# Stochastic Geometry Analysis of Spatial-Temporal Performance in Wireless
Networks: A Tutorial

Xiao Lu, Member, IEEE, Mohammad Salehi, Martin Haenggi, Fellow, IEEE,
Ekram Hossain, Fellow, IEEE, and Hai Jiang, Senior Member, IEEE X. Lu and H.
Jiang are with the Department of Electrical and Computer Engineering,
University of Alberta, Canada. M. Salehi and E. Hossain are with the
Department of Electrical and Computer Engineering, University of Manitoba,
Canada. M. Haenggi is with the Department of Electrical Engineering,
University of Notre Dame, Notre Dame, USA and with the Department of
Information Technology and Electrical Engineering, Swiss Federal Institute of
Technology, Zürich (ETHZ), Switzerland.

###### Abstract

The performance of wireless networks is fundamentally limited by the aggregate
interference, which depends on the spatial distributions of the interferers,
channel conditions, and user traffic patterns (or queueing dynamics). These
factors usually exhibit spatial and temporal correlations and thus make the
performance of large-scale networks environment-dependent (i.e., dependent on
network topology, locations of the blockages, etc.). The correlation can be
exploited in protocol designs (e.g., spectrum-, load-, location-, energy-aware
resource allocations) to provide efficient wireless services. For this,
accurate system-level performance characterization and evaluation with
spatial-temporal correlation are required. In this context, stochastic
geometry models and random graph techniques have been used to develop
analytical frameworks to capture the spatial-temporal interference correlation
in large-scale wireless networks. The objective of this article is to provide
a tutorial on the stochastic geometry analysis of large-scale wireless
networks that captures the spatial-temporal interference correlation (and
hence the signal-to-interference ratio (SIR) correlation). We first discuss
the importance of spatial-temporal performance analysis, different parameters
affecting the spatial-temporal correlation in the SIR, and the different
performance metrics for spatial-temporal analysis. Then we describe the
methodologies to characterize spatial-temporal SIR correlations for different
network configurations (independent, attractive, repulsive configurations),
shadowing scenarios, user locations, queueing behavior, relaying,
retransmission, and mobility. We conclude by outlining future research
directions in the context of spatial-temporal analysis of emerging wireless
communications scenarios.

###### Index Terms:

Large-scale wireless access networks, signal-to-interference ratio (SIR),
spatial-temporal correlation, point process modeling, stochastic geometry.

##  I Introduction

Wireless communications systems are evolving toward a heterogeneous
architecture (e.g., multi-tier and cell-free) with the dense deployment of
different types of access points (e.g., smallcells and hotspots) to enable
pervasive wireless Internet access [1]. The evolving wireless networks are
expected to provide seamless connectivity to ubiquitous and/or high-mobility
devices and users with millisecond delay and gigabits per second data rate
[2]. The ever-increasing demand for low-latency high-reliability services from
pervasive terminals will lead to an explosive increase in mobile traffic. To
accommodate the massive traffic volume, high network densification and
aggressive spatial frequency reuse will be required, which will result in high
levels of interference in the network.

###  I-A Background of Stochastic Geometry and Objective

Signal propagation over a radio link is impaired by large-scale path loss and
shadowing, small-scale fading, as well as co-channel interference from
concurrent transmissions. Since all of these effects are heavily location-
dependent, the network spatial configurations become a dominant factor that
determines the system-level performance. Hence, developing tractable
approaches for modeling large-scale wireless systems and analyzing their
statistical performance taking into account the randomness (due to the above-
mentioned factors) have become compelling.

In this context, stochastic geometry [3] (also referred to as geometric
probability), a probabilistic analytical approach to study (random) point
configurations, has become a necessary theoretical tool for the analysis and
characterization of large-scale wireless systems, including heterogeneous
cellular networks [6, 5, 4], dynamic spectrum access systems [7, 8], wireless
ad hoc networks [9, 12, 10, 11], drone networks [13, 14, 15], vehicular
networks [16, 17, 18], and low earth orbit satellite networks [19, 20].
Spatial-temporal aspects of mobile communications, including the spatial
distribution of network nodes, wireless channels and traffic patterns have to
be considered for system development, resource allocation, performance
evaluation and optimization. The purpose of this paper is to provide a
tutorial on how to quantitatively analyze the effects of spatial and temporal
fluctuations of interference (resulting from the factors above) on the system-
level network performance.

(a) Example 1: Temporal interference correlation in a static network.  (b)
Example 2: Spatial interference correlation in a full-duplex communication
system.  (c) Example 3: Spatial-temporal interference correlation in a mobile
network.

Figure 1: Examples of interference correlation.

###  I-B Importance of Characterization of Signal-to-Interference-plus-Noise
Ratio Correlation

Wireless communications systems need to preserve the quality of radio links in
time-varying environments. The signal-to-interference-plus-noise ratio (SINR)
statistics is commonly used as the main measure of the quality of links [21],
and most of the performance metrics for system-level evaluation (to be
introduced in Section II-C) are based on the SINR. The variation of the
wireless environment (and hence the SINR) is mainly attributed to two causes.
On the one hand, the changes of the relative positions of communication
devices and surrounding obstructions affect the multipath propagation and thus
the received power of both desirable and interfering signals. Also, variations
in traffic patterns cause fluctuations in the interference and hence the SINR.

Owing to the spatial-temporal fluctuations of network distributions, wireless
channels and traffic patterns, the interferences and hence SINRs (at different
locations and time instants) are correlated. Although rapid channel
fluctuations due to small-scale fading can result in reduced interference
correlation [22], the correlation still exists due to large-scale path-loss
and shadowing [23]. For example, in a static network, as shown in Fig. 1(a),
if the ambient transmitters have data packets to send during two time
intervals, the interferences at the receiver are temporally correlated.
Besides, in a full-duplex communication system as shown in Fig. 1(b), the
interferences at the transceiver pair are spatially correlated at any time
instance. In a mobile network, as shown in Fig. 1(c), the interference at the
receiver is spatially and temporally correlated. The causes of the temporal
and/or spatial interference correlation in the above examples all arise from
the fact that the interference comes from the same group of transmitters. The
interference correlation results in the spatial-temporal correlation of
transmission outage/success [24, 25, 26], throughput [27, 28], mean local
delay [29, 30], etc., thus needs to be treated carefully in the designs of
mobile systems.

Since SINR correlation affects the performance at different transmission
attempts (e.g., when using an error recovery method) and/or locations (e.g.,
in a relay-based system), an accurate characterization of it is essential to
the understanding of wireless network performance. Information about SINR
correlation can be exploited to optimize the performance and design of the
system accordingly.

###  I-C Related Work

Several survey and tutorial papers have focused on the stochastic geometry
analysis of wireless communication networks. In particular, reference [31]
provides a survey of point process models and stochastic geometry tools that
have been used to analyze static wired, wireless, ad hoc and cellular networks
prior to 2009. Reference [32] overviews the impact of spatial modeling on the
SINR-based performance metrics, i.e., connectivity, coverage area, and
capacity of different types of systems, including ad hoc, cellular and
cognitive networks. The survey in [33] comprehensively reviews the works on
stochastic geometry analysis of multi-tier and cognitive cellular systems
prior to 2013.

In addition to the above survey papers, tutorial papers on the mathematical
tools used for stochastic geometry analysis of large-scale systems have also
been written. Reference [34] is the first tutorial on point process theory,
random geometry graphs, and percolation theory for interference
characterizations in ad hoc networks. Targeting cellular networks, [35]
provides a tutorial on stochastic geometry analysis of both downlink and
uplink networks based on Poisson point process (PPP) modeling. The focus is on
characterizing interference in different scenarios by exploiting the
properties of the PPP under Rayleigh fading assumption. As PPP modeling fails
to capture the spatial correlation among the random points, the authors in
[36] emphasize the use of repulsive point processes to model cellular
networks, where the base station locations are usually planned with a moderate
degree of irregularity due to different development issues. To this end, the
authors present a tutorial on the SINR distribution analysis of downlink
cellular networks based on the $\beta$-GPP (Ginibre point process), which is a
fairly tractable model for random points with spatial repulsion. The tutorial
in [37] focuses on a unifying analysis of bit/symbol error probability,
coverage outage probability, and ergodic capacity in cellular networks. More
recently, reference [40] systematically tutors the analytical techniques to
characterize interference, success probability and capacity in Poisson
networks. However, none of the existing survey and tutorial papers focus on
the stochastic geometry techniques to characterize the spatial and temporal
correlations in their considered systems. Moreover, this is the first article
to include the refined-grained analysis tool of the signal-to-interference
ratio (SIR) meta distribution.

Figure 2: The suggested learning path for stochastic geometry analysis in
wireless networks.

For a better understanding of the contents of this tutorial, we recommend
taking some prerequisite knowledge of stochastic geometry regarding network
analysis which has been comprehensively reviewed in the literature. The
suggested learning path and the recommended readings corresponding to the
prerequisite knowledge are shown in Fig. 2, which includes the following.

  * •

Primers [32, 34, 38, 39, 40, 31, 33, 41] that cover the technical concepts of
spatial models and their applications in modeling wireless networks;

  * •

Intermediate tutorials [33, 34, 35, 39, 40, 37] that tutor the fundamental
properties of spatial point processes (e.g., counting measure, superposition,
thinning, and transforms) and analytical methods to characterize the
interference distribution in different types of wireless networks (e.g.,
cellular and ad hoc networks);

  * •

Advanced tutorials [40, 42, 43, 44, 45] that lays out the higher-order
statistics of point processes for fine-grained analysis (e.g., product/joint
meta distribution) and up-to-date analytical techniques (e.g., approximation
and bounding methods) to analyze complicated scenarios (e.g., with spatial-
temporal correlation) where the exact performance characterization is of low
tractability or unavailable.

###  I-D Contributions and Organization

This tutorial aims to concisely present the analytical approaches for
performance evaluation of large-scale wireless networks taking into account
various correlation causes and effects, such as shadowing, traffic queueing,
and spatial distribution of network nodes. We focus on the characterization of
the interference distribution and signal-to-interference ratio (SIR)-based
performance metrics111Since in large-scale cellular networks, the impact of
aggregate interference typically dominates that of noise [6, 46], this
tutorial focuses on analyzing the interference-limited cases with the noise
ignored. However, without loss of generality, the same analytical approaches
can also be applied to characterize SINR-based performance metrics., e.g.,
success probability, joint success probability, and moments of conditional
success probability (CSP) given the point process. For spatial point process
models, we additionally derive the asymptotic SIR gain, which is the
horizontal gap between a target SIR distribution and a reference SIR
distribution. This metric directly reflects the variation of SIR due to the
changes in the network model with respect to (w.r.t.) the reference model.
Furthermore, this metric can be utilized to simplify the analysis of non-
Poisson networks based on the PPP [47] (to be introduced in Section III-B).

This tutorial considers the following correlation effects in wireless
networks.

  * •

The spatial correlation (i.e., attraction and repulsion) among the locations
of the transmitters;

  * •

The spatially-correlated shadowing experienced by the links that traverse
common obstacles (e.g., buildings);

  * •

The spatially and temporally correlated queue status among the transmitters
due to the cross-interference imposed on each other over space and time;

  * •

The spatial correlation experienced by users located in the cell-center and
cell-boundary regions;

  * •

The temporal interference correlation between multiple transmission attempts
due to the correlation in the locations of the interferers over time;

  * •

The spatial-temporal interference correlation among relaying nodes in a
multihop network due to the correlation in the locations of the interferers
over space and time;

  * •

The spatial-temporal interference correlation in mobile systems (i.e., where
the users and/or base stations (BSs) are mobile) due to the correlation among
interferers’ locations over space and time.

TABLE I: Main differences between other survey and tutorial papers Reference (year) |  Type of Review | Examined systems | Spatial models |  Target performance metrics |  Spatial-temporal correlation analysis
---|---|---|---|---|---
|  |  Ad hoc |  Cellular |  Multihop |  Mobile | Poisson |  Non-Poisson |  |
[31] (2009) |  Survey |  ✓ |  ✓ |  |  | ✓ |  |  Success probability, paging, handover |  No
[34] (2009) |  Tutorial |  ✓ |  |  |  | ✓ |  |  Interference characterization, outage probability, capacity, and area spectral efficiency |  No
[32] (2010) |  Survey |  ✓ |  ✓ |  |  | ✓ |  |  Success probability, coverage area, capacity |  No
[33] (2013) |  Survey |  ✓ |  ✓ |  |  |  |  |  Success probability, capacity |  No
[35] (2016) |  Tutorial |  |  ✓ |  |  | ✓ |  |  Success probability |  No
[36] (2016) |  Tutorial |  |  ✓ |  |  |  |  ✓ |  Success probability |  No
[37] (2017) |  Tutorial |  |  ✓ |  ✓ |  | ✓ |  |  Interference characterization, error probability, error rate, outage probability, capacity, and handover |  No
[40] (2021) |  Tutorial |  |  ✓ |  |  | ✓ |  ✓ |  Interference characterization, success probability, capacity |  No
This work |  Tutorial |  ✓ |  ✓ |  ✓ |  ✓ | ✓ |  ✓ |  Success probability, joint success probabilities, conditional success probabilities, moments of conditional success probability given the point process, SIR meta distribution, SIR gain, interference correlation coefficient |  Yes

This tutorial presents the methodologies of analyzing the interference (and
SIR) correlation and the effects of spatial-temporal SIR correlations. For a
thorough exposition of the analytical techniques, we demonstrate the step-by-
step derivations as well as numerical results222Note that the analytical
methodologies introduced in this tutorial come from the referenced literature.
The analytical results are either well-established or direct extensions of the
ones derived in the references.. The main differences between our tutorial and
the state-of-the-art discussed in the previous subsection are summarized in
Table I333Herein, success probability refers to the complementary cumulative
distribution function (CDF) of the SINR/SIR.. Herein, Poisson spatial models
refer to the PPP and binomial point process, while non-Poisson spatial models
refer to any point process whose points are not independently distributed.

The organization of this tutorial and the relations among different sections
are shown in Fig. 3. We restrict the point processes to the Euclidean spaces
$\mathbb{R}$ and $\mathbb{R}^{2}$. However, the same methodologies can be
applied to analyze point processes in higher dimensions without loss of
generality. Section I introduces the role of stochastic geometry analysis and
highlights the importance of spatial-temporal correlation characterization.
Section II explains the correlation effects of different network factors that
may affect the SIR-based network performance. Sections III-VI present exact
methodologies to analyze network performance with correlations in node
distribution, link distance distribution, shadowing, and queueing,
respectively. Moreover, Sections VII and VIII present a performance
characterization with multiple transmission attempts (i.e., retransmission)
and multihop relaying, respectively, for correlated and independent
interference. Section IX characterizes the spatial-temporal performance with
mobility. Future directions and research challenges are then discussed in
Section X followed by the conclusion in Section XI. Additionally, for
convenience, we list the abbreviations used in Table II.

TABLE II: List of abbreviations Abbreviation |  Description
---|---
PPP |  Poisson point process
PLP |  Poisson line process
BPP |  Binomial point process
MCP |  Matérn cluster process
GPP |  Ginibre point process
RDP |  Relative distance process
SINR |  Signal-to-interference-plus-noise ratio
SIR |  Signal-to-interference ratio
CSP |  Conditional success probability
JSP |  Joint success probability
MISR |  Mean interference-to-signal ratio
LSU |  Location-specific user
ASAPPP |  Approximate SIR analysis based on the PPP
MIMO |  Multiple-input and multiple-output
PDF |  Probability density function
CDF |  Cumulative distribution function
PCF |  Pair correlation function
PGFL |  Probability generating functional
MAC |  Medium access control
CSMA |  Carrier-sense multiple access
BS |  Base station
DF |  Decode-and-forward
HARQ |  Hybrid automatic repeat request
QSI |  Quasi-static interference
FVI |  Fast-varying interference
IoT |  Internet of Things
IoNT |  Internet of Nanothings
IoST |  Internet of Space Things
ML |  Machine learning

Notations: The notations defined in Table III are used throughout this
tutorial.

Figure 3: Organization of this paper.  TABLE III:  Notations Symbol |  Definition
---|---
$\jmath=\sqrt{-1}$ |  The imaginary unit
$\equiv$ |  Equivalence relation
$\mathbb{N}$, $\mathbb{R}$, $\mathbb{R}^{+}$ |  Natural numbers, real numbers, and positive real numbers, respectively
$\mathbb{R}^{d}$ |  $d$-dimensional Euclidean space
$o$ |  The origin of $\mathbb{R}^{d}$
$\mathbb{B}(x,R)$ |  Disk of radius $R$ centered at $x$
$\Phi$ |  Point process representing the nodes in the network
$\Phi^{!}$ |  The set of interferers
$\Phi(A)$ |  The number of elements in $\Phi\cap A$
$\triangleq$ |  The definition operator
$\overset{(\rm d)}{=}$ |  Equivalence in distribution
$\mathbbm{1}_{\\{\cdot\\}}$ |  Indicator function which equals 1 and 0 if the statement $\\{\cdot\\}$ is true and false, respectively
$\mathbb{E}[\cdot]$ |  Expectation operator
$\mathbb{E}^{!}_{x}[\cdot]$ |  Expectation operator w.r.t. the reduced Palm measure at $x$
$\mathbb{P}[\cdot]$ |  Probability measure
$\mathbb{P}^{!}_{x}[\cdot]$ |  Reduced Palm probability measure at $x$
$\mathbb{V}[\cdot]$ |  Variance
$|\cdot|$ |  Modulus operator
$\|\cdot\|$ |  Euclidean norm
$\Gamma(\cdot)$ |  Gamma function
$\gamma(s,x)$ |  Lower incomplete gamma function, i.e., $\gamma(s,x)=\int^{x}_{0}u^{s-1}e^{-u}\mathrm{d}u$
$\mathcal{E}(a)$ |  Exponential distribution with rate parameter $a$
$\mathcal{P}(a)$ |  Poisson distribution with rate parameter $a$
$\mathcal{G}(a,b)$ |  Gamma distribution with shape parameter $a$ and scale parameter $b$
$\mathrm{Det}$ |  Determinant operator
${{}_{2}}F_{1}(.,.;.;z)$ |  The Gauss hypergeometric function
$f_{X}(\cdot)$, $F_{X}(\cdot)$, $\mathcal{L}_{X}(\cdot)$ |  The probability density function (PDF), the cumulative distribution function, and Laplace transform of random variable $X$.
$\mathcal{W}(\cdot)$ |  The Lambert-$\mathcal{W}$ function, i.e., the inverse function of $f(x)=xe^{x}$
TABLE IV: SIR components Component |  Representation |  Characterization
---|---|---
$\Phi$ |  Spatial distribution of interferers |  Spatial point process models (in Section III)
$r_{1}=\|x_{1}\|$ |  Contact distance distribution |  Location-dependent analysis of cellular models (in Section IV)
$S_{x}$ |  Blockage |  Shadowing models (in Section V)
$\iota_{x}$ |  Buffer status |  Queueing models (in Section VI)

##  II Overview of Spatial-Temporal Performance Correlation

This section first discusses the impact of SIR components and then elaborates
on the SIR correlation effects in wireless networks and defines the
performance metrics for spatial-temporal performance analysis.

