a large remote area with
a limited number of devices at a manageable cost is challenging. Hence,
studying the coverage probability under different platform distributions is
necessary to design a robust NTN system. Furthermore, access equity in remote
areas ensures fairness among all the users in the network. Access equity can
be measured by availability, i.e., ensuring that at least one platform is
available to access the network via an NTN platform.

###  IV-B Post-disaster Reconstructed Networks

When a disaster destroys terrestrial communication infrastructures, NTNs play
a pivotal role in providing coverage for the disaster area. Considering that,
satellite networks can achieve seamless and ubiquitous global coverage; the
post-disaster network will utilize satellites as the primary NTN platform for
communication. LAPs and HAPs can form a temporary network in a short period of
time to increase the capacity of the post-disaster network.

When the original communication networks are destroyed by a disaster, the
affected areas will face a high demand for communication services in a short
period. Therefore, the channel capacity of NTN is a key system-level metric to
assess the quality of communications in post-disaster scenarios. Furthermore,
satellites rely on solar power for recharging, which limits the amount of
energy they can store. On the other hand, HAPs and LAPs require frequent
recharging, which is challenging in post-disaster areas. Therefore, improving
energy efficiency is critical in post-disaster scenarios.

###  IV-C Military Operations

During military operations, communication and surveillance are of the utmost
importance and play a decisive role in warfare. Communication consists of the
transmission of commands, orders, and instructions to their soldiers. It also
consists of the transmission of information gathered through surveillance.
Surveillance, on the other hand, consists of persistent and prolonged
monitoring of a target or a region. Communication through terrestrial networks
especially in an enemy-occupied area is difficult and dangerous during
military actions. This is because the network can be easily compromised,
hijacked, or eavesdropped on. In addition, surveillance requires visual
observation and can be conducted using soldiers on the ground or a UAV. NTNs
are a preferable alternative for terrestrial networks for this type of
operation. NTNs have a larger coverage than terrestrial networks, perform
surveillance with a large visual range, are easy and fast to deploy during a
military operation, and have a higher LoS probability.

When monitoring fast-moving military equipment, a low propagation latency is
necessary so that the target will not leave the detection range, making it
critical to form a high-quality network. In addition, high bit rate images and
videos help military commanders complete military deployments, take strategic
decisions, and locate enemy forces. These high-resolution images or video
streams rely heavily on high channel capacity. When the military has to
transmit messages, orders, or reports (both in the field and at sea) or is
faced with a demanding surveillance mission, the $k$-coverage probability is a
more pertinent system-level metric than the coverage probability for these
types of operations. The $k$-coverage is achieved when an object or region is
covered by at least $k$ platforms at the same time [14]. NTN platforms have
advantageous anti-reconnaissance ability and mobility compared with the
terrestrial network platforms to form the high-quality network and satisfy the
$k$-coverage probability to reliably support military operations.

##  V Simulation Results

In this section, we provide comprehensive simulation results that include
several system-level metrics, NTN platform types, communication analysis, and
application scenarios. The models used to characterize NTN platforms and users
are based on the existing SG models. To enhance the richness of the simulation
results, we choose communication analysis with system-level metrics that have
not been well-investigated in the literature. In the simulations, we consider
decoding and forward relaying scheme, and the power consumption at each node
is not considered.

###  V-A Relay Communications Availability in Remote Areas

Figure 2: Availability analysis of relay communications in remote areas.

(a) Capacity

(b) Energy Efficiency

Figure 3: Capacity and energy efficiency analysis in post-disaster
reconstructed network. Figure 4: $k$-coverage probability under different
thresholds ($k=4$).

Considering that availability is a key system-level metric in both relay
communications and remote area networks, we constructed a hybrid HAP and LEO
satellite network to carry out an availability analysis. Ground users are
modeled as a homogeneous PPP on a spherical cap with density $1/$km2. The area
of the spherical cap is $1.55\times 10^{5}$ km2, its dome angle is
$\frac{1}{45}\pi$, and its radius is $6371$ km, which is equal to the Earth
radius.

Assuming that the nearest terrestrial base station from a core network is
$500$ km away from the center of the spherical cap so that there is no
terrestrial base station serving this area. Therefore, users in a remote area
cannot establish direct communication links with the core network and must be
relayed by HAPs or LEO satellites. The satellites form a homogeneous spherical
BPP in the whole sphere with a radius of 6921 km, which is the same radius as
the recently deployed Starlink constellation. Unlike satellites, HAPs can be
deployed stationary relative to the ground, thus we deploy HAPs directly above
the user. HAPs are modeled by a homogeneous BPP on a spherical cap with a
radius of 6391 km (the HAPs altitude is 20 km above Earth’s mean sea level)
and the same dome angle as users.

