#  A Dominant Interferer plus Mean Field-based Approximation for SINR Meta
Distribution in Wireless Networks

Yujie Qin, Mustafa A. Kishk, Member, IEEE, and Mohamed-Slim Alouini, Fellow,
IEEE Yujie Qin and Mohamed-Slim Alouini are with Computer, Electrical and
Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah
University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia
Arabia. Mustafa Kishk is with the Department of Electronic Engineering,
Maynooth University, Maynooth, W23 F2H6, Ireland. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>slim.alouini@kaust.edu.sa).

###### Abstract

This paper proposes a novel approach for computing the meta distribution of
the signal-to-interference-plus-noise ratio (SINR) for the downlink
transmission in a wireless network with Rayleigh fading. The novel approach
relies on an approximation mix of exact and mean-field analysis of
interference (dominant interferer-based approximation) to reduce the
complexity of analysis and enhance tractability. In particular, the proposed
approximation omits the need to compute the first or the second moment of the
SINR that is used in the beta approximation typically adopted in the
literature but requires of computing the joint distance distributions. We
first derive the proposed approximation based on a Poisson point process (PPP)
network with a standard path-loss and Rayleigh fading and then illustrate its
accuracy and operability in another four widely used point processes: Poisson
bipolar network, Matérn cluster process (MCP), $K$-tier PPP and Poisson line
Cox process (PLCP). Specifically, we obtain the SINR meta distribution for
PLCP networks for the first time. Even though the proposed approximation looks
simple but it shows good matching in comparison to the popular beta
approximation as well as the Monte-Carlo simulations, which opens the door to
adopting this approximation in more advanced network architectures.

###### Index Terms:

Meta distribution, approximation, stochastic geometry, Poisson network,
reliability, PLCP

##  I Introduction

###  I-A Motivation

In wireless communication, the accurate modeling of the locations of the base
stations (BSs) is essential to characterize the system performance and obtain
the critical design insights [1]. Traditionally, the locations of BSs are
modeled by lattices, which are intractable in analysis [2]. However, to
enhance the spatial reuse and meet an exponential growth in mobile traffic,
the deployment of BSs becomes irregular and heterogeneous. For instance,
macro, pico, and femto BSs can coexist, and UAVs or other high altitude
platforms are deployed to help offload the ground BSs, which yield different
path loss exponents [3, 4] and more complicated system models. In this case,
stochastic geometry provides the tools which are widely utilized in modeling,
characterizing, and obtaining design insights of the wireless networks with
randomly placed nodes [5, 6]. Among many point processes, the Poisson point
process (PPP) is a widely used model due to its analytical tractability and
stationarity. This property results in a simple expression for the probability
generating functional (PGFL) [7]. While most of the stochastic geometry-based
analyses are confined to the spatial average, the performance from the
perspective of each user is ignored. Taking coverage probability, for example,
this performance is obtained by averaging over the channel fading and the
point process by utilizing the Laplace transform of the interference with the
aid of PGFL [8]. Such performance metric quantifies the overall signal-to-
interference-plus-noise ratio (SINR) performance, however, limited information
about the individual links. For example, a cellular network with a coverage
probability of $0.8$ does not mean that all the links have the same success
probability, some may have $0.95$ while the others may have $0.6$, and two
networks with the same coverage probabilities would differ greatly. In other
words, it is crucial to obtain key information about ”the distribution of
success probability of the individual link in a given network” [9], which
reveals the reliability and quality of service (QoS) of the network and is a
fundamental design objective for cellular operators. This new and fundamental
performance metric is called the SIR meta distribution [5], defined as a
complementary cumulative distribution function (CCDF) of SIR success
probability given the realization [10, 11].