###  II-A Parameters Impacting SIR Correlation

Let $o$ denote the origin of $\mathbb{R}^{d}$. The SIR at a target receiver
located at $o$ expressed as

$\displaystyle\textup{SIR}=\frac{P_{y}h_{y}S_{y}\ell(\|y\|)}{\sum_{x\in\Phi^{!}}\iota_{x}P_{x}h_{x}S_{x}\ell(\|x\|)},$
(1)

where $\Phi^{!}$ denotes the set of all the interfering transmitters, $y$ the
location of the transmitter associated with the target receiver, $P_{x}$ the
transmit power of transmitter at $x$, $h_{x}$ ($S_{x}$) the fading (shadowing)
coefficient between the transmitter located at $x$ and receiver located at
$o$, $\ell$ the path loss function, and $\iota_{x}$ the state indicator of the
transmitter located at $x$ which equals 1 and 0 when the transmitter is on and
off, respectively. The physical implications of the SIR components are shown
in Table IV.

The SIR varies due to the fading and shadowing effects and system factors such
as timing-varying traffic loads and positions of transmitters and receivers.
Their impacts are discussed below.

####  II-A1 Spatial Distribution

The spatial distribution of the network nodes can be categorized into three
types: independent, repulsive, and attractive.

Figure 4: Examples of wireless systems with spatially repulsive node
locations.

  * •

With independent distribution, the locations of the transmitters are
independent of each other. Spatial point process models to characterize
independent distributions include PPP and binomial point process (BPP) [48]
models.

  * •

With repulsive distribution, the wireless transmitters transmitting
simultaneously are not too close to each other. The repulsion may arise from
planned deployment, physical restrictions (e.g., geographic exclusion and
terrain occlusion) and channel access control (e.g., carrier sense multiple
access (CSMA) [49, 50, 51, 52] and licensed-user activity [7, 53]) as shown in
Fig. 4. Lattice processes [39], Ginibre point processes [54, 36, 55], and
Matérn hardcore point processes [49, 56] are some well-known examples of point
process models to characterize this repulsive behavior.

  * •

Attractive distributions can be observed when wireless transmitters are only
clustered in certain regions, i.e., not identically distributed over the
entire plane, as illustrated in Fig. 5. This can be caused by the base station
(BS)-centric user gathering [57], (e.g., around open-access WiFi spots) or due
to user-centric BS deployment [58]. The Matérn cluster process [60, 59],
Thomas cluster process [57], Gauss-Poisson process [61, 62], Cox process [64,
63, 16], and the Poisson hole process [7, 53] are some representative point
processes to model attractive spatial distributions.

Additionally, a point process can be a mix of the above three types. Whether a
point process is of any type may depend on the distance between the locations
considered. For example, the type I user point process introduced in [65] (for
modeling user distribution in cellular networks) is repulsive at short
distances, attractive at intermediate distances and eventually approaches
independence at larger distances.

Figure 5: Illustration of a wireless system with spatially attractive
deployment.

For network performance analysis, a significant body of literature adopts the
independent distribution assumption for simplicity and tractability (see [33]
and references therein). However, this assumption is too idealized to hold in
practice as most of the cellular networks are deployed under system-level
planning [66]. Due to factors such as geographical restriction, access control
and resource allocation, active network nodes can exhibit a spatial pattern
with a degree of correlation (i.e., repulsion or attraction) which cannot be
ignored in performance characterization.

####  II-A2 Locations of Receivers

In cellular networks, the transmission performance of a mobile user is highly
dependent on its location w.r.t. the serving BS and interfering BSs [67].
Specifically, for a cell-edge user, the desired signal is weaker and the
interference signal is stronger compared to those for a cell-center user (Fig.
6). Different techniques have been developed to strengthen the desirable
signals and/or to mitigate the interference of cell-edge users, e.g., through
coordinated beamforming [68] and intercell interference coordination [70, 69].
With these techniques, resource allocation strongly depends on the spatial
variation of mobile users. Therefore, location-dependent modeling of user
performance is fundamental to the understanding of spatial-temporal
performance of these systems.

Figure 6: Illustration of location-specific users.

####  II-A3 Small-Scale Fading

Owing to small-scale fading, over only a fraction of a wavelength, the signal
power variation may reach up to 40 dB [71]. The correlation of the signal
strength can be both temporal and spatial and is frequency dependent [21]. The
temporal correlation occurs due to movement of environmental scatters. The
spatial correlation is caused by correlated multipath components due to the
common propagation environment. However, small-scale fading is only correlated
over a short time duration and a small distance (e.g., several wavelengths) in
environments with moving scattering objects that change the multipath
propagation. Hence it is reasonable to adopt independent and identically
distributed (i.i.d.) small-scale fading models for receiving antennas with
wavelength separation. This assumption is widely used in the existing
literature (see [33, 37] and references therein).

####  II-A4 Shadowing

Figure 7: Illustration of correlated shadowing.

Compared to small-scale fading, shadowing is correlated at a much larger time
and space scale [72]. For example, in an urban environment, shadowing caused
by obstacles in communication paths can be geographically correlated on a
scale from 50 to 200 meters [21]. Moreover, temporal shadow fades with
variations below 1 dB (i.e., highly correlated) over an IEEE 802.11ad channel
can be commonly measured in urban streets [73]. Fig. 7 demonstrates an example
of correlated shadowing. As the two links between the BS and users experience
a similar propagation environment, their shadowing attenuations tend to be
correlated. As evidenced by experimental studies in [74] and [75], path
attenuations in a region are correlated due to shadowing effects. Therefore,
correlated shadowing should be carefully treated when assessing system
performance that is heavily impacted by environmental blockages.

####  II-A5 Transmission Buffer Status

The buffer status (i.e., queueing status) of each transmitter depends on the
packet arrival and service processes, and it impacts the activation of the
transmitters, and therefore, the mutual interference. In view of this, the
buffer statuses of the transmitters are interdependent, leading to interacting
queues. The queues are spatially coupled since the mutual interference
directly affects the transmission success probability (i.e., service processes
of the queues). Also, the queues are temporally coupled since the current
buffer status is affected by the previous departure process. Due to the random
nature of channel fading and aggregate interference, the service processes can
be very dynamic.

Figure 8: Illustration of the interacting queues.

Fig. 8 depicts an example of the correlation between the queues at two
transmitters. Transmitter T1, which serves receiver R1, has a longer
transmission link and a shorter interference link compared to those of
transmitter T2, which serves receiver R2. On the one hand, the channel
disparity between the two communication links results in different packet
service rates. Compared to T2, T1 suffers from more severe path loss and
interference. Correspondingly, given a similar traffic load T1 tends to vacate
its queue more slowly and thus remain active more frequently than T2. On the
other hand, the queue lengths of T1 and T2 determine the activation of T1 and
T2. In particular, if both transmitters are busy, their transmissions will
cause mutual interference, which slows down the departure process. If one of
the transmitters has an empty queue, the other enjoys a speedy departure
process. Due to the interacting queues, transmissions in a large-scale system
inherently experience spatial and temporal correlation.

Figure 9: Sources of SINR correlation.

Finally, Fig. 9 summarizes the sources of SINR correlation and their spatial-
temporal impact discussed in this subsection.

###  II-B Spatial-Temporal SIR Correlation and Network Scenarios

Even over independent fading channels, aggregated interference is spatially
correlated when it comes from the same group of interferers. Similarly,
interference becomes temporally correlated when common static interferers
exist across time. Spatially and temporally-correlated interference causes
spatial-temporal SIR correlation. When different transmission attempts are
affected by the same group of static interferers (e.g., the same point
process), it gives rise to a quasi-static interference (QSI) scenario. By
contrast, a fast-varying interference (FVI) [76] scenario results when
different transmission attempts are affected by different sets of
interferers444The QSI and FVI assumptions can be used for modeling static and
highly-mobile network scenarios, where the interferers across different
transmissions remain the same and become completely different, respectively.
(e.g., independent point processes). As two extreme cases, QSI and FVI,
respectively, render the highest and lowest interference correlation caused by
the interferer locations.

Spatial-temporal SIR correlation affects the performance of many practical
systems, where the transmission performance depends on the SIR measured over
different space and time spots. Typical examples of these systems include the
following:

  * •

Multihop relaying: The end-to-end performance of a multihop relaying system
depends on the SIRs at the receivers at different hops. The SIRs at different
hops can be correlated due to spatial-temporal interference correlation (e.g.,
common interferers exist from one transmission to another).

  * •

Retransmission: The reliability of a retransmission scheme or a multi-packet
transmission scheme depends on the SIR at the receiver during multiple
transmissions. The SIRs from different transmissions can are correlated if
they are subject to the interference from the same group of transmitters.

  * •

Mobile networks: For a mobile user, transmissions could occur at different
spatial locations and times. The correlation between the SIRs at the receiver
for different transmissions are dependent on the mobility.

The SIR correlation affects the performance metrics such as temporal joint
success probability (JSP) [77], end-to-end success probability [78], the local
delay [29], and joint SIR meta distribution [43].

###  II-C Metrics for Spatial-Temporal Performance Analysis

######  Definition 1.

(Success Probability): The (average) success probability of a user can be
defined as the probability that the user’s received SIR is greater than a
threshold denoted by $\theta$ [66], i.e.,

$\displaystyle P_{\rm
s}=\bar{F}_{\mathrm{SIR}}(\theta)\triangleq\mathbb{P}[\mathrm{SIR}>\theta],\quad\theta\in\mathbb{R}^{+}.$
(2)

where $\bar{F}_{\mathrm{SIR}}$ represents the complementary CDF of SIR.

######  Definition 2.

(Moments of Conditional Success Probability given the Point Process): Let
$P_{\rm
s\mid\Phi}(\theta)=\mathbb{P}\big{[}\mathrm{SIR}>\theta\mid\Phi\big{]}$
represent the CSP given the point process $\Phi$ (abbreviated as CSPΦ), which
is averaged over the fading of all the links and the random channel access (if
applicable). The $b$-th moment of $P_{\rm s|\Phi}$ is defined as [79]

$\displaystyle\mathcal{M}_{P_{\rm
s}}(b)\triangleq\mathbb{E}\Big{[}\big{(}P_{\rm
s\mid\Phi}(\theta)\big{)}^{b}\Big{]},\quad b\in\mathbb{C}.$ (3)

Note that the first moment of the CSPΦ is the average success probability
defined in (2), i.e., $\mathcal{M}_{P_{\rm s}}(1)\equiv P_{\rm s}$, and the
variance is $\mathbb{V}(P_{\rm s})=\mathcal{M}_{P_{\rm
s}}(2)-\mathcal{M}^{2}_{P_{\rm s}}(1)$. Besides, the mean local delay, defined
as the average number of transmission attempts to accomplish a success [80],
is given by $\mathcal{M}_{P_{\rm s}}(-1)$. It is worth mentioning that for
static random networks and $b\in\mathbb{N}$, the $b$-th moment of the CSPΦ is
equivalent to the JSP of $b$ transmissions of the same link [81] and JSP that
the $b$ antennas of a multiple-antenna receiver all succeed in reception
(i.e., SIR exceeds $\theta$) [82].

######  Definition 3.

(SIR Meta Distribution): The SIR meta distribution is the complementary CDF of
the CSPΦ, defined as [44, 79]

$\displaystyle\bar{F}_{P_{\rm s}}(\theta,s)\triangleq\mathbb{P}\big{[}P_{\rm
s\mid\Phi}(\theta)>s\big{]},\,\,\,s\in[0,1],$ (4)

where $s$ represents the target success probability.

In ergodic point processes, the SIR meta distribution indicates the fraction
of links that can achieve successful transmission with probabilities greater
than $x$ in any realization of $\Phi$. Compared to the average success
probability defined in (2), the SIR meta distribution also characterizes the
variability of link success probabilities. For example, the inverse function
of the SIR meta distribution $\bar{F}^{-1}_{P_{\rm s}}(p)$, $p\in[0,1]$,
yields the success probability $x$ that a fraction $1-p$ of the links achieve
while the rest do not.

Example: With $\theta=1$ and $s=0.9$, $\bar{F}_{P_{\rm s}}(1,0.9)$ represents
the fraction of links/transmissions that achieve a receive SIR greater than 1
with probability at least 0.9.

######  Definition 4.

(SIR Gain): The SIR gain is the horizontal gap between the complementary CDF
of two SIR distributions. Evaluated at the target success probability $P_{\rm
t}$, the SIR gain is defined as [83, Eq. (1)]

$\displaystyle G(P_{\rm t})\triangleq\frac{\bar{F}^{-1}_{\mathrm{SIR}_{\rm
tm}}(P_{\rm t})}{\bar{F}^{-1}_{\mathrm{SIR}_{\rm rm}}(P_{\rm t})},\quad P_{\rm
t}\in(0,1),$ (5)

where $\bar{F}^{-1}_{\mathrm{SIR}}$ represents the inverse function of the
complementary CDF of the SIR, and $\mathrm{SIR}_{\rm tm}$ and
$\mathrm{SIR}_{\rm rm}$ denote the SIRs of a target model and a reference
model, respectively.

The SIR gain can be used to quantify the impact of a target model on the SIR
distribution w.r.t. that of a reference model. An SIR gain greater or less
than 1 indicates that the target model can achieve the same success
probability with a larger or smaller SIR threshold, respectively, than the
reference model.

In wireless networks, the SIR gain is usually not sensitive to the target
success probability to be evaluated [83, 84]. Therefore, the SIR gain can be
approximated by the asymptotic SIR gain evaluated in the high-reliability
regime, i.e., $P_{\rm t}\to 1$ or $\theta\to 0$, defined as

$\displaystyle G_{0}\triangleq\lim_{P_{\rm t}\to 1}G(P_{\rm t}),$ (6)

whenever the limit exists.

Note that a necessary and sufficient condition for the asymptotic SIR gain to
exist is that the slopes of the two CDFs of the SIR are asymptotically the
same as $\theta\to 0$ [83].

######  Definition 5.

(Joint Success Probability): Let
$\mathbf{x}=\\{x_{k}\\}_{k\in\\{1,\ldots,K\\}}\in(\mathbb{R}^{d})^{K}$ and
$\mathbf{t}=\\{t_{k}\\}_{k\in\\{1,\ldots,K\\}}\in\mathbb{N}^{K}$ denote
deterministic vectors of locations and time instances, respectively, and
$\textup{SIR}_{k}$ denote the SIR measured at location $x_{k}$ and time
$t_{k}$. The JSP is defined as the probability that $\textup{SIR}_{i}$ is
greater than the corresponding SIR threshold $\theta_{k}$ for all
$k\in\\{1,\ldots,K\\}$. Given the locations $\mathbf{x}$, times $\mathbf{t}$,
and the target SIR thresholds $\mbox{\boldmath$\theta$
}=\\{\theta_{k}\\}_{k\in\\{1,2,\ldots,K\\}}\in(\mathbb{R}^{+})^{K}$, the JSP
is defined as

$\displaystyle\mathcal{J}_{K}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})$
$\displaystyle\triangleq\mathbb{P}[\mathrm{SIR}_{1}>\theta_{1},\mathrm{SIR}_{2}>\theta_{2},\ldots,\mathrm{SIR}_{K}>\theta_{K}],$
(7)

where $\mathrm{SIR}_{k}$ denotes the SIR of the $k$-th transmission.

The SIR corresponding to the different transmissions can exhibit spatial
and/or temporal correlation, the effect of which will be reflected in the JSP.
The JSP can be for temporal, spatial, or spatial-temporal transmission events.
In particular, we have

  * •

Temporal JSP of multiple transmission attempts occurring at the same location
but different time instances (e.g., multiple transmissions [81] and
retransmissions [85] by a transmitter);

  * •

Spatial JSP of multiple transmission attempts occurring at the same time but
different locations (e.g., joint uplink and downlink transmissions [87, 86]);

  * •

Spatial-temporal JSP of multiple transmission events occurring at different
space and time intervals (e.g., for the same mobile user [77]).

For a multihop communication scenario, the probability that the transmissions
at each hop are successful for a given packet is referred to as the end-to-end
success probability.

######  Definition 6.

(Conditional Success Probability): The CSP is the probability of achieving a
successful transmission given that $K-1$ ($K\geq 2$) such events have
occurred. Given the locations $\mathbf{x}$, times $\mathbf{t}$, and the target
SIR thresholds $\theta$, the CSP is given by

$\displaystyle\mathcal{C}_{K}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})=\frac{\mathcal{J}_{K}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})}{\mathcal{J}_{K-1}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})},$
(8)

where $\mathcal{J}_{K}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})$ is
defined in (5).