A randomly selected user in a remote area initiates a communication request.
The relay communication link is available when there is at least one NTN
platform in the common LoS region between the user and the ground base
station. Fig. 2 shows the ratio of the number of available communication links
to the number of communication requests under the different numbers of HAPs
and satellites. Even though LEO satellites are distributed on the whole
sphere, they still contribute more to availability than HAP. Satellites have
always been the first choice to promote equal access to networks. A small-
scale LEO satellite constellation of $40$ satellites assisted by $50$ HAPs is
sufficient to provide $95\%$ coverage over a region with $1.55\times 10^{5}$
km2 area.

###  V-B Capacity and Energy Efficiency of Post-disaster Networks

We consider an aftermath scenario in which a ground network in a 2-dimensional
plane with a radius of 30 km was destroyed by a disaster. The locations of the
ground residents in this disaster area follow a homogeneous PPP with a density
of 100/km2. LAPs form a homogeneous BPP at an altitude of 80 m in the disaster
area. HAPs and satellites follow the same distribution as subsection V-A.
Therefore, the deployment area of HAP will be larger than the disaster area.

In this NTN system composed of LAPs, HAPs, and satellites, all the platforms
transmit in an RF frequency of 2 GHz. LAPs adopt the same channel model as in
[9], and the channel model adopted by HAPs and satellites is given in [2]. The
transmission powers of LAPs, HAPs, and satellites are 1 dBW, 8 dBW, and 15
dBW, respectively, and their antenna gains are 10 dB, 20 dB, and 70 dB,
respectively. The small-scale fading parameters of the LAP channel are the
same as [9], and the small-scale fading parameters of the HAP and satellite
channels are given in [2]. The environmental noise is set to -174 dBm/Hz.

As shown in the lines with circle and star marks in Fig. 3(a), when the number
of LAPs increases, the channel capacity increases significantly. The
cumulative distribution function (CDF) curves corresponding to the red and
green lines converge, indicating that almost only LAPs can achieve a channel
capacity greater than 300 Mbps. With the same number of LAPs, increasing the
number of HAPs brings more benefits to channel capacity than increasing the
number of satellites. The curves in Fig. 3(b) show the same convergence
characteristics as in Fig. 3(a), which means that only LAPs can achieve an
energy efficiency of more than 250 Mb/J. The slope of the curves in the
interval of $80-200$ Mb/J is steep, further indicating that the energy
efficiency that HAP and satellites can achieve is lower than 80 Mb/J. When the
number of UAVs is 1000, HAPs are more likely to be associated with satellites.

###  V-C $k$-Coverage Probability in Military Operations

This subsection investigates the $k$-coverage probability when $k=4$ during a
military operation, that is, the probability that a military target in a given
ground region is covered by at least four NTN platforms simultaneously. The
distribution model and channel model for the three types of platforms in NTN
are similar to those in the previous two subsections. It is assumed that the
target is distributed in the same spherical cap as the users in subsection
V-A.

Fig. 4 shows the simulation results of 4-coverage probability as a function of
the SNR threshold. NTN platforms that are closer to the target can achieve
better communication performance. The average SNR that LAP can provide is
10–20 dB, while the SNR of the HAP signal is greater than 5dB. When the
military targets can demodulate signals with SNR over 5 dB, deploying more
than 40 HAPs and 1000 LAPs over an area of $1.5-2\times 10^{5}$km2 can achieve
4-coverage for almost all military targets.

##  VI Conclusion

In this paper, we started with an introductory discussion about the concepts
of NTNs and system-level metrics, and the motivation to apply SG tools. Then,
we discussed suitable SG models for different platforms and compared their
performance. Three typical communication issues, namely coverage, relay, and
routing, were introduced and the system-level metrics of each issue were
elaborated. The requirements for system-level metrics were further discussed
for the three representative application scenarios of NTNs. Finally, we
conducted some new combinations of system-level metrics with communication
issues. The detailed SG-based analysis framework and numerical results of the
metrics are presented.

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## biographies

Qi Huang received his B.Eng degree from SWPU, China in 2017, and is pursuing
his master’s degree in KAUST.
---
Baha Eddine Youcef Belmekki received the M.Sc. degree in wireless
communications and networking from the USTHB, Algeria, in 2013, and the Ph.D.
degrees in wireless communications from INPT, France, in 2020. He is currently
a postdoctoral fellow at KAUST.
---
Ahmed M. Eltawil received his Ph.D. degree from UCLA in 2003. He was with UCI
since 2015 as a professor and now is a professor of electrical engineering at
KAUST.
---
Mohamed-Slim Alouini received his Ph.D. degree in electrical engineering from
the California Institute of Technology, Pasadena, in 1998. He joined KAUST as
a professor of electrical engineering in 2009.
---
  *[SG]: stochastic geometry
  *[LAP]: Low altitude platform
  *[HAP]: high altitude platform
  *[LoS]: line-of-sight
  *[SINR]: signal-to-interference plus noise ratio