The CSP reveals the dependence between two successful transmission events. If
the two events are positively correlated (spatially or temporally), one has
$\mathcal{C}_{2}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})>\mathcal{J}_{1}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})$
[81]. Moreover, if the events are independent, one has
$\mathcal{C}_{2}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})=\mathcal{J}_{1}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})=\sqrt{\mathcal{J}_{2}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})}$.
Similar to the definition of the JSP, the CSP of spatial events, temporal
events, and spatial-temporal events are referred to as spatial CSP, temporal
CSP, and spatial-temporal CSP, respectively.

######  Definition 7.

(Product SIR Meta Distribution): Let
$\mathcal{J}_{K|\Phi}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})\triangleq\mathbb{P}[\mathrm{SIR}_{1}>\theta_{1},\ldots,\mathrm{SIR}_{K}>\theta_{K}\mid\Phi]$
denote the JSP of $K$ transmissions given the point process. The product SIR
meta distribution is defined as the complementary CDF of the JSP at locations
$\mathbf{x}$ and times $\mathbf{t}$ given the point process $\Phi$ [43, Eq.
(11)], i.e.,

$\displaystyle\bar{F}_{\mathcal{J}_{K}}\\!(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t},s)\triangleq\mathbb{P}\Big{[}\mathcal{J}_{K|\Phi}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t})>s\Big{]},\,s\\!\in\\![0,1].$
(9)

Example: Let $K=2$, ${\bm{\theta}}=\\{\theta_{1},\theta_{2}\\}$,
$\mathbf{x}=\\{x_{1},x_{2}\\}$, $\mathbf{t}=\\{t_{1},t_{2}\\}$. Considering a
stationary and ergodic point process $\Psi=\\{u_{j}\\}_{j\in\mathbb{N}}$
representing the users, the product SIR meta distribution measures the
fraction of users for which the joint probability that the SIR at their
location $u$ at time $t_{1}$ exceeds $\theta_{1}$ and the SIR at location
$u+(x_{2}-x_{1})$ at time $t_{2}$ exceeds $\theta_{2}$ is larger than $s$.

######  Definition 8.

(Joint SIR Meta Distribution): Let $P^{(k)}_{\rm
s|\Phi}(\theta_{k})=\mathbb{P}\big{[}\textup{SIR}_{k}>\theta_{k}\mid\Phi\big{]}$
represent the CSPΦ at $x_{k}$. The joint SIR meta distribution is defined as
the joint distribution of CSPΦ at locations $\mathbf{x}$ and times
$\mathbf{t}$ [43, Eq. (10)], i.e.,

$\displaystyle\bar{F}^{(K)}_{P_{\rm
s}}(\mbox{\boldmath$\theta$},\mathbf{x},\mathbf{t},\mathbf{s})$
$\displaystyle\triangleq\mathbb{P}\bigg{[}\bigcap^{K}_{k=1}\big{\\{}P^{(k)}_{s|\Phi}(\theta_{k})>s_{k}\big{\\}}\bigg{]},$
(10)

where $\mathbf{s}=\\{s_{1},s_{2},\ldots,s_{K}\\}\in[0,1]^{K}$ is the vector of
target success probabilities corresponding to the locations $\mathbf{x}$.

Example: Let $K=2$, $\bm{\theta}=\\{\theta_{1},\theta_{2}\\}$,
$\mathbf{x}=\\{x_{1},x_{2}\\}$, $\mathbf{t}=\\{t_{1},t_{2}\\}$,
$\mathbf{s}=\\{s_{1},s_{2}\\}$. Considering a stationary and ergodic point
process $\Psi=\\{u_{j}\\}_{j\in\mathbb{N}}$ representing the users, the joint
SIR meta distribution measures the fraction of users that meet the following
conditions: 1) the probability that the SIR at the user location $u$ at time
$t_{1}$ exceeds $\theta_{1}$ is larger than $s_{1}$ and 2) the probability
that the SIR at location $u+(x_{2}-x_{1})$ at time $t_{2}$ exceeds
$\theta_{2}$ is larger than $s_{2}$.

######  Definition 9.

(Interference Correlation Coefficient): Let $I^{(t)}_{x}$ denote the
aggregated interference received at location $x$ at time $t$. The correlation
degree of interference at two locations and times can be quantified by the
Pearson correlation coefficient defined as [23, Eq. (2)]

$\displaystyle\zeta_{t_{1},t_{2}}(u_{1},u_{2})\triangleq$
$\displaystyle\frac{\mathbb{E}\big{[}I^{(t_{1})}_{u_{1}}I^{(t_{2})}_{u_{2}}\big{]}-\mathbb{E}\big{[}I^{(t_{1})}_{u_{1}}\big{]}\mathbb{E}\big{[}I^{(t_{2})}_{u_{2}}\big{]}}{\sqrt{\mathbb{E}\big{[}\big{(}I^{(t_{1})}_{u_{1}}\big{)}^{\\!2}\big{]}\\!-\\!\mathbb{E}\big{[}I^{(t_{1})}_{u_{1}}\big{]}^{\\!2}}\sqrt{\mathbb{E}\big{[}\big{(}I^{(t_{2})}_{u_{2}}\big{)}^{\\!2}\big{]}\\!-\\!\mathbb{E}\big{[}I^{(t_{2})}_{u_{2}}\big{]}^{\\!2}}},$
(11)

where the numerator computes the covariance of $I^{(t_{1})}_{u_{1}}$ and
$I^{(t_{2})}_{u_{2}}$ and the denominator is the product of the standard
deviations of $I^{(t_{1})}_{u_{1}}$ and $I^{(t_{2})}_{u_{2}}$.

The interference correlation coefficient is a statistical measure that
quantifies the extent to which the interferences at $u_{1}$ and $u_{2}$ are
associated. The value of the interference correlation coefficient ranges
between 0 and 1. The coefficients of 0 and 1, respectively, indicate no linear
relationship and full correlation between the interferences at $u_{1}$ and
$u_{2}$.

Note that when the interferers are motion-invariant, one has
$I^{(t_{1})}_{o}\overset{(d)}{=}I^{(t_{1})}_{u}$. In this case, $\zeta(\|u\|)$
can be simplified to [22, Eq. (12)]

$\displaystyle\zeta_{t_{1},t_{2}}(\|u_{1}-u_{2}\|)\triangleq\frac{\mathbb{E}\Big{[}I^{(t_{1})}_{u_{1}}I^{(t_{2})}_{u_{2}}\Big{]}-\mathbb{E}\Big{[}I^{(t_{1})}_{u_{1}}\Big{]}^{\\!2}}{\mathbb{E}\Big{[}\big{(}I^{(t_{1})}_{u_{1}}\big{)}^{2}\Big{]}\\!-\\!\mathbb{E}\Big{[}I^{(t_{1})}_{u_{1}}\Big{]}^{2}}.$
(12)

######  Definition 10.

(Interference Coherence Time): The interference coherence time is defined as
the minimum time lag such that the interference correlation coefficient is
below a threshold $\zeta_{\rm th}$ [88, Eq. (35)], i.e.,

$\displaystyle\tau_{\rm
ct}\\!\triangleq\\!\min\big{\\{}\tau=t_{2}\\!-\\!t_{1}\in\mathbb{N}\mid\zeta_{t_{1},t_{2}}\leq\zeta_{\rm
th}\big{\\}},\,\zeta_{\rm th}\in\mathbb{R}^{+}.$ (13)

Fig. 10 summarizes the above performance metrics for spatial-temporal
performance analysis.

Figure 10: Metrics for spatial-temporal performance analysis.

##  III Spatial Point Process Models

This section formally defines some common point processes, also referred to as
random point fields, and illustrates how the spatial distribution of the
random points affects the network performance. To evaluate the impacts of
independent, attractive and repulsive spatial configurations, we choose three
representative point processes, namely, the Poisson point process (PPP), the
Matérn cluster process (MCP), and the $\beta$-Ginibre point process (GPP), due
to their tractability.

  * •

In a PPP, each point is located independently from the others.

  * •

In an MCP, the locations of the random points have a propensity to be
clustering.

  * •

In a $\beta$-GPP, the random points are scattered with repulsion.

This section considers both ad hoc and downlink cellular networks modeled
based on the above point processes and analyzes the interference correlation
coefficient, success probability, SIR meta distribution and SIR gain.

###  III-A System Models

####  III-A1 System Configurations

We focus on analyzing the performance of a target in both ad hoc and downlink
cellular networks. The target receiver is considered to be at the origin
$o\in\mathbb{R}^{2}$ and attempts to decode the transmitted signals from the
associated transmitter subject to the aggregate interference.

  * •

Ad hoc networks: The target receiver at $o$ is served by the transmitter
located at $x_{\rm t}\in\mathbb{R}^{2}$, where $\|x_{\rm t}\|=r_{\rm t}$. The
target link is impaired by the interference from a random field of interferers
modeled by a stationary point process $\Phi$ of intensity $\lambda$. $\Phi$
does not contain the serving transmitter at $x_{\rm t}$.

  * •

Downlink cellular networks: The transmitters (i.e., BSs) form a stationary
point process $\Phi$ with intensity $\lambda$. The users are located according
to a stationary point process independent of the BS process. Each user is
served by the nearest BS in $\Phi$.

In the networks considered, the transmitters stay active with unit transmit
power, i.e., there is no MAC scheme. All the transmitters and receivers are
equipped with one antenna. The system employs universal frequency reuse. The
channels of the links experience i.i.d. block Rayleigh fading and power-law
path loss, i.e., $d^{-\alpha}$, where $d$ and $\alpha>2$ are the link distance
and the path-loss exponent, respectively.

For the convenience of notation, the points in $\Phi$ are assumed to be
ordered from nearest to farthest to the origin, i.e., $\|x_{j}\|<\|x_{j+1}\|$.
Subsequently, the set of interferers in the ad hoc networks and downlink
cellular networks is $\Phi^{!}=\Phi=\\{x_{j}\\}_{j\in\mathbb{N}}$ and
$\Phi^{!}=\Phi\backslash\\{x_{1}\\}$, respectively. The distance from the
$j$-th nearest point in $\Phi$ to the origin is denoted as $r_{j}=\|x_{j}\|$.

The SIR at the target receiver in an ad hoc and downlink cellular network is
given by, respectively, as

$\displaystyle\eta=\frac{h_{\rm t}\|x_{\rm
t}\|^{-\alpha}}{\sum_{j\in\mathbb{N}}h_{j}\|x_{j}\|^{-\alpha}}$ (14)

and

$\displaystyle\eta=\frac{h_{1}\|x_{1}\|^{-\alpha}}{\sum^{\infty}_{j=2}h_{j}\|x_{j}\|^{-\alpha}},$
(15)

where $h_{\rm t}$ and $h_{j}$ denote the power fading coefficients from the
transmitters at $x_{\rm t}$, $x_{j}$ to the target receiver, which are
exponential random variables with unit mean, i.e., $h_{\rm
t},h_{j}\sim\mathcal{E}(1)$. It is noted that (14) represents the SIR of a
specific receiver instead of the typical receiver in ad hoc networks, while
(15) is the SIR of a user at the origin that, upon averaging over the base
station and user point processes, becomes the typical user in the downlink
cellular network, for any stationary point process of users that is
independent of the base station point process.

####  III-A2 Spatial Configurations

We consider the homogeneous PPP, MCP, and $\beta$-GPP, formally defined as
follows.

######  Definition 11.

(Homogeneous Poisson point process): A point process
$\Phi=\\{x_{j}\\}_{j\in\mathbb{N}}\subset\mathbb{R}^{d}$ is a homogenous PPP
if it satisfies two conditions: i) for any compact set
$\mathcal{B}\subset\mathbb{R}^{d}$, the number of points inside it follows a
Poisson distribution with average value being $\lambda|\mathcal{B}|$; and ii)
disjoint sets are independent in terms of the number of points inside them.

Properties:

1) Density: For a homogeneous PPP, the first moment density (or first-order
density) and second moment density (or second-order product density) are
given, respectively, as [89]

$\displaystyle\rho^{(1)}(x)=\lambda,$ (16)

and

$\displaystyle\rho^{(2)}(x,y)=\big{(}\rho^{(1)}(x)\big{)}^{2}=\lambda^{2}.$
(17)

The $n$-th moment density is the density pertaining to the $n$-th order
factorial moment measure, which, for $n>1$, indicate the spatial correlation.
For example, $\rho^{(1)}(x)\mathrm{d}x$ measures the probability of having a
point at $x$ in some infinitesimal region $\mathrm{d}x$.

2) Contact Distance Distribution: Contact distance refers to the distance
between a reference location and the nearest point in $\Phi$ [90]. The
probability density function (PDF) and CDF of the contact distance in a
homogeneous PPP are given, respectively, as [91]

$\displaystyle f_{r_{1}}(r)=2\pi\lambda\exp(-\lambda\pi r^{2})$ (18)

and

$\displaystyle F_{r_{1}}(r)=\exp(-\lambda\pi r^{2}).$ (19)

3) Joint Distance Distribution: The joint PDF of the distances to $n$ nearest
points is [92, Eq. (30)]

$\displaystyle
f_{r_{1},r_{2},\ldots,r_{n}}(x_{1},x_{2},\ldots\\!,x_{n})\\!=\\!e^{-\lambda\pi
x^{2}_{n}}(2\lambda\pi)^{n}x_{1}x_{2}\ldots x_{n}.$ (20)

4) Distance Ratio Distribution: Let
$\varrho_{j}=\frac{r_{1}}{r_{j}},j\in\mathbb{N}$, denote the distance ratio of
the nearest point to the $j$-th nearest point. The CDF and PDF of
$\varrho_{j}$ are given, respectively, by [93, Lemma 3]

$\displaystyle F_{\varrho_{j}}(\varrho)=1-(1-\varrho^{2})^{j-1},\hskip
28.45274pt\varrho\in[0,1]$ (21)

and

$\displaystyle
f_{\varrho_{j}}(\varrho)=2(j-1)\varrho(1-\varrho^{2})^{j-2},\hskip
7.11317pt\varrho\in[0,1].$ (22)

5) Probability Generating Functional (Product Functional): For
$\upsilon(x)\in[0,1]$ and
$\int_{\mathbb{R}^{2}}(1-\upsilon(x))\mathrm{d}x<\infty$, the probability
generating functional (PGFL) for the PPP is given by [3, Eq. (4.8)]

$\displaystyle\mathbb{E}\bigg{[}\prod_{j\in\mathbb{N}}\upsilon(x_{j})\bigg{]}=\exp\bigg{(}-\lambda\int_{\mathbb{R}^{d}}\big{(}1-\upsilon(x)\big{)}\mathrm{d}x\bigg{)}.$
(23)

According to Slivnyak’s theorem [3, Theorem 8.10], the reduced Palm
distribution of the PPP is the same as its ordinary distribution. Therefore,
the conditional PGFL for the PPP can also be expressed as (23).

Let $\Phi^{\rm
R}=\big{\\{}x\in\Phi\backslash\\{x_{1}\\}:\|x_{1}\|/\|x\|\big{\\}}\subset(0,1)$
denote the relative distance process (RDP) of a PPP $\Phi$. The PGFL of
$\Phi^{\rm R}$ is given by [84, Lemma 1]

$\displaystyle\mathbb{E}\bigg{[}\prod_{\varrho\in\Phi^{\rm
R}}\upsilon(\varrho)\bigg{]}=\frac{1}{1+2\int^{1}_{0}\big{(}1-\upsilon(r)\big{)}r^{-3}\mathrm{d}r},$
(24)

for functions $\upsilon(r)\in[0,1]$ such that
$\int_{\mathbb{R}^{+}}(1-\upsilon(r))r^{-3}\mathrm{d}r$ is finite.

6) Sum Functional: For any measurable function $g\geq 0$ on $\mathbb{R}^{d}$,
the sum functional for a stationary point process is given by Campbell’s
Theorem [3, Theorem 4.1] as

$\displaystyle\mathbb{E}\bigg{[}\sum_{j\in\mathbb{N}}g(x_{j})\bigg{]}=\lambda\int_{\mathbb{R}^{d}}g(x)\mathrm{d}x.$
(25)

7) Sum-Product Functional: Let
$\bm{p}\triangleq(p_{1},p_{2},\ldots,p_{q})\in\mathbb{N}^{q}$,
$\|\bm{p}\|_{1}\triangleq\sum^{q}_{k=1}p_{k}$ and
$\\{M_{j}\\}_{j\in\mathbb{N}}$ denote a set of i.i.d. random marks associated
with the points in $\Phi=\\{x_{j}\\}_{j\in\mathbb{N}}$. If
$g_{k}(x)\in[0,\infty),1\leq k\leq q$, and $\upsilon(x)\in[0,1]$ are
measurable functions for all $x\in\mathbb{R}^{d}$, the sum-product functional
for the PPP with $\|\bm{p}\|_{1}>0$ is given by [42, Theorem 1]

$\displaystyle\mathbb{E}\Bigg{[}\prod^{q}_{k=1}\bigg{(}\sum_{j\in\mathbb{N}}g(x_{j},M_{j})\bigg{)}^{p_{i}}\prod_{j\in\mathbb{N}}\upsilon(x_{j},M_{j})\Bigg{]}$
$\displaystyle=\exp\bigg{(}-\lambda\int_{\mathrm{R}^{d}}\Big{(}1-\mathbb{E}_{M_{j}}\Big{[}\upsilon(x,M_{j})\Big{]}\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\hskip
14.22636pt\times\sum^{\|\bm{p}\|_{1}}_{l=1}\sum_{N\in\mathcal{N}^{p}_{l}}\frac{D_{N}}{l!}\prod^{l}_{k=1}\lambda\int_{\mathbb{R}^{d}}\mathbb{E}_{\bm{M}}\bigg{[}\upsilon(x,M_{j})$
$\displaystyle\hskip
14.22636pt\times\prod^{q}_{i=1}f^{n_{ki}}_{i}(x,M_{j})\bigg{]}\mathrm{d}x,$
(26)

where $D_{N}\triangleq\prod^{q}_{k=1}\frac{p_{k}!}{\prod^{l}_{i=1}m_{ik}!}$
and $\mathcal{N}^{\bm{p}}_{l}\subset\mathbb{N}^{q\times l}$ is the class of
all $q\times l$ matrices with the columns $\|n_{\cdot i}\|_{1}>0$, $\forall
i\in\\{1,2,\ldots,l\\}$ and the rows $\|n_{k\cdot}\|_{1}=p_{k}$, $\forall
k\in\\{1,2\dots,q\\}$.

In the special case $q=1$ and $p=1$, the sum-product functional for the PPP is
given by the Campbell-Mecke Theorem [3, Theorem 8.2] as

$\displaystyle\mathbb{E}\Bigg{[}\sum_{j\in\mathbb{N}}g(x_{j},M_{j})\prod_{j\in\mathbb{N}}\upsilon(x_{j},M_{j})\Bigg{]}$
$\displaystyle=\exp\bigg{(}-\lambda\int_{\mathbb{R}^{d}}\Big{(}1-\mathbb{E}_{M_{j}}\Big{[}\upsilon(x,M_{j})\Big{]}\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\hskip
28.45274pt\times\lambda\int_{\mathbb{R}^{d}}\mathbb{E}_{M_{j}}\Big{[}g(x,M_{j})\upsilon(x,M_{j})\Big{]}\mathrm{d}x.$
(27)

######  Definition 12.

(Matérn cluster process): The MCP $\Phi^{\rm M}$ is a doubly Poisson cluster
process constructed from a parent PPP $\Phi_{\rm
p}=\\{x_{j}\\}_{j\in\mathbb{N}}$ with intensity $\lambda_{\rm p}$ with each
point of $\Phi_{\rm p}$ substituted by a daughter cluster consisting of a PPP
with an average number of points $\bar{c}$ within a disk of radius $R_{\rm d}$
centered at that point.

Properties:

1) Density: For an MCP, the first moment and second moment densities are
given, respectively, by [3]

$\displaystyle\rho^{(1)}(x)=\lambda=\lambda_{\rm p}\bar{c},$ (28)
$\displaystyle\rho^{(2)}(x,y)=\lambda^{2}_{\rm p}\bar{c}^{2}+\lambda_{\rm
p}\frac{\bar{c}^{2}}{\pi^{2}R^{4}_{\rm d}}A_{R_{\rm d}}(\|x-y\|),$ (29)

$A_{R_{\rm d}}$ represents the area of the intersection of two disks with
radius $R_{\rm d}$ at a distance $r>0$ given as [3, Eq. (6.4)]

$\displaystyle A_{R_{\rm d}}(r)=\begin{dcases}2R^{2}_{\rm d}\arccos(r/2R_{\rm
d})-r\sqrt{R^{2}_{\rm d}-r^{2}/4},\quad\\\ \hskip 108.12047pt0\leq r\leq
2R_{\rm d},\\\ 0\hskip 102.42992pt\textup{otherwise.}\end{dcases}$ (30)

2) Daughter Point Distribution: Each daughter point is located uniformly
within a disk of radius $R$ around the origin with the PDF given by [9, Eq.
(5)]

$\displaystyle f^{\rm M}(y)=\begin{cases}\frac{1}{\pi R^{2}_{\rm
d}},\quad\text{if}\,\,\|y\|\leq R_{\rm d}\\\
0,\quad\quad\text{otherwise}.\end{cases}$ (31)

3) Contact Distance Distribution: The PDF and CDF of the contact distance in
the MCP are given as (32) and (33), respectively [90].

$\displaystyle f^{\rm M}_{r_{1}}(r|x)=\begin{dcases}\frac{2r}{R^{2}_{\rm
d}},\hskip 110.96574pt\textup{if}\,\,0\leq r\leq R_{\rm
d}-x\,\,\textup{for}\,\,\|x\|<R_{\rm d}\\\ \frac{2r}{\pi R^{2}_{\rm
d}}\cos^{-1}\Big{(}\frac{r^{2}+x^{2}-R^{2}_{\rm d}}{2rx}\Big{)},\hskip
5.69054pt\textup{if}\,\,R_{\rm d}-\|x\|\leq r\leq R_{\rm
d}-\|x\|\,\,\textup{for}\,\,\|x\|<R_{\rm
d},\,\,\textup{and}\,\,\textup{if}\,\,\|x\|>R_{\rm d},\end{dcases}$ (32)
$\displaystyle F^{\rm M}_{r_{1}}(r)$
$\displaystyle=1-\exp\Bigg{(}-\lambda_{\rm p}\bigg{(}\int_{\mathbb{B}(0,R_{\rm
d})}\bigg{[}1-\exp\bigg{(}-\bar{c}\bigg{(}\int^{\min(r,R_{\rm
d}-x)}_{0}\frac{2y}{R^{2}_{\rm d}}\mathrm{d}y$ $\displaystyle\hskip
28.45274pt+\int^{\min(r,R_{\rm d}+x)}_{\min(r,R_{\rm d}-x)}\frac{2y}{\pi
R^{2}_{\rm d}}\cos^{-1}\Big{(}\frac{y^{2}+x^{2}-R^{2}_{\rm
d}}{2yx}\Big{)}\mathrm{d}y\bigg{)}\bigg{)}\bigg{]}\mathrm{d}x$
$\displaystyle\hskip
56.9055pt+\int_{\mathbb{R}^{2}\backslash\mathbb{B}(0,R_{\rm
d})}\bigg{[}1-\exp\bigg{(}-\bar{c}\int^{\min(r,R_{\rm d}+x)}_{\min(r,R_{\rm
d}-x)}\frac{2y}{\pi R^{2}_{\rm d}}\cos^{-1}\Big{(}\frac{y^{2}+x^{2}-R^{2}_{\rm
d}}{2yx}\mathrm{d}y\Big{)}\bigg{)}\bigg{]}\mathrm{d}x\bigg{)}\Bigg{)},$ (33)

4) Probability Generating Functional: The PGFL for an MCP is given as follows
[9]:

$\displaystyle\mathbb{E}\bigg{[}\prod_{j\in\mathbb{N}}\upsilon(x_{j})\bigg{]}$
$\displaystyle=\exp\Bigg{(}\\!-\lambda\int_{\mathbb{R}^{2}}\bigg{[}1-M\bigg{(}\int_{\mathbb{R}^{2}}\upsilon(x+y)f^{\rm
M}(y)\mathrm{d}y\bigg{)}\bigg{]}\mathrm{d}x\Bigg{)}\mathrm{d}y,$ (34)

where $M(t)=e^{-\bar{c}(1-t)}$ is the moment generating function of the
representative cluster in an MCP.

Let $\mathbb{E}^{!}_{o}[\cdot]$ denote the expectation operation based on the
reduced Palm measure [89] which takes the expectation for a point process
conditioned at a point of the process at $o$ without including the point. The
conditional PGFL of an MCP is [9, Lemma 1]

$\displaystyle\mathbb{E}^{!}_{o}\bigg{[}\prod_{j\in\mathbb{N}}\upsilon(x_{j})\bigg{]}$
$\displaystyle\\!=\exp\\!\Bigg{(}\\!-\lambda_{\rm
p}\int_{\mathbb{R}^{2}}\\!\\!\bigg{[}1-M\bigg{(}\int_{\mathbb{R}^{2}}\\!\upsilon(x+y)f(y)\mathrm{d}y\bigg{)}\\!\bigg{]}\mathrm{d}x\\!\Bigg{)}$
$\displaystyle\hskip 62.59605pt\times\int_{\mathbb{R}^{2}}G^{\rm M}_{\rm
d}\Big{(}\upsilon(x-y)\Big{)}f^{\rm M}(y)\mathrm{d}y,$ (35)

where $G^{\rm M}_{\rm d}$ is the PGFL for the representative cluster given by

$\displaystyle G^{\rm M}_{\rm
d}(\upsilon)=M\bigg{(}\int_{\mathbb{R}^{2}}\upsilon(x)f^{\rm
M}(x)\mathrm{d}x\bigg{)}.$ (36)

######  Definition 13.

($\beta$-Ginibre point process): A $\beta$-GPP $\Phi^{\rm
G}_{\beta}=\\{x_{j}\\}_{j\in\mathbb{N}}$ is a determinantal point process with
the kernel555The kernel represents the interaction force among the points of
the process. given by [54]

$\displaystyle\mathcal{K}_{\beta,\lambda}(x,y)=\lambda
e^{-\frac{\pi\lambda|x-y|^{2}}{2\beta}},\quad x,y\in\mathbb{C},$ (37)

w.r.t. the Lebesgue measure on $\mathbb{C}$ [94].

Properties:

1) Density: For a $\beta$-GPP, the first moment density and the second moment
density of the $\beta$-GPP are given, respectively, by [54]

$\displaystyle\rho^{(1)}(x)=\det[K_{\beta,\lambda}(x,x)]=\lambda,$ (38)
$\displaystyle\quad\rho^{(2)}(x,y)=\det\left[\begin{array}[]{cc}\mathcal{K}_{\beta,\lambda}(x,\bar{x})&\mathcal{K}_{\beta,\lambda}(x,\bar{y})\\\
\mathcal{K}_{\beta,\lambda}(\bar{x},y)&\mathcal{K}_{\beta,\lambda}(y,\bar{y})\end{array}\right]$
(41)
$\displaystyle=\lambda^{2}\bigg{(}1-\exp\bigg{(}\\!\\!-\\!\frac{\pi\lambda|x-y|^{2}}{\beta}\bigg{)}\bigg{)}.$
(42)

2) Link Distance Property: Let $\lambda$ represents the intensity of
$\Phi^{\rm G}_{\beta}$. Let $\\{Q_{j}\\}_{j\in\mathbb{N}}$ be a set of
independent gamma random variables with PDF

$\displaystyle
f_{Q_{j}}(q)=\frac{q^{j-1}e^{-\frac{\pi\lambda}{\beta}q}}{\Big{(}\frac{\beta}{\pi\lambda}\Big{)}^{j}\Gamma(j)},$
(43)

i.e., $Q_{j}\sim\mathcal{G}(j,\beta/\pi\lambda)$. Then the set
$\\{\|x_{j}\|^{2}\\}_{j\in\mathbb{N}}$ is equivalent in distribution with the
set constructed by retaining each element from $\\{Q_{j}\\}_{j\in\mathbb{N}}$
independently with probability $\beta$ [96, Theorem 4.7.1].

(a) Realization of an MCP ($\lambda_{\rm p}=0.1$, $\bar{c}=5$ and $R_{\rm
d}=1$).  (b) Realization of a homogeneous PPP ($\lambda=0.5$).

(c) Realization of a $\beta$-GPP ($\lambda=0.5$, $\beta=0.5$).

(d) Pair correlation functions ($\lambda=1$, $R_{\rm d}=1$).

Figure 11: Illustration of spatial point processes.

Fig. 11(a), (b), (c), respectively, illustrate realizations of MCP, PPP, and
$\beta$-GPP. As shown by the realizations of MCP and $\beta$-GPP, the point
sets are attractive and repulsive, respectively.

The correlation between the spatial points can be measured by the pair
correlation function (PCF). For a point process $\Phi\subset\mathbb{R}^{d}$,
the PCF is defined as
$g(x,y)\triangleq\frac{\rho^{(2)}(x,y)}{\rho^{(1)}(x)\rho^{(2)}(y)}$. If
$\Phi$ is motion-invariant, the first moment density is the constant intensity
and the second moment density $\rho^{(2)}(x,y)$ only depends on the difference
$r=\|x-y\|$. Hence, the PCF can be expressed as
$g(r)=\frac{\rho^{(2)}(r)}{\lambda^{2}}$.

Let $\Phi(A)$ denote the number of points in $\Phi\cap A$. The PCF quantifies
the degree of correlation between the random variables $\Phi(A)$ and $\Phi(B)$
in a non-centered way. If $g(x,y)\equiv 1$ then for disjoint $A$ and $B$ the
covariance of $\Phi(A)$ and $\Phi(B)$ is zero, which means the two variables
are uncorrelated. For spatial point processes, the PCF describes how a point
is surrounded by others. The PCF equals one if the points are uncorrelated
(e.g., as in the PPP), and is greater (smaller) than 1 if the points are
attractive (repulsive).

The PCFs of the MCP, PPP and $\beta$-GPP are given, respectively, as [3, Page
153], [54]

$\displaystyle g(r)=\begin{dcases}1+\frac{\bar{c}A_{R_{\rm
d}}(r)}{\lambda\pi^{2}R^{4}_{\rm d}},\hskip 68.2866pt\textup{MCP}\\\ 1,\hskip
119.50157pt\textup{PPP}\\\ 1-\frac{\exp(-r^{2}/\beta)}{\pi\lambda},\hskip
51.21495pt\textup{$\beta$-GPP},\end{dcases}$ (44)

where $A_{R_{\rm d}}(r)$ is given in (30).

Properties:

1) For $r\geq 2R_{\rm d}$, for an MCP, the PCF is the same as that of the PPP,
because two points with a distance greater than $2R_{\rm d}$ must belong to
different clusters and thus are independent.

2) Given the intensity $\lambda$, the PCF of an MCP approaches that of the PPP
as $\bar{c}\to 0$, since the points are less likely to belong to the same
cluster.

3) For a PPP, the PCF is not affected by the intensity since the points are
independently distributed.

4) For a $\beta$-GPP, the PCF approaches that of the PPP as $\beta\to 0$ or
$\lambda\to\infty$.

Fig. 11(d) shows the PCFs of the spatial point processes and demonstrates the
properties discussed above.

####  III-A3 Validity

The validity of the point process approximation of real-world cellular
networks has been demonstrated in the literature. In particular, an early
study in [95] found that the success probability of an actual cellular network
(with a degree regularity) is lower-bounded by that of a Poisson network (with
complete randomness). More importantly, reference [83] revealed that adding a
horizontal SIR threshold shift from 0 dB to 3.4 dB to the success probability
in a Poisson network yields a tight approximation of that in any network with
a spatial layout from complete randomness to perfect regularity (i.e.,
triangular lattice). Hence, the analysis of an actual cellular network can be
performed based on that of a Poisson network with a horizontally shifted SIR
threshold calibrated to the regularity of the topology.

The validity of the $\beta$-GPP to model actual cellular networks has been
rigorously explored. The studies in [54, 97] show that the repulsion parameter
$\beta$ that adjusts the degree of repulsion among points can be numerically
fitted with the data set from actual network deployments. The simulations
demonstrate that the success probabilities from $\beta$-GPP analysis closely
approximates the ones from simulations with the actual deployment topologies.
Hence, it is valid to employ either a horizontally shifted PPP or the fitted
$\beta$-GPP to analyze actual cellular networks.

###  III-B Performance Analysis

####  III-B1 Spatial-Temporal Interference Correlation in Ad Hoc Networks

We first investigate the correlation of interference observed at two locations
$o$ and $u$ in two time slots $t_{1}$ and $t_{2}$, respectively, with the
interferers distributed as the three types of point processes considered. As
PPP, MCP and $\beta$-GPP are motion-invariant, the spatial-temporal
interference correlation can be measured by the Pearson correlation
coefficient defined in (12).

As can be seen from (12), the expectation and the second moments of
$I^{(t_{1})}_{o}$ and the mean product of $I^{(t_{1})}_{o}$ and
$I^{(t_{2})}_{u}$ are needed to quantify the interference correlation.
However, these two quantities do not exist because of the singularity of the
path-loss function $\ell(x)$. To cope with this issue, we follow the approach
in [9] by defining
$\ell_{\epsilon}(x)=\frac{1}{\epsilon+\|x\|^{\alpha}},\alpha>2,\epsilon>0$,
such that $\ell(x)=\lim_{\epsilon\to 0}\ell_{\epsilon}(x)$.

The expectation of $I^{(t_{1})}_{o}$ can be derived as

$\displaystyle\mathbb{E}\big{[}I^{(t_{1})}_{o}\big{]}$
$\displaystyle\overset{(a)}{=}\mathbb{E}\big{[}I_{o}\big{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sum_{j\in\mathbb{N}}\\!h_{j}\ell_{\epsilon}(x_{j})\bigg{]}$
$\displaystyle=\mathbb{E}\big{[}h\big{]}\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\rho^{(1)}(x)\mathrm{d}x$
$\displaystyle\overset{(b)}{=}2\pi\lambda\int^{\infty}_{0}\frac{x}{\epsilon+x^{\alpha}}\mathrm{d}x$
$\displaystyle\overset{(c)}{=}\delta\pi^{2}\lambda\epsilon^{\delta-1}\csc(\delta\pi),$
(45)

where $(a)$ follows since MCP, PPP, and $\beta$-GPP are all motion-invariant
and the superscript $(t_{1})$ is dropped for conciseness, $(b)$ applies the
conversion from Cartesian to polar coordinates, and $(c)$ substitutes
$\frac{2}{\alpha}$ with $\delta$.

If the densities of the interferers following a PPP, MCP, and $\beta$-GPP as
given in (16), (28), and (38), respectively, are the same, we can observe from
(45) that the three point processes cause the same mean interference at an
arbitrary location. This indicates that spatial attraction and repulsion do
not affect the first moment of the interference.

We then continue to derive the second moment of $I^{(t_{1})}_{o}$ as

$\displaystyle\mathbb{E}\Big{[}\Big{(}I^{(t_{1})}_{o}\Big{)}^{2}\Big{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\bigg{(}\sum_{j\in\mathbb{N}}h_{j}\ell_{\epsilon}(x_{j})\bigg{)}^{\\!2}\Bigg{]}$
$\displaystyle=\\!\mathbb{E}\Bigg{[}\sum_{j\in\mathbb{N}}h^{2}_{j}\ell^{2}_{\epsilon}(x_{j})+\sum^{j\neq
i}_{j,i\in\mathbb{N}}h_{j}h_{i}\ell_{\epsilon}(x_{j})\ell_{\epsilon}(x_{i})\Bigg{]}$
$\displaystyle=\mathbb{E}[h^{2}]\int_{\mathbb{R}^{2}}\ell^{2}_{\epsilon}(x)\rho^{(1)}(x)\mathrm{d}x$
$\displaystyle\hskip
28.45274pt+\mathbb{E}[h]^{2}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(y)\rho^{(2)}(x,y)\mathrm{d}x\mathrm{d}y$
$\displaystyle=2\delta\lambda\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)$
$\displaystyle\hskip
42.67912pt+\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(y)\rho^{(2)}(x,y)\mathrm{d}x\mathrm{d}y.$
(46)

By plugging in the second moment density $\rho^{(2)}(x,y)$ for the MCP, PPP,
and $\beta$-GPP given, respectively, in (29), (17), (42), we obtain the second
moment of the interference in (47),

$\displaystyle\mathbb{E}\bigg{[}\Big{(}I^{(t_{1})}_{o}\Big{)}^{2}\bigg{]}\overset{(a)}{=}\\!\begin{dcases}\bar{I}^{2}_{\textup{PPP}}+\frac{\lambda\bar{c}}{\pi^{2}R_{\rm
d}^{4}}\int_{\mathbb{R}^{2}}\\!\\!\bigg{(}\\!\\!\int_{\mathbb{R}^{2}}\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)A_{R_{\rm
d}}(\|x-y\|)\mathrm{d}x\bigg{)}\mathrm{d}y,\hskip 36.98857pt\textup{MCP}\\\
\bar{I}^{2}_{\textup{PPP}},\hskip 241.84842pt\textup{PPP}\\\
\bar{I}^{2}_{\textup{PPP}}-\lambda^{2}\\!\int_{\mathbb{R}^{2}}\\!\\!\\!\bigg{(}\\!\int_{\mathbb{R}^{2}}\\!\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)\bigg{(}\exp\bigg{(}\\!-\frac{\pi\lambda\|x-y\|^{2}}{\beta}\bigg{)}\\!\bigg{)}\mathrm{d}x\bigg{)}\mathrm{d}y,\quad\\!\\!\beta\textup{-GPP}.\end{dcases}$
(47)

where $\bar{I}^{2}_{\textup{PPP}}$ is

$\displaystyle\bar{I}^{2}_{\textup{PPP}}=2\delta\lambda\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)$
$\displaystyle\hskip
85.35826pt+\delta^{2}\pi^{4}\lambda^{2}\epsilon^{2\delta-2}\csc(\delta\pi)^{2}.$
(48)

Table V shows the variance of the interference with different fields of
interferers. It can be observed that the MCP and the $\beta$-GPP cause larger
and smaller interference variance than the PPP, respectively, as illustrated
in Fig. 12. Moreover, it can be found that, for an MCP, given the interference
density $\lambda=\lambda_{\rm p}\bar{c}$, the variance increases when the
points are more densely clustered (i.e., with smaller cluster density
$\lambda_{\rm p}$ and a larger average number of points $\bar{c}$ within each
cluster).

Figure 12: Interference variance in ad hoc networks with different fields of interferers ($\ell(x)=\frac{1}{1+x^{\alpha}}$ and $R_{\rm d}=1$). TABLE V: Variance of Interference Point Process |  $\mathbb{E}[I^{2}_{o}]-\mathbb{E}[I_{o}]^{2}$
---|---
MCP |  $\delta\pi^{2}\lambda(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)+\frac{\lambda\bar{c}}{\pi^{2}R_{\rm d}^{4}}\int_{\mathbb{R}^{2}}\big{(}\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(y)A_{R_{\rm d}}(\|x-y\|)\mathrm{d}x\big{)}\mathrm{d}y$
PPP |  $\delta\pi^{2}\lambda(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)$
$\beta$-GPP |  $\delta\pi^{2}\lambda(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)-\lambda^{2}\int_{\mathbb{R}^{2}}\Big{(}\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(y)\exp\big{(}-\pi\lambda\|x-y\|^{2}/\beta\big{)}\mathrm{d}x\Big{)}\mathrm{d}y$

Similarly, we have the mean product of $I^{(t_{1})}_{o}$ and
$I^{(t_{2})}_{u}$, $t_{1}\neq t_{2}$, as

$\displaystyle\mathbb{E}\Big{[}I^{(t_{1})}_{o}I^{(t_{2})}_{u}\Big{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\sum_{j\in\mathbb{N}}h^{(t_{1})}_{j}\ell_{\epsilon}(x_{j}-o)\sum_{i\in\mathbb{N}}h^{(t_{2})}_{i}\ell_{\epsilon}(x_{i}-u)\Bigg{]}$
$\displaystyle=\\!\mathbb{E}\Bigg{[}\sum_{j\in\mathbb{N}}h^{(t_{1})}_{j}h^{(t_{2})}_{j}\ell_{\epsilon}(x_{j})\ell_{\epsilon}(x_{j}-u)$
$\displaystyle\hskip 28.45274pt+\\!\sum^{j\neq
i}_{j,i\in\mathbb{N}}h^{(t_{1})}_{j}h^{(t_{2})}_{i}\ell_{\epsilon}(x_{j})\ell_{\epsilon}(x_{i}-u)\Bigg{]}$
$\displaystyle=\mathbb{E}[h]^{2}\\!\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\rho^{(1)}(x)\mathrm{d}x$
$\displaystyle\hskip
28.45274pt+\mathbb{E}[h]^{2}\\!\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)\rho^{(2)}(x,y)\mathrm{d}x\mathrm{d}y,$
(49)

which is an integral function of the first and second moment densities.
Subsequently, (49) can be obtained by following the derivation of the second
moment of interference as in (50).

$\displaystyle\mathbb{E}\Big{[}I^{(t_{1})}_{o}I^{(t_{2})}_{u}\Big{]}=\\!\begin{dcases}\lambda\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x+\delta^{2}\pi^{4}\lambda^{2}\epsilon^{2\delta-2}\csc(\delta\pi)^{2}+\frac{\lambda\bar{c}}{\pi^{2}R_{\rm
d}^{4}}\\\ \hskip
128.0374pt\times\int_{\mathbb{R}^{2}}\\!\\!\bigg{(}\\!\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(y)A_{R_{\rm
d}}(\|x-y\|)\mathrm{d}x\bigg{)}\mathrm{d}y,\hskip 11.38109pt\textup{MCP}\\\
\lambda\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x+\delta^{2}\pi^{4}\lambda^{2}\epsilon^{2\delta-2}\csc(\delta\pi)^{2},\hskip
119.50157pt\textup{PPP}\\\
\lambda\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x+\delta^{2}\pi^{4}\lambda^{2}\epsilon^{2\delta-2}\csc(\delta\pi)^{2}\\\
\hskip
93.89409pt-\lambda^{2}\\!\int_{\mathbb{R}^{2}}\\!\\!\\!\bigg{(}\\!\int_{\mathbb{R}^{2}}\\!\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)\exp\bigg{(}\\!-\frac{\pi\lambda\|x-y\|^{2}}{\beta}\bigg{)}\mathrm{d}x\bigg{)}\mathrm{d}y,\quad\\!\\!\beta\textup{-GPP}.\end{dcases}$
(50)

Finally, by inserting the expectation, second moment, and mean interference
product given in (45), (46), and (49), respectively, into (12), we have the
spatial-temporal interference correlation coefficient in the following
theorem.

######  Theorem 1.

The spatial-temporal correlation coefficient with interferers distributed as
MCP, PPP, and $\beta$-GPP and path-loss function
$\ell_{\epsilon}(x)=\frac{1}{\epsilon+x^{\alpha}}$ is given by (51),

$\displaystyle\zeta(\|u\|)=\begin{dcases}\frac{\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x+\varpi^{\textup{MCP}}(R_{\rm
d},\bar{c})}{2\delta\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)+\varpi^{\textup{MCP}}(R_{\rm
d},\bar{c})},\hskip 51.21495pt\,\textup{MCP}\\\
\frac{\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x}{2\delta\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)},\hskip
119.50157pt\textup{PPP}\\\
\frac{\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x-\varpi^{\beta\textup{-GPP}}(\lambda)}{2\delta\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)-\varpi^{\beta\textup{-GPP}}(\lambda)},\hskip
62.59605pt\beta\textup{-GPP},\end{dcases}$ (51)

where $\varpi^{\textup{MCP}}(R_{\rm d},\bar{c})$ and
$\varpi^{\beta\textup{-GPP}}(\lambda)$ are given, respectively, by

$\displaystyle\varpi^{\textup{MCP}}(R_{\rm d},\bar{c})$
$\displaystyle=\frac{\bar{c}}{\pi^{2}R_{\rm
d}^{4}}\int_{\mathbb{R}^{2}}\\!\\!\bigg{(}\\!\int_{\mathbb{R}^{2}}\\!\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)A_{R_{\rm
d}}(\|x-y\|)\mathrm{d}x\bigg{)}\mathrm{d}y,$

and

$\displaystyle\varpi^{\beta\textup{-GPP}}(\lambda)$
$\displaystyle=\\!\lambda\\!\int_{\mathbb{R}^{2}}\\!\\!\\!\bigg{(}\\!\int_{\mathbb{R}^{2}}\\!\\!\ell_{\epsilon}(x)\ell_{\epsilon}(y)\exp\bigg{(}\\!-\\!\frac{\pi\lambda\|x\\!-\\!y\|^{2}}{\beta}\bigg{)}\mathrm{d}x\bigg{)}\mathrm{d}y.$

Remark 1: The interference correlation coefficient for the MCP is greater than
for the PPP, since

$\displaystyle\zeta^{\textup{MCP}}(\|u\|)-\zeta^{\textup{PPP}}(\|u\|)=\frac{(C_{2}-C_{1})\varpi^{\textup{MCP}}(R_{\rm
d},\bar{c})}{C_{2}(C_{2}+\varpi^{\textup{MCP}}(R_{\rm
d},\bar{c}))}\overset{(a)}{>}0,$

where $(a)$ holds as $\varpi^{\textup{MCP}}(R_{\rm d},\bar{c})>0$ and
$C_{2}-C_{1}\geq 1/2$ [9], and
$C_{1}=\int_{\mathbb{R}^{2}}\ell_{\epsilon}(x)\ell_{\epsilon}(x-u)\mathrm{d}x$
and $C_{2}=2\delta\pi^{2}(1-\delta)\epsilon^{\delta-2}\csc(\delta\pi)$.

Remark 2: The interference correlation coefficient for the $\beta$-GPP is
smaller than for the PPP, since

$\displaystyle\zeta^{\beta\textup{-GPP}}(\|u\|)-\zeta^{\textup{PPP}}(\|u\|)=\frac{(C_{1}-C_{2})\varpi^{\beta\textup{-GPP}}(\lambda)}{C_{2}(C_{2}-\varpi^{\beta\textup{-GPP}}(\lambda))}\overset{(a)}{<}0,$

where $(a)$ holds as $C_{1}<C_{2}$ and $C_{2}>\varpi^{\beta\textup{-GPP}}$
(since $\exp\big{(}\\!-\pi\lambda\|x-y\|^{2}/\beta\big{)}<1$).

Fig. 13 shows the spatial-temporal correlation coefficient for different
fields of interferers, which illustrates the properties discussed above.

Figure 13: Correlation coefficient in ad hoc networks with different fields of
interferers ($\lambda=0.1$, $\epsilon=1$ and $R_{\rm d}=1$).

####  III-B2 Moments of the CSPΦ in Ad Hoc Networks

Next, we derive the moments of the CSPΦ in three different random fields of
interferers by following the methodology in [79]. For this,

  * •

We start by obtaining the CSPΦ by averaging out the randomness of channel
gains for a given point process $\Phi$ and a link distance.

  * •

We then derive the moments of the CSPΦ by averaging over the spatial
distributions of the interferers in the different random fields. Specifically,

    * –

after averaging over the channel gains of the contact link and interfering
links based on their distributions, the moments of the CSPΦ can be represented
by the expectation of the product of a function w.r.t. the locations of the
interferers, i.e., $\mathbb{E}\big{[}\prod_{x\in\Phi}f(x)\big{]}$.

    * –

the expectation for Matérn cluster and Poisson fields of interferers can be
derived based on the PGFLs of MCP and PPP given in (III-A2) and (23),
respectively, and that for the $\beta$-Ginibre field of interferers can be
derived based on the distributions of the distances of the interfering links
given in (43).

######  Theorem 2.

The moments of the CSPΦ for a Matérn cluster field, Poisson field and
$\beta$-Ginibre fields of interferers are given by (52),

$\displaystyle\mathcal{M}_{P_{\rm
s}}(b)=\begin{dcases}\exp\\!\Bigg{(}-\lambda_{\rm
p}\\!\int_{\mathbb{R}^{2}}\\!\\!\bigg{(}\\!1\\!-\exp\\!\bigg{(}\\!-\bar{c}\\!+\\!\frac{\bar{c}V_{b}(x,\theta)}{\pi
R^{2}_{\rm d}}\\!\bigg{)}\\!\bigg{)}\mathrm{d}x\\!\Bigg{)},\hskip
71.13188pt\textup{MCP}\\\ \exp\Bigg{(}-\pi\lambda\theta^{\delta}r^{2}_{\rm
t}\frac{\Gamma(1-\delta)\Gamma(b+\delta)}{\Gamma(b)}\Bigg{)},\hskip
113.81102pt\,\,\,\textup{PPP}\\\ \int^{\infty}_{0}\\!\\!e^{-\pi\lambda
q/\beta}\bigg{(}\frac{\beta}{1+\theta r^{\alpha}_{\rm
t}q^{-\alpha/2}}+1-\beta\bigg{)}^{b}\prod_{j\geq
1}\frac{(\pi\lambda/\beta)^{j}}{\Gamma(j)}q^{j-1}\mathrm{d}q,\hskip
17.07164pt\textup{$\beta$-GPP}\end{dcases}$ (52)

where

$\displaystyle V_{b}(x,\theta)$ $\displaystyle=\int_{\mathbb{B}(0,R_{\rm
d})}\\!\\!\bigg{(}\frac{1}{1\\!+\\!\theta r^{\alpha}_{\rm
t}\|x-y\|^{-\alpha}}\bigg{)}^{\\!b}\mathrm{d}y.$ (53)

###### Proof.

See Appendix A. ∎

Fig. 14 shows the average success probabilities (i.e., $\mathcal{M}_{P_{\rm
s}}(1)$) in ad hoc networks with different fields of interferers. In order to
reveal the entire SIR distribution, we plot the success probabilities in the
Möbius homeomorphic (MH) scale. The conversion from linear scale to MH scale
is given by the function $x\,\text{MH}=\frac{x}{1-x}$ [98], which maps the
one-sided infinite support $[0,\infty)$ to the unit interval $[0,1)$. It can
be observed that, in ad hoc networks, spatial repulsion and attraction among
the interferers result in lower and higher success probabilities,
respectively, compared to the independently located interferers. This can be
intuitively understood from the fact that stronger spatial attraction
(repulsion) increases the chance that the interferers are located further away
(closer to) the target receiver (as shown in Fig. 11).

Figure 14: Success probability in different ad hoc networks ($\lambda=0.1$,
$r_{\rm t}=1$, and $R_{\rm d}=1$. The curves and markers correspond to the
analytical and simulation results, respectively.).

(a) MCP ($\lambda=0.1$)  (b) $\beta$-GPP ($\lambda=0.1$)

Figure 15: Temporal CSP in ad hoc networks with non-Poisson fields of
interferers ($\alpha=4$).

Subsequently, we investigate the temporal dependence of successful
transmissions by evaluating the temporal CSP, i.e., $\mathcal{M}_{P_{\rm
s}}(2)/\mathcal{M}_{P_{\rm s}}(1)$, in Fig. 15. It can be found that the CSP
increases when the interferers are more clustered (i.e., with smaller $R_{\rm
d}$ or larger $\bar{c}$) and decreases when the interferers are more scattered
(i.e., with larger $\beta$). The reason is that spatial attraction and
repulsion cause a lower and higher intensity of aggregated interference at the
target receiver, and thus a transmission attempt is more likely to succeed
given a previous successful transmission.

Furthermore, we investigate the SIR meta distribution for a target link in ad
hoc networks. According to the Gil-Pelaez theorem [99], the exact SIR meta
distribution can be represented as an integral function of the moments of the
CSPΦ as

$\displaystyle\bar{F}(\theta,x)=\frac{1}{2}\\!+\\!\frac{1}{\pi}\int^{\infty}_{0}\frac{\Im\big{(}e^{-\jmath
u\log x}M_{\jmath u}(\theta)\big{)}}{u}\mathrm{d}u,$ (54)

herein $\Im(z)$ is the imaginary part of $z$ and $\jmath=\sqrt{-1}$ denotes
the imaginary unit.

Fig. 16 shows the SIR meta distribution in ad hoc networks with different
fields of interferers. Compared with the success probability in Fig. 14, the
SIR meta distribution gives the entire distribution of the CSPΦ. For example,
for a $0.5$ MH SIR threshold, although the average success probability for PPP
is slightly greater than that of $\beta$-GPP, the percentage of the links
achieving $90\%$ reliability in PPP is higher than twice of that in
$\beta$-GPP.

####  III-B3 Moments of the CSPΦ in Downlink Cellular Networks

With Rayleigh fading, the exact moments of the CSP given $\Phi^{\rm M}$ and
$\Phi^{\rm G}_{\beta}$ can be obtained using the following steps [79]:

  * •

Deriving the CSPΦ by averaging over the randomness of the channel gains of the
contact link and interfering links based on the PDF of the exponential
distribution due to Rayleigh fading;

  * •

Deriving the moments of the CSPΦ by deconditioning on the spatial
distributions of the interferers in the different random fields and contact
distance. Specifically,

    * –

The spatial randomness of the interferers in Matérn cluster and Poisson
downlink networks is averaged out based on the reduced PGFL of MCP and PPP,
respectively, given in (III-A2) and (23), and that in Ginibre downlink
networks is averaged out based on the PDF of the distances of the interfering
links given in (43);

    * –

Based on the nearest-BS associated rule, the contact distances in Matérn
cluster, Poisson, and Ginibre downlink networks are averaged out based on
their PDFs given in (32), (18), and (43), respectively.

Figure 16: SIR meta distribution in ad hoc networks with different fields of
interferers. ($\theta=0.5$ MH, $r_{\rm t}=1$, $\lambda=0.1$, and $R_{\rm
d}=1$. The curves and markers correspond to the analytical and simulation
results, respectively.)

For non-Poisson networks, since deriving the exact downlink success
probability is tedious (if not impossible) and the resulting expressions are
cumbersome, we use an approximation method, referred to as Approximate SIR
analysis based on the PPP (ASAPPP) method [100, 47, 101], to simplify the
evaluation of the SIR distribution. ASAPPP provides an approximate SIR
distribution in a non-Poisson network, obtained from the SIR distribution in a
Poisson network along with a horizontal shift (in the dB scale). This method
is based on the insight that when the disparity between the target system
model and the Poisson model purely lies in the spatial configuration of the
points, shifting the SIR threshold $\theta$ of the Poisson model by a
coefficient $G_{0}$ results in a close approximation of the success
probability and the meta distribution of the target system model. The
subscript in $G_{0}$ indicates that the shift is calculated when $\theta$
approaches $0$. In other words, the ASAPPP method becomes exact as $\theta\to
0$. As shown in [84], ASAPPP yields a very good approximation across the whole
SIR distribution and is barely susceptible to the fading and path-loss models.

The asymptotic gain $G_{0}$ can be obtained by taking the ratio of the mean
interference-to-signal ratios (MISRs) of the considered point process to that
of the PPP as [83]

$\displaystyle
G_{0}=\frac{\textup{MISR}_{\textup{PPP}}}{\textup{MISR}}=\frac{2}{\alpha-2}\frac{1}{\textup{MISR}},$
(55)

where MISR is defined as

MISR
$\displaystyle\triangleq\mathbb{E}\Bigg{[}\frac{\sum^{\infty}_{k=2}\|x_{k}\|^{-\alpha}}{\|x_{1}\|^{-\alpha}}\Bigg{]},$
(56)

and $\textup{MISR}_{\textup{PPP}}$ is the MISR in Poisson downlink networks,
which can be obtained by utilizing the distance ratio distribution as follows
[83].

$\displaystyle\textup{MISR}_{\textup{PPP}}$
$\displaystyle=\sum^{\infty}_{j=2}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{1}}{r_{j}}\bigg{)}^{\\!\alpha}\bigg{]}$
$\displaystyle=\sum^{\infty}_{j=2}\mathbb{E}\big{[}\varrho_{j}^{\alpha}\big{]}$
$\displaystyle\overset{(a)}{=}\sum^{\infty}_{j=2}\int^{1}_{0}\varrho^{\alpha}2(j-1)\varrho(1-\varrho^{2})^{j-2}\mathrm{d}\varrho$
$\displaystyle=\sum^{\infty}_{j=2}\frac{\Gamma(1+\alpha/2)\Gamma(j)}{\Gamma(j+\alpha/2)}$
$\displaystyle=\frac{2}{\alpha-2},$ (57)

where $(a)$ follows the PDF of $\varrho_{j}$ given in (22).

The numerical value of $G_{0}$ can be obtained easily from Monte Carlo
simulations for a given network geometry, i.e., $\lambda_{\rm p}$, $\bar{c}$,
and $R_{\rm d}$ for the MCP and $\lambda$ and $\beta$ for the $\beta$-GPP.
Note that, as found in [4], the MISR-based gain of the $\beta$-GPP can be
accurately approximated as

$\displaystyle G_{0}\approx 1+\beta/2,$ (58)

which is insensitive to the network density and path-loss exponent [84].
Therefore, the success probability of the typical user in a $\beta$-GPP
network is approximately identical to that in a Poisson network with the SIR
threshold scaled from $\theta$ to $\theta/(1+\beta/2)$.

Fig. 18 and Fig. 18, respectively, confirm the effectiveness of the ASAPPP
method for approximating the SIR distribution and the SIR meta distribution in
MCP and $\beta$-GPP downlink networks. As expected, for both types of
networks, the ASAPPP method yields a more accurate approximation of SIR
distribution as the SIR threshold decreases. For the MCP model, we can observe
that the ASAPPP method is more accurate when $\Phi^{\rm M}$ is less clustered
(e.g., with smaller $\bar{c}$ given $\lambda$). When the network is more
clustered (e.g., when $\bar{c}=10$), there exists an observable gap between
the approximation and the simulation results for both success probability and
SIR meta distribution. The reason is that a higher degree of clustering
results in a slowly growing rate of the asymptotics [101]. Moreover, for the
$\beta$-GPP, the insignificant disparity between the simulation and the
approximation results can be ascribed to the approximation of the MISR-based
gain given in (58).

Figure 17: The ASAPPP approximation of non-Poisson cellular networks
($\alpha=4$, $\lambda=0.1$, $R_{\rm d}=4$).

Figure 18: The ASAPPP approximation of SIR meta distribution in non-Poisson
cellular networks ($\theta=0.5$ MH, $\alpha=4$, $\lambda=0.1$, $R_{\rm d}=5$).

Next, in Fig. 19(a) and Fig. 19(b), we evaluate the temporal CSPs for MCP and
$\beta$-GPP downlink networks, respectively. We observe that the CSP decreases
slightly when the network becomes more clustered (i.e., with larger $\bar{c}$
and/or smaller $R_{\rm d}$ with fixed $\lambda$) and increases slightly when
the network becomes more repulsive (i.e., with larger $\beta$). However, the
curves are almost flat, i.e., the temporal CSPs in Matérn cluster and Ginibre
downlink networks are generally non-sensitive to attractiveness and
repulsiveness of the spatial points.

(a) MCP  (b) $\beta$-GPP

Figure 19: Temporal CSP in non-Poisson cellular networks ($\alpha=4$,
$\lambda=0.1$).

###  III-C Summary and Discussion

We have analyzed the impact of the spatial distribution of transmitters on the
interference correlation coefficient and success probability. Specifically, we
have presented the derivations of spatial-temporal interference correlation
coefficient and success probability and SIR meta distribution of a target link
in MCP, PPP, and $\beta$-GPP fields of interferers. We have also introduced an
approximation of the success probabilities and SIR meta distributions in non-
Poisson networks based on the analysis of a Poisson network by scaling the SIR
threshold. The main lessons learned are as follows.

  * •

With non-Poisson fields of interferers, the spatial-temporal interference is
more and less correlated when the interferers are distributed more
attractively and repulsively, respectively.

  * •

Compared with a Poisson field of interferers, the Matérn cluster and
$\beta$-Ginibre field of interferers increases and decreases the success
probability of a target link, respectively. Moreover, the successful
transmission events are more and less temporally-correlated when the locations
of the interferers are more clustered and scattered, respectively.

  * •

Compared with a Poisson downlink network, the Matérn cluster and Ginibre
downlink networks render a lower and a higher success probability,
respectively. Additionally, the successful transmission events are less and
more temporally-correlated when the interferers are more attractive and
repulsive, respectively.

Open Technical Issues: The point process models presented in Section III have
been adopted to characterize the general spatial features (e.g., independence,
repulsion and clustering) of wireless networks. In addition to these generic
models, some point process models have been established for some specialized
network scenarios. For example, reference [102] proposes an energized point
process to model the spatial distribution of wireless-powered devices. The
model exhibits spatial correlation of the RF-powered nodes that can harvest
sufficient energy from a Poisson field of RF sources. Besides, reference [103]
introduces a Poisson rain process to model cellular networks with spatial-
temporal traffic. Specifically, the model employs a space-time PPP to model
traffic arrives which are then assigned to PPP-distributed BSs based on
different allocation schemes. Future efforts should be dedicated to developing
point processes that incorporate spatial-temporal features (e.g., fading,
shadowing, blockage, and renewable energy arrivals) for more specific system
models.

##  IV Location-Dependent Analysis of Cellular Models

The locations of the users play a crucial part in their performance.
Conditioning the regions (e.g., cell center and boundary) of the user of
interest allows a location-dependent analysis [105, 104] for a location-
specific user (LSU) in cellular networks. The objective of this section is to
demonstrate the effect of the location of the target user on the SIR
distribution. For this, we derive the moments of the CSPΦ and the SIR gain for
an LSU in different regions.

###  IV-A System Models

For a stationary point process $\Phi\subset\mathbb{R}^{2}$ of BSs, let
$x_{j}(u)\in\Phi$ represent the location of the $j$-th nearest BS to the LSU
at $u$. Under the nearest-BS association principle, $x_{1}(u)$ and $x_{2}(u)$
are the locations of the associated (serving) BS and the nearest interfering
BS to the LSU, respectively. The region of a user’s location can be defined by
the relationship between the link distances to the associated BS and to the
nearest interfering BS.

For $\rho\in[0,1]$, the regions of cell-center user and cell-boundary user are
defined, respectively, as

$\displaystyle\mathcal{R}_{\rm
c}\triangleq\\{u\in\mathbb{R}^{2}:\|x_{1}(u)-u\|\leq\rho\|x_{2}(u)-u\|\\},$
$\displaystyle\mathcal{R}_{\rm
b}\triangleq\\{u\in\mathbb{R}^{2}:\|x_{1}(u)-u\|>\rho\|x_{2}(u)-u\|\\}.$

The area fraction of a region can be determined by the probability that an
arbitrary location, say $o$, falls into that region, which solely depends on
$\rho$. For the analysis, we focus on the case where $\Phi$ is a PPP.

The area fractions of $\mathcal{R}_{\rm c}$ and $\mathcal{R}_{\rm b}$ can be
obtained based on the CDF of $\varrho_{2}$ (given in (21)) as

$\displaystyle\mathbb{P}[o\in\mathcal{R}_{\rm
c}]=\mathbb{P}[\varrho_{2}\leq\rho]=F_{\varrho_{2}}(\rho)=\rho^{2}$

and

$\displaystyle\mathbb{P}[o\in\mathcal{R}_{\rm
b}]=1-\mathbb{P}[o\in\mathcal{R}_{\rm c}]=1-\rho^{2}.$

Moreover, in the special cases when a cell-boundary user has two or three
equidistant closest BSs, it is referred to as an edge user or a vertex user,
respectively. The regions of the edge users (one-dimensional) and vertex users
(zero-dimensional) are defined, respectively, as

$\displaystyle\mathcal{R}_{\rm
e}=\\{u\in\mathbb{R}^{2}:\|x_{1}(u)-u\|=\|x_{2}(u)-u\|\\},$
$\displaystyle\mathcal{R}_{\rm
v}=\\{u\in\mathbb{R}^{2}:\|x_{1}(u)-u\|=\|x_{2}(u)-u\|$ $\displaystyle\hskip
113.81102pt=\|x_{3}(u)-u\|\\}.$ (59)

Note that in a two-dimensional homogeneous PPP there exists no location with
more than three nearest BSs almost surely [89].

In the following, we explore and the performance at five different types of
typical users: (i) The standard typical user, whose performance represents the
average of all users; it is referred to in this section as the typical general
user; (ii) the typical cell-center user, whose performance represents the
average of all cell-center users; (iii) the typical cell-boundary user, whose
performance represents the average of all cell-boundary users; (iv) the
typical cell-edge user whose performance represents the average of all
locations on the cell edges; and (v) the typical vertex user, whose
performance represents the average of all vertex points of the cells.

In Poisson cellular networks, the PDF of the contact distance of the typical
vertex user is given by [106, 107]

$\displaystyle f^{\rm v}_{r_{1}}(r)=2(\lambda\pi)^{2}r^{3}\exp(-\lambda\pi
r^{2}).$ (60)

The success probability of the typical cell-center and cell-boundary users can
be defined, respectively, as

$\displaystyle\bar{F}^{\rm c}_{\eta}(\theta)=\mathbb{P}[\eta>\theta\mid
o\in\mathcal{R}_{\rm c}],$ and $\displaystyle\bar{F}^{\rm
b}_{\eta}(\theta)=\mathbb{P}[\eta>\theta\mid o\in\mathcal{R}_{\rm b}].$ (61)

Fig. 20 shows a realization of network partitions of a Poisson network with
unit intensity, where the locations of the BSs are represented by the black
circles. White and cyan regions represent the cell-center regions
$\mathcal{R}_{\rm c}$ and cell-boundary regions $\mathcal{R}_{\rm b}$,
respectively. The blue triangle markers represent the region of the vertex
users and blue lines represent the region of the edge users.

Figure 20: Illustration of network partitions in a two-dimensional Poisson
network for $\rho=0.8$.

###  IV-B Performance Analysis

####  IV-B1 Cell-Center and Cell-Boundary User

We start by showing how to obtain the $b$-th moments of the CSPΦ for cell-
center and cell-boundary users, denoted as $\mathcal{M}^{\rm c}_{P_{\rm
s}}(b)$ and $\mathcal{M}^{\rm b}_{P_{\rm s}}(b)$, respectively. The
methodology of obtaining the former is to derive the SIR meta distribution for
the typical general user by conditioning its location to be in the cell-center
and cell-boundary regions [105, 104]. Such conditioning restricts the
distribution regions of interferers which will be reflected in the lower
bounds of the integrals after applying the PGFL of PPP.

Based on this approach, we obtain $\mathcal{M}^{\rm c}_{P_{\rm s}}(b)$ in the
following theorem.

######  Theorem 3.

The moments of the CSPΦ for the typical cell-center user, i.e., the typical
general user conditioned on $\mathcal{R}_{\rm c}$, in a Poisson downlink
network with Rayleigh fading are

$\displaystyle\mathcal{M}^{\rm c}_{P_{\rm
s}}(b)=\frac{1}{{{}_{2}}F_{1}(b,-\delta;1-\delta;-\rho^{\alpha}\theta)},$ (62)

where $\delta=2/\alpha$.

###### Proof.

See Appendix B. ∎

Remark 3: The SIR gain of the typical cell-center user compared to the typical
general user is $\rho^{-\alpha}$, which is easily obtained by comparing (62)
with the moments of the CSPΦ for the typical general user in a Poisson
downlink network given as [79]:

$\displaystyle\mathcal{M}_{P_{\rm
s}}(b)=\frac{1}{{{}_{2}}F_{1}(b,-\delta;1-\delta;-\theta)}.$ (63)

Remark 4: A fraction $x=\rho^{2}$ of users who have the lowest distance ratio
of the associated BS to the interfering BS has an SIR gain of
$-5\alpha\log_{10}x$ dB.

Applying the law of total probability, the moments of the CSPΦ for the typical
general user can be expressed as follows:

$\displaystyle\mathcal{M}_{P_{\rm s}}(b)=$
$\displaystyle\mathbb{E}\Big{[}\mathbb{P}\big{[}\eta>\theta\big{]}^{b}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\mathbb{P}\big{[}\eta>\theta\big{]}^{b}\bigm{|}o\in\mathcal{R}_{\rm
c}\Big{]}\mathbb{P}\big{[}o\in\mathcal{R}_{\rm c}\big{]}$
$\displaystyle+\mathbb{E}\Big{[}\mathbb{P}\big{[}\eta>\theta\big{]}^{b}\bigm{|}o\in\mathcal{R}_{\rm
b}\Big{]}\mathbb{P}\big{[}o\in\mathcal{R}_{\rm b}\big{]}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\rm c}_{P_{\rm
s}}(b)\mathbb{P}\big{[}o\in\mathcal{R}_{\rm c}\big{]}+\mathcal{M}^{\rm
b}_{P_{\rm s}}(b)\Big{(}1-\mathbb{P}\big{[}o\in\mathcal{R}_{\rm
c}\big{]}\Big{)}.$

Then, the moments of the CSPΦ for the typical cell-boundary user are given by

$\displaystyle\mathcal{M}^{\rm b}_{P_{\rm s}}(b)$
$\displaystyle=\frac{\mathcal{M}_{P_{\rm s}}(b)-\mathcal{M}^{\rm
c}_{b}(\theta)\mathbb{P}[o\in\mathcal{R}_{\rm
c}]}{1-\mathbb{P}[o\in\mathcal{R}_{\rm c}]}$
$\displaystyle=\frac{\mathcal{M}_{P_{\rm s}}(b)-\rho^{2}\mathcal{M}^{\rm
c}_{P_{\rm s}}(b)}{1-\rho^{2}}.$ (64)

We then have the following corollary by using (63) and (62) in (1).

######  Corollary 1.

The moments of the CSPΦ of the typical cell-boundary user, i.e.,
$o\in\mathcal{R}_{\rm b}$, in a Poisson downlink network with Rayleigh fading
are

$\displaystyle\mathcal{M}^{\rm b}_{P_{\rm s}}(b)=$
$\displaystyle\frac{1}{(1-\rho^{2})\,{{}_{2}}F_{1}(b,-\delta;1-\delta;-\theta)}$
$\displaystyle-\frac{\rho^{2}}{(1-\rho^{2})\,{{}_{2}}F_{1}(b,-\delta;1-\delta;-\rho^{\alpha}\theta)},$
(65)

where $\delta=2/\alpha$.

####  IV-B2 Edge User

An edge user can be considered as an asymptotic case of a cell-boundary user
when $\rho\to 1$, i.e., $\mathbb{P}[o\in\mathcal{R}_{\rm b}]\to 0$. In this
case, the cell-boundary user is equidistant from the serving BS and the
nearest interferer, i.e., on the cell edge. Therefore, the performance of the
typical cell-edge user can be obtained from the asymptotics of the typical
cell-boundary user as $\rho\to 1$.

By taking the limit $\rho\to 1$ of $\mathcal{M}^{\rm b}_{P_{\rm s}}(b)$ in
(1), we have the moments of the CSPΦ for the typical edge user as follows:

$\displaystyle\mathcal{M}^{\rm e}_{P_{\rm s}}(b)$
$\displaystyle\overset{(a)}{=}\lim_{\rho\to 1}\mathcal{M}^{\rm b}_{P_{\rm
s}}(b)$ $\displaystyle=\frac{1}{{{}_{2}}F_{1}(b,-\delta;1-\delta;-\theta)}$
$\displaystyle\hskip
14.22636pt-\frac{b\theta\,{{}_{2}}F_{1}(b+1,1-\delta;2-\delta;-\theta)}{(1-\delta)\,{{}_{2}}F_{1}(-b,-\delta;1-\delta;-\theta)^{2}},$
(66)

where $(a)$ follows from L’Hôpital’s rule.

After some mathematical manipulations, we have $\mathcal{M}^{\rm e}_{P_{\rm
s}}(b)$ in the following corollary.

######  Corollary 2.

With Rayleigh fading, the moments of the CSPΦ for the typical edge user in a
Poisson downlink network are

$\displaystyle\mathcal{M}^{\rm e}_{P_{\rm
s}}(b)=\frac{1}{(1+\theta)^{b}{{}_{2}}F_{1}(b,-\delta;1\\!-\\!\delta;-\theta)^{2}},\quad
b\in\mathbb{C}$ (67)

where $\delta=2/\alpha$.

Remark 5: Compared with $\mathcal{M}_{P_{\rm s}}(b)$ in (62),
$\mathcal{M}^{\rm e}_{P_{\rm s}}(b)$ in (67) can be expressed as
$\mathcal{M}^{\rm e}_{P_{\rm s}}(b)=\frac{\mathcal{M}_{P_{\rm
s}}(b)^{2}}{(1+\theta)^{b}}$, $\forall b\in\mathbb{R}^{+}$. $\mathcal{M}^{\rm
e}_{P_{\rm s}}(b)$ is smaller than $\mathcal{M}_{P_{\rm s}}(b)$ and their gap
depends only on $\theta$ and $b$.

(a) Typical general user and cell-center/boundary user  (b) Typical edge user
and vertex user

Figure 21: Success probability of LSUs in Poisson downlink networks
($\lambda=1$, $\alpha=4$).

####  IV-B3 Vertex User

Finally, we compute the performance of the typical vertex user, which
represents the worst-case performance. To obtain the moments of the CSPΦ,
denoted as $\mathcal{M}^{\rm v}_{P_{\rm s}}(b)$, we follow the derivation
steps similar to those for the typical general user with the condition that
the user is equidistant from the serving BS and the two nearest interfering
BSs [105]. Under this condition, the contact distance is averaged out based on
its PDF given in (60). With this methodology, we have $\mathcal{M}^{\rm
v}_{P_{\rm s}}(b)$ given in the following theorem.

######  Theorem 4.

With Rayleigh fading, the moments of the CSPΦ for the typical vertex user in a
Poisson downlink network are

$\displaystyle\mathcal{M}_{P_{\rm
s}}(b)=\frac{1}{(1+\theta)^{2b}{{}_{2}}F_{1}(b,-\delta;1-\delta;-\theta)^{2}},$
(68)

where $\delta=2/\alpha$.

###### Proof.

See Appendix C. ∎

Remark 6: Compared with (120), the success probability of the typical vertex
user in (68) can be expressed as $\mathcal{M}^{\rm v}_{P_{\rm
s}}(b)=\frac{\mathcal{M}^{\rm e}_{P_{\rm s}}(b)}{(1+\theta)^{b}}$.
$\mathcal{M}^{\rm v}_{P_{\rm s}}(b)$ is smaller than $\mathcal{M}^{\rm
e}_{P_{\rm s}}(b)$ due to the two equidistant closest interfering BSs.

Fig. 21 depicts the average success probabilities (i.e., $\mathcal{M}_{P_{\rm
s}}(1)$) of the LSUs. It is straightforward that the success probability
monotonically decreases with $\rho$. The weighted mean of the success
probabilities for the typical cell-center user and the typical cell-boundary
user with the same value of $\rho$ are equal to that of the typical general
user.

####  IV-B4 Temporal Effect of Location Dependence

Next, we investigate how location dependence affects the temporal correlation
among different successful transmission events by evaluating the CSP
$\frac{\mathcal{M}_{P_{\rm s}}(2)}{\mathcal{M}_{P_{\rm s}}(1)}$. Fig. 22 shows
how the CSP varies with $\rho$. It is found that the CSP for the typical cell-
center user is greater than the typical cell-boundary user. Moreover, for both
typical cell-center and cell-boundary users, their CSPs decrease with $\rho$.
The reason is that the successful transmission events are more temporally
correlated when the received signal dominates the interference. Such temporal
correlation decreases when the interference becomes stronger.

Figure 22: The temporal CSP of LSUs.

####  IV-B5 SIR Gain

Next, we investigate the asymptotic SIR gain to illustrate the SIR improvement
over the typical general user due to location dependence. The MISR of the
typical cell-center user can be derived as

$\displaystyle\textup{MISR}_{\mathcal{R}_{\rm c}}$
$\displaystyle\\!=\sum^{\infty}_{i=2}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{1}}{r_{i}}\bigg{)}^{\\!\alpha}\mid
o\in\mathcal{R}_{\rm c}\bigg{]}$
$\displaystyle\\!\overset{(b)}{=}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{1}}{r_{2}}\bigg{)}^{\\!\alpha}\mid
o\in\mathcal{R}_{\rm
c}\bigg{]}\sum^{\infty}_{i=2}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{2}}{r_{i}}\bigg{)}^{\\!\alpha}\bigg{]},$
$\displaystyle\\!=\frac{\mathbb{E}\Big{[}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\\!\alpha},o\in\mathcal{R}_{\rm
c}\Big{]}}{\rho^{2}}\sum^{\infty}_{i=2}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{2}}{r_{i}}\bigg{)}^{\\!\alpha}\bigg{]},$
(69)

where $(b)$ holds as the condition that the typical general user is in the
cell-center region, i.e., $o\in\mathcal{R}_{\rm c}$, only affects the relative
distance between $r_{1}$ and $r_{2}$.

The first expectation in (69) can be derived based on the joint distribution
of $r_{1}$ and $r_{2}$ as

$\displaystyle\mathbb{E}\Big{[}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\\!\alpha},o\in\mathcal{R}_{\rm
c}\Big{]}$
$\displaystyle=\int^{\rho}_{0}f_{v_{2}}(v)v^{\alpha}\mathrm{d}v=\frac{2\rho^{2+\alpha}}{2+\alpha}.$
(70)

The second expectation in (69) can be obtained by exploiting the distribution
of the distance ratios of a PPP as

$\displaystyle\sum^{\infty}_{j=2}\mathbb{E}\bigg{[}\bigg{(}\frac{r_{2}}{r_{i}}\bigg{)}^{\\!\alpha}\bigg{]}$
$\displaystyle=\frac{\sum^{\infty}_{j=2}\mathbb{E}\Big{[}\Big{(}\frac{r_{1}}{r_{i}}\Big{)}^{\\!\alpha}\Big{]}}{\mathbb{E}\Big{[}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\\!\alpha}\Big{]}}$
$\displaystyle\overset{(c)}{=}\frac{2+\alpha}{2}\sum^{\infty}_{j=2}\int^{1}_{0}v_{i}^{\alpha}f_{i}(v)\mathrm{d}v$
$\displaystyle=\frac{2+\alpha}{2}\int^{1}_{0}v^{\alpha}2v^{-3}\mathrm{d}v$
$\displaystyle=\frac{\alpha+2}{\alpha-2},$ (71)

where $(c)$ follows the distribution of $\varrho_{2}$.

Plugging (70) and (71) into (69) yields

$\displaystyle\textup{MISR}_{\mathcal{R}_{\rm
c}}=\frac{2\rho^{\alpha}}{\alpha-2}.$ (72)

Subsequently, with $\textup{MISR}_{\textup{PPP}}$ and
$\textup{MISR}_{\mathcal{R}_{\rm c}}$ given in (III-B3) and (72),
respectively, the SIR gain of the typical cell-center user can be obtained as

$\displaystyle G_{\mathcal{R}_{\rm
c}}=\frac{\textup{MISR}_{\textup{PPP}}}{\textup{MISR}_{\mathcal{R}_{\rm
c}}}=\rho^{-\alpha}.$ (73)

Similarly, the SIR gains of different LSUs can be obtained as in Table VI.

Remark 7: Since $\alpha>2$ and $\rho\in[0,1]$, the SIR gain of the typical
cell-center user is greater than or equal to one, and the equality holds when
$\rho=1$. Moreover, the SIR gain of the typical cell-boundary user is smaller
than or equal to one and the equality holds when $\rho=0$. It can be found
that the influence of $\rho$ on the SIR gain of the typical cell-center user
is much more significant than that of the typical cell-boundary user.
Moreover, the gap between the SIR gains of the typical edge user and vertex
user increases with $\alpha$.

TABLE VI: MISR and SIR gain of location-specific users User Location |  MISR |  SIR Gain
---|---|---
Typical general user |  $2/(\alpha-2)$ |  1
Typical cell-center user |  $2\rho^{\alpha}/(\alpha-2)$ |  $\rho^{-\alpha}$
Typical cell-boundary user |  $2(1-\rho^{\alpha+2})/(\alpha-2)(1-\rho^{2})$ |  $(1\\!-\\!\rho^{2})/(1-\rho^{\alpha+2})$
Typical edge user |  $(\alpha+2)/(\alpha-2)$ |  $2/(\alpha+2)$
Typical vertex user |  $2\alpha/(\alpha-2)$ |  $1/\alpha$

###  IV-C Summary and Discussion

We have discussed the impact of the relative distance of a random user on its
performance in Poisson downlink networks. Based on the relationship between
the contact distance and the distance to the two nearest interferers, a user
can be categorized as cell-center, cell-boundary, cell-edge, and vertex user.
We present the derivations of the moments of the CSPΦ for the typical cell-
center, cell-boundary, cell-edge and vertex users in two-dimensional Poisson
downlink networks. To show the direct impact of the location on the SIR, we
also derive the SIR gain of the four types of users. The major lessons learned
are as follows.

  * •

The gaps among the $b$-th moments of the CSPΦ for the typical cell-center,
cell-edge and vertex users depend only on $b$ and the SIR threshold $\theta$,
i.e., $\mathcal{M}^{\rm c}_{P_{\rm s}}(b)=(1+\theta)^{b}\mathcal{M}^{\rm
e}_{P_{\rm s}}(b)=(1+\theta)^{2b}\mathcal{M}^{\rm v}_{P_{\rm s}}(b)$, $\forall
b\in\mathbb{R}^{+}$.

  * •

The successful transmission events are more temporally correlated for the
typical cell-center user than the typical cell-boundary user.

  * •

The SIR gains for the typical cell-center and cell-boundary user depend on the
path-loss exponent and location dependence coefficient $\rho$ while those for
the typical cell-edge and vertex user only depend on the path-loss exponent.

Open Technical Issues: In the literature, a location-dependent analysis has
been carried out [105], and BS cooperation schemes have been designed based on
the location-dependent modeling [104]. However, the location-dependent
performance for Poisson uplink networks remains unexplored. Characterizing the
location-dependent uplink user is more challenging as the uplink model is not
fully tractable [65]. Another interesting direction is to analyze the
performance of LSUs in non-Poisson cellular networks, e.g., when the
transmitters exhibit repulsive or attractive distribution.

Additionally, cell-free communication systems where densely deployed BSs could
simultaneously serve a number of users have emerged as a practical solution
for future-generation communication systems. In such an infrastructure, the BS
cooperation schemes need to take into account the user locations. For example,
it is more meaningful to assign more resources for cell-boundary users to
improve their success probabilities. Thus, location-dependent analysis is the
key to the design of BS cooperation schemes in cell-free massive MIMO systems.

##  V  Spatially Correlated and Independent Shadowing

The main objective of this section is to investigate the effect of correlated
and independent shadowing in large-scale systems based on stochastic geometry
analysis.

###  V-A System Model

To model the effect of shadowing, we consider that the whole plane is
partitioned into a set of deterministic shadowing cells
$\mathbf{S}=\\{\mathbb{S}_{k}\subset\mathbb{R}^{2}\\}_{k\in\mathbb{N}}$ in
which the transmitters experience similar shadowing. We consider an ad hoc
network with a Poisson field of interferers $\Phi$ introduced in Section III-A
as an example system model for the analysis in this section.

The channel gain from $x_{j}\in\Phi$ to $o$ is $h_{j}S_{j}\ell(\|x_{j}\|)$,
where $h_{j}$ and $S_{j}$ are the fading coefficient and shadowing
coefficient, respectively, from $x_{j}$ to $o$, and
$\ell(\|x_{j}\|)=\|x_{j}\|^{-\alpha}$ is the path-loss function. With a
Poisson field of interferers, the aggregated interference at the receiver of
interest is

$\displaystyle I_{o}$
$\displaystyle=\sum_{j\in\mathbb{N}}h_{j}S_{j}\ell(\|x_{j}\|),$ (74)

and the SIR follows as

$\displaystyle\eta$ $\displaystyle=\frac{h_{\rm t}S_{\rm t}\ell(\|x_{\rm
t}\|)}{I_{o}},$ (75)

where $S_{\rm t}$ denotes the shadowing coefficient of the link under
consideration.

Let $c(j)$ be the cell index that $x_{j}$ resides in, i.e., $c(j)=k$ if
$x_{j}\in\mathbb{S}_{k}$. We consider the extreme cases of (fully) correlated
and independent intra-cell shadowing. In independent shadowing, the $S_{c(j)}$
are all independent and distributed with CDF $F_{c(j)}$. In correlated
shadowing, $S_{c(j)}$ and $S_{c(i)}$ are independent only if $c(j)\neq c(i)$,
and $S_{c(j)}=T_{k}$ for all $x_{j}\in\mathbb{S}_{k}$. By choosing $F_{k}$,
different shadowing properties can be assigned to the individual cells.

###  V-B Performance Analysis

We first characterize the Laplace transform of the interference. Then, based
on it, we derive the mean and variance of the interference. Furthermore, we
obtain the moments of the CSPΦ under both correlated and independent
shadowing.

####  V-B1 Laplace Transform of the Interference

To analyze the interference distribution, we start by characterizing the
Laplace transform of the aggregated interference at the target receiver at the
origin with correlated and independent shadowing, denoted by $I^{\rm Cor}_{o}$
and $I^{\rm Ind}_{o}$, respectively, by following the methodology in [108].
Similar to the derivation steps of the moments of the CSP given a Matérn
cluster field of interferers presented in Appendix A, we first express the
Laplace transform as the expectation of the product of a function of the
locations of the interferers given the distribution of the shadowing
coefficients. Then, by the PGFLs of MCP given in (III-A2), we convert the
expectation into an integral expression as a function of the shadowing
coefficients. Subsequently, the randomness of the shadowing can be averaged
out by conditioning on that the daughter points in each cluster are associated
with the same and independent shadowing coefficients with correlated shadowing
and independent shadowing, respectively.

Based on this methodology, the Laplace transforms of $I^{\rm Cor}_{o}$ and
$I^{\rm Ind}_{o}$ are given in the following theorem.

######  Theorem 5.

In a Rayleigh fading environment with a Poisson field of interferers, the
conditional Laplace transform of the interference at the target receiver is

$\displaystyle\mathcal{L}_{I^{\rm Cor}_{o}}(s)=$
$\displaystyle\prod_{k\in\mathbb{N}}\mathbb{E}_{T_{k}}\Bigg{[}\exp\bigg{(}\\!-\\!\lambda\int_{\mathbb{S}_{k}}\\!\\!\bigg{(}1-\frac{1}{1\\!+\\!s\ell(\|x\|)T_{k}}\\!\bigg{)}\mathrm{d}x\bigg{)}\Bigg{]},$
(76)

with correlated shadowing, and

$\displaystyle\mathcal{L}_{I^{\rm Ind}_{o}}(s)=$
$\displaystyle\prod_{k\in\mathbb{N}}\exp\Bigg{(}\\!-\\!\lambda\int_{\mathbb{S}_{k}}\\!\\!\bigg{(}1-\mathbb{E}_{T_{k}}\bigg{[}\frac{1}{1\\!+\\!s\ell(\|x\|)T_{k}}\bigg{]}\\!\bigg{)}\mathrm{d}x\Bigg{)},$
(77)

with independent shadowing.

###### Proof.

See Appendix D. ∎

According to Jensen’s inequality, i.e., the convex transformation of an
expectation equals or exceeds the expectation over the convex transformation,
we have the inequality in (78).

$\displaystyle\mathbb{E}_{T_{k}}\bigg{[}\exp\bigg{(}-\bar{c}\int_{\mathbb{S}_{k}}\Big{(}1-\frac{1}{1+s\ell(\|x\|)T_{k}}\Big{)}\mathrm{d}x\bigg{)}\bigg{]}$
$\displaystyle\hskip
128.0374pt>\exp\bigg{(}-\bar{c}\int_{\mathbb{S}_{k}}\bigg{(}1-\mathbb{E}_{T_{k}}\bigg{[}\frac{1}{1+s\ell(\|x\|)T_{k}}\bigg{]}\mathrm{d}x\bigg{)}\bigg{)}.$
(78)

Furthermore, since $\exp(-x)$ is a completely monotone function, we readily
obtain the following observation by comparing (5) and (5).

Remark 8: For all $s>0$, $\mathcal{L}_{I^{\rm Cor}_{o}}(s)>\mathcal{L}_{I^{\rm
Ind}_{o}}(s)$.

####  V-B2 Mean and Variance of Interference

Due to the singularity of the path-loss function $\ell(x)$, we adopt
$\ell_{\epsilon}(x)=\frac{1}{\epsilon+\|x\|^{\alpha}},\alpha>2,\epsilon>0$, to
evaluate the mean and variance of interference, similar to Sec. III-B1. As the
mean of a random variable can be derived by taking the first-order derivative
of its Laplace transform w.r.t. $s$ and setting $s=0$, i.e.,
$\mathbb{E}[X]=-\frac{\mathrm{d}\mathcal{L}_{X}(s)}{\mathrm{d}s}|_{s=0}$, we
can easily obtain the mean interference as

$\displaystyle\mathbb{E}\Big{[}I^{\rm Ind}_{o}\Big{]}=\mathbb{E}\Big{[}I^{\rm
Cor}_{o}\Big{]}=\lambda\sum_{k\in\mathbb{N}}\mathbb{E}[T_{k}]\int_{\mathbb{S}_{k}}\frac{1}{\epsilon+\|x\|^{\alpha}}\mathrm{d}x,$

which shows that the mean interference at the typical receiver with
independent shadowing and correlated shadowing are identical, i.e., $I^{\rm
Ind}_{o}=I^{\rm Cor}_{o}$.

The same result can also be obtained from Campbell’s Theorem [3, Thm. 4.1],
which shows that it holds for all stationary point processes with an arbitrary
fading model.

As the second moment of a random variable can be derived by evaluating the
second derivative of its Laplace transform w.r.t. $s$ at $s=0$, the variance
of $I^{\rm Ind}_{o}$ can be obtained based on the Laplace transform as [108]

$\displaystyle\mathbb{V}[I^{\rm Ind}_{o}]$
$\displaystyle=\mathbb{E}\Big{[}\big{(}I^{\rm
Ind}_{o}\big{)}^{2}\Big{]}-\mathbb{E}\Big{[}I^{\rm Ind}_{o}\Big{]}^{2}$
$\displaystyle=\frac{\mathrm{d}^{2}\mathcal{L}_{I^{\rm
Ind}_{o}}(s)}{\mathrm{d}^{2}s^{2}}\big{|}_{s=0}-\bigg{(}\frac{\mathrm{d}\mathcal{L}_{I^{\rm
Ind}_{o}}(s)}{\mathrm{d}s}\big{|}_{s=0}\bigg{)}^{2}$
$\displaystyle=2\lambda\sum_{k\in\mathbb{N}}\mathbb{E}\big{[}T^{2}_{k}\big{]}\int_{\mathbb{S}_{k}}\frac{1}{\big{(}\epsilon+\|x\|^{\alpha}\big{)}^{2}}\mathrm{d}x$
$\displaystyle\hskip
5.69054pt+\lambda^{2}\sum_{k\in\mathbb{N}}\mathbb{E}[T_{k}]^{2}\bigg{(}\int_{\mathbb{S}_{k}}\frac{1}{\epsilon+\|x\|^{\alpha}}\mathrm{d}x\bigg{)}^{2}.$
(79)

In the same manner, the variance of $I^{\rm Cor}_{o}$ can be derived as

$\displaystyle\mathbb{V}[I^{\rm Cor}_{o}]$
$\displaystyle=2\lambda\sum_{k\in\mathbb{N}}\int_{\mathbb{S}_{k}}\frac{\mathbb{E}\big{[}T^{2}_{k}\big{]}}{\big{(}\epsilon+\|x\|^{\alpha}\big{)}^{2}}\mathrm{d}x$
$\displaystyle\hskip
14.22636pt+\lambda^{2}\sum_{k\in\mathbb{N}}\mathbb{E}[T_{k}]^{2}\bigg{(}\int_{\mathbb{S}_{k}}\frac{1}{\epsilon+\|x\|^{\alpha}}\mathrm{d}x\bigg{)}^{2}$
$\displaystyle\hskip
14.22636pt+\lambda^{2}\sum_{k\in\mathbb{N}}\mathbb{V}[T_{k}]\bigg{(}\int_{\mathbb{S}_{k}}\frac{1}{\epsilon+\|x\|^{\alpha}}\mathrm{d}x\bigg{)}^{2}$
$\displaystyle\overset{(a)}{=}\\!\mathbb{V}[I^{\rm
Ind}_{o}]\\!+\\!\underbrace{\lambda^{2}\\!\sum_{k\in\mathbb{N}}\\!\mathbb{V}[T_{k}]^{2}\bigg{(}\int_{\mathbb{S}_{k}}\\!\frac{1}{\epsilon\\!+\\!\|x\|^{\alpha}}\mathrm{d}x\bigg{)}^{2}}_{>0},$
(80)

where $(a)$ follows from (79). Comparing (79) and (80) yields the following
observation.

Remark 9: The variance of the interference at the typical receiver with
correlated shadowing is greater than that with independent shadowing, i.e.,
$\mathbb{V}[I^{\rm Cor}_{o}]>\mathbb{V}[I^{\rm Ind}_{o}]$.

####  V-B3 Moments of the CSPΦ

Given the point process $\Phi$, the CSP is

$\displaystyle\mathbb{P}[\eta>\theta\mid\Phi]$
$\displaystyle=\mathbb{P}\bigg{[}\frac{h_{\rm t}S_{\rm t}r^{-\alpha}_{\rm
t}}{I_{o}}>\theta\Bigm{|}\Phi\bigg{]}$
$\displaystyle\overset{(a)}{=}\mathbb{E}_{S_{\rm
t},(S_{j})}\Bigg{[}\prod_{j\in\mathbb{N}}\bigg{(}1+\frac{\theta
r^{\alpha}_{\rm t}r_{j}^{-\alpha}S_{j}}{S_{\rm t}}\bigg{)}^{\\!-1}\Bigg{]},$

where $(a)$ follows the derivations of (112) in Appendix A.

Then, the moments of the CSPΦ can be represented as

$\displaystyle\mathcal{M}_{P_{\rm
s}}(b)=\mathbb{E}\Bigg{[}\prod_{j\in\mathbb{N}}\\!\bigg{(}1\\!+\\!\frac{\theta
r^{\alpha}_{\rm t}r_{j}^{-\alpha}S_{j}}{S_{\rm t}}\bigg{)}^{\\!-b}\Bigg{]}.$

By following the same methodology to the proof of Theorem 5, we have
$\mathcal{M}_{P_{\rm s}}(b)$ derived in the following theorem.

######  Theorem 6.

In a Rayleigh fading environment, the conditional moments of the CSPΦ under
correlated shadowing and independent shadowing are given, respectively, by
(81) and (82), shown on the top of the next page.

$\displaystyle\mathcal{M}^{\rm Cor}_{P_{\rm
s}}(b)=\prod_{k\in\mathbb{N}}\\!\mathbb{E}_{T_{k},T_{c(\rm
t)}}\\!\Bigg{[}\exp\\!\bigg{(}\\!-\\!\lambda\\!\int_{\mathbb{S}_{k}}\\!\\!\bigg{(}\\!1-\bigg{(}\frac{1}{1\\!+\\!\theta
r^{\alpha}_{\rm t}\|x\|^{-\alpha}T_{k}/T_{c(\rm
t)}}\bigg{)}^{\\!b}\bigg{)}\mathrm{d}x\\!\bigg{)}\\!\Bigg{]},$ (81)
$\displaystyle\mathcal{M}^{\rm Ind}_{P_{\rm
s}}(b)=\prod_{k\in\mathbb{N}}\mathbb{E}_{T_{c(\rm
t)}}\Bigg{[}\exp\\!\Bigg{(}\\!-\\!\lambda\\!\int_{\mathbb{S}_{k}}\\!\\!\bigg{(}\\!1-\mathbb{E}_{T_{k}}\\!\bigg{[}\bigg{(}\frac{1}{1\\!+\\!\theta
r^{\alpha}_{\rm t}\|x\|^{-\alpha}T_{k}/T_{c(\rm
t)}}\bigg{)}^{\\!b}\bigg{]}\\!\bigg{)}\mathrm{d}x\\!\Bigg{)}\Bigg{]}.$ (82)

Remark 10: It follows from Remark 8 that with Rayleigh fading, the moments of
the CSPΦ under correlated shadowing are greater than that with independent
shadowing.

Next, we numerically evaluate the network performance under correlated and
independent shadowing assumptions. We assume $T_{k}=\kappa^{N_{k}(x)}$ [108],
where $\kappa$ is the attenuation factor accounting for the signal loss while
penetrating a blockage (e.g., a wall) and $N_{k}(x)$ represents the number of
blockages between $x$ and the receiver of interest. $N_{k}(x)$ is assumed to
be a Poisson random variable for each link [109], i.e.,
$N_{k}(x)\sim\mathcal{P}(\lambda_{\rm b}\|x\|)$, where $\lambda_{\rm b}$
denotes the blockage density. With correlated shadowing, the transmitters in
the same cell $\mathbb{S}_{k}$ are associated with the same shadowing
coefficient $T_{k}=\kappa^{N_{k}(d_{k,o})}$,
$N_{k}(d_{k,o})\sim\mathcal{P}(\lambda_{\rm b}d_{k,o})$, where $d_{k,o}$
represents the distance between the center of $\mathbb{S}_{k}$ and the origin.
Differently, with independent shadowing, the transmitters at
$x_{j}\in\mathbb{S}_{k}$ are associated with i.i.d. shadowing coefficients
$S_{j}=\kappa^{N_{k}(d_{k,o})}$, $N_{k}(d_{k,o})\sim\mathcal{P}(\lambda_{\rm
b}d_{k,o})$. Besides, we assume that the link between the target receiver and
the serving transmitter is subject to no shadowing, i.e., $S_{\rm t}=1$. For
the spatial configuration of the shadowing cells, we consider the example in
Fig. 23, where $\mathbf{S}=\\{\mathbb{S}_{k}\\}_{k\in\mathbb{N}}$ is a set of
squares with length $L$. It is worth noting that the analytical framework can
be extended to other types of point processes by changing the distribution of
points as long as $|\mathbb{S}_{k}|<\infty$, i.e., the point process has a
finite number of points in each shadowing cell almost surely.

Figure 25 illustrates the success probability, i.e., $\mathcal{M}^{\rm
Cor}_{P_{\rm s}}(1)$ and $\mathcal{M}^{\rm Ind}_{P_{\rm s}}(1)$, as a function
of the SIR threshold. It can be observed that the success probability with
correlated shadowing is greater than that with independent shadowing, which
agrees with Remark 10. Moreover, the gap between the success probabilities
with correlated and independent shadowing decreases as the cell size (i.e.,
$L^{2}$) becomes smaller, as in this case the signals from the interferers
experience nearly independent shadowing even with the correlated shadowing
model.

Figure 23: Configuration of shadowing cells for simulations.

Figure 24: Success probability versus SIR threshold under correlated and
independent shadowing ($\lambda=1$, $\kappa=0.5$, $\alpha=4$, $r_{\rm t}=1$).

Figure 25: CSP versus $L$ under correlated and independent shadowing
($\lambda=1$, $\alpha=4$, $r_{\rm t}=1$, $\theta=1$).

Furthermore, to evaluate the temporal effect of shadowing, we evaluate the
temporal CSP with both correlated and independent fading, i.e.,
$\frac{\mathcal{M}^{\rm Cor}_{P_{\rm s}}(2)}{\mathcal{M}^{\rm Cor}_{P_{\rm
s}}(1)}$ and $\frac{\mathcal{M}^{\rm Ind}_{P_{\rm s}}(2)}{\mathcal{M}^{\rm
Ind}_{P_{\rm s}}(1)}$, as shown in Figure 25. We can observe that correlated
shadowing results in more temporal dependence between two successful
transmission events. The gap of CSPs between correlated and independent
shadowing decreases as the cell size $L^{2}$ becomes larger. Moreover, the
dependence between two successful temporal transmission events increases as
the cell size becomes smaller, especially when the SIR threshold is small. The
reasons for the above observations can be ascribed to the adopted shadowing
model, which essentially leads to exponential path loss.

###  V-C Summary and Discussion

In this section, we have established an analytical framework to model the
interference distribution and success probabilities in networks with
spatially-correlated shadowing and spatially-independent shadowing. In
particular, the analytical framework characterizes deterministic shadowing
cells where the transmitters inside are associated with the same shadowing
effects. The key lessons learned are as follows.

  * •

Spatially-correlated shadowing generates the same interference as but higher
variance than spatially-independent shadowing.

  * •

The moments of the CSPΦ under spatially-correlated shadowing is greater than
that with spatially-independent shadowing. The performance increase reduces
when the cell size shrinks.

  * •

The successful transmission events are more temporally correlated with
spatially-correlated shadowing than with spatially-independent shadowing.

Open Technical Issues: Most of the existing literature, e.g., [110, 111, 112,
108, 113], only investigates network performance under the assumptions that
the shadowing coefficients of the interferers in the same shadowing cells are
either fully spatially correlated or independent. However, in practice, the
interferers in the same shadowing cell may only experience partially
correlated shadowing effect due to their location difference. More accurate
shadowing models need to be developed by taking into account the properties
(e.g., shape, density and mobility) of the obstacles that may exhibit
location-dependency (e.g., urban and rural areas). Moreover, temporal
shadowing variation due to mobility of obstacles (e.g., vehicles), access
points (e.g., drone hot spots) or users is an important factor to be
investigated in practical systems.

Different from the deterministic shadowing cells introduced in this section,
reference [114] introduces a correlated shadowing model dependent on the
Poisson Voronoi cell. This model is suitable for studying coverage-oriented
cellular networks where BSs deployment is configured to guarantee the cell-
boundary users to acquire sufficient signal strength. An intriguing future
direction is to extend the study of cell-dependent correlated shadowing to
capacity-oriented cellular networks.

##  VI Spatial-Temporal Interactions Between Queues

The majority of the existing literature heavily relies on the assumption that
each transmitter always has packets in the buffer to send out, which does not
characterize random traffic flows. While the temporal randomness of the
traffic flows complicates the analysis, it is nevertheless essential to
understanding system-level performance. This is due to the fact that the
traffic patterns in the evolving wireless networks are getting increasingly
more dynamic and heterogeneous. The main difficulty of random traffic
characterization originates from the correlation among the buffer statuses of
different transmitters, often referred to as interacting queues [115, 116].
Since the queues interact spatially and temporally, an exact analysis of the
mutual interference is quite challenging.

###  VI-A System Model

We consider both Poisson downlink networks (as introduced in Section III-A)
and Poisson bipolar networks [3, Def. 5.8]. In a Poisson downlink network, the
transmitters (i.e., BSs) and receivers (i.e., users) are distributed following
independent homogeneous PPPs, denoted as $\Phi_{\rm
B}=\\{x_{j}\\}_{j\in\mathbb{N}}$ and $\Phi_{\rm u}$ with intensity
$\lambda_{\rm B}$ and $\lambda_{\rm u}$, respectively. The points in
$\Phi_{\rm B}$ are assumed to be ordered from nearest to farthest to the
origin, i.e., $\|x_{j}\|<\|x_{j+1}\|$. It is assumed that each user is
associated with its nearest BS for downlink transmission. In a Poisson bipolar
network, the transmitters are distributed as a homogeneous PPP $\Phi$ with
intensity $\lambda$. Each transmitter is paired with one receiver in a
uniformly random direction with a link distance $r_{\rm t}$. Without loss of
generality, we study the performance of the typical receiver, conditioned to
be at the origin, in both models.

We consider a discrete-time transmission and queueing model. Specifically, the
data transmissions are divided into equal-duration time slots. We consider
fixed-length data packets and assume it takes exactly one time slot to send
out one packet. If a transmitter is scheduled for transmission, it can only
send out the accumulated packet(s) that arrived prior to the transmission. In
light of queueing, we assume that the incoming packets at each queue are
stored in a buffer with infinite size and sent out on a first-in-first-out
basis. A transmitted packet is removed from the head of the queue only if it
is successfully decoded at the target receiver.

For Poisson downlink networks, each BS maintains an individual queue for the
arrived packets of each associated user [30]. Hence, each BS has a number of
queues equal to the number of users in its Voronoi cell. The temporal arrival
of traffic at each queue follows an i.i.d. Bernoulli process with arrival rate
$\xi_{\rm u}$ representing the probability of a new arrival per time slot. The
users associated with the same BS are served based on random scheduling [30],
i.e., each BS randomly selects one user within its Voronoi cell with equal
probability to serve in each time slot. If the selected user has a non-empty
queue at the BS, the BS is scheduled for transmission. Otherwise, the BS is
muted.

For Poisson bipolar networks, the packet arrival at each transmitter
$x_{j}\in\Phi$ follows an i.i.d. Bernoulli process with arrival rate $\xi$. At
each time slot, the transmitters with non-empty buffer are all scheduled for
transmission [117].

Given that the typical receiver is receiving data, its SIR in Poisson downlink
and Poisson bipolar networks are given, respectively, by

$\displaystyle\eta=\frac{h_{1}\|x_{1}\|^{-\alpha}}{\sum^{\infty}_{j=2}\iota_{j}h_{j}\|x_{j}\|^{-\alpha}},\,\,x_{j}\in\Phi_{\rm
B},$ (83)

and

$\displaystyle\eta=\frac{h_{\rm t}\|x_{\rm
t}\|^{-\alpha}}{\sum_{j\in\mathbb{N}}\iota_{j}h_{j}\|x_{j}\|^{-\alpha}},\,\,x_{j}\in\Phi,$
(84)

where $\iota_{j}$ denote the state indicator of the transmitter located at $j$
which equals 1 and 0 when the transmitter is on and off, respectively.

###  VI-B Performance Analysis

In large random networks, characterizing the exact queue interaction among the
transmitters is quite challenging. Fortunately, in a large-scale network, the
correlation among the interacting queues tends to be “weak” and “global”
[118]. Therefore, the impact of interacting queues tends to be negligible. In