While SIR/SINR meta approximation is an important performance metric, it is
difficult to compute since the exact integral expression is derived by using
Gil-Pelaez theorem [12] which requires the imaginary moments of the
conditional success probability. Generally, beta approximation is applied to
obtain the approximated SIR/SINR meta distribution, which only requires the
first and the second moments of the conditional success probability. However,
even deriving the first and the second moments of the conditional success
probability is not always simple. Therefore, given the importance of SIR/SINR
meta distribution and the difficulties of deriving it, in this work, we
provide an alternative approximation to SIR/SINR meta which mainly requires
distance distributions.

We propose a novel approach to compute SINR meta distribution which relies on
an approximation mix of exact and mean-field analysis of interference and
reduces the complexity of the analysis and enhances tractability. In this
paper, we call the proposed approximation ’dominant interferer-based
approximation’, however, we would like to clarify that this approximation is
obtained by considering the dominant interferer(s) exactly while the rest are
in the average sense. In other words, we considered all the interference.

###  I-B Related Work

Literature related to this work can be categorized into: (i) the concept of
meta distribution and existing approximations and (ii) the applications of
meta distribution in different system models. A brief discussion on related
works in each of these categories is discussed in the following lines.

Stochastic geometry is a strong mathematical tool that enables characterizing
the statistics of various large-scale wireless networks. In the analysis of
wireless networks with randomly deployed nodes based on stochastic geometry,
the PPP is the most widely used model. The authors in [8, 13] presented a
tutorial on the fundamental concepts of the point process, modeling the
interference in large-scale networks, and a comprehensive survey on single-
tier, multi-tier, and cognitive cellular networks. Most of the existing
studies focus on spatial averaging performance metrics. However, it yields
limited information about the individual links.

The concept of the meta distribution can be traced back to [14], where the
authors computed the distribution of the outage probability in a large-scale
wireless network instead of the spatial average. They obtained the outage
distribution bound by calculating the moments. Similar works about individual
links can be found in [15, 16, 17, 18, 19, 20]. The definition and examples,
Poisson bipolar networks and PPP networks, of meta distribution in wireless
networks was provided in [10]. Since computing the meta distribution requires
calculating the moments of conditional success probability, the exact equation
is hard to derive.Therefore, statistical inequalities and approximations are
extremely useful. The beta approximation is the most common one, which only
requires the first two moments [5]. In [11], the author provided a closed-form
result for the SIR meta distribution by only considered the nearest interferer
in Poission and Poisson bipolar networks. Fourier-Jacobi expansion of moments
to reconstruct meta distribution is used in [21]. The separability of
computing SIR meta distribution for any independent fading is analyzed in
[22], their results show that the separable form behaves as a good
approximation of the SIR meta distribution in Ginibre and triangular lattice
networks when the SINR threshold is chosen large enough. The authors in [9]
study the asymptotics of the moments as the SINR threshold approaches $0$ for
general networks. They provide the meta distribution of general networks by
shifting the meta distribution of Poisson networks, and the shift gain is
obtained by computing the ratio of the mean interference-to-signal ratios
between the point process under consideration and the PPP. Based on the
moments of the conditional success probability, the authors in [23, 24]
provide the numerical methods to calculate the meta distribution.

The meta distribution of both Poisson and non-Poisson cellular networks has
already been addressed in few works. For instance, authors in [25] analyzed
the binomial Poisson process (BPP)-based networks, and [26, 9] utilized the
deployment gain to approximate the meta distribution for general networks. The
SIR meta distribution of a 1-D hardcore point process was investigated in
[27]. Authors in [28] studied a Poisson line Cox bipolar network and a Poisson
stick line Cox bipolar network, and proposed a transdimensional PPP
approximation, which highly reduces the complexity and improves the
tractability of vehicular network analysis. SIR meta distribution for cellular
networks with BS cooperation is analyzed in [29, 30, 31, 32, 33], with power
control, offloading, or with non-orthogonal multiple access (NOMA),
respectively. SIR meta distribution of the moving networks is analyzed in
[34]. In [35], the authors characterized the meta distribution of the downlink
SIR for the typical cell in the case of the BSs are modeled by PPP. The meta
distribution of downlink SIR in a Poisson cluster process-based HetNet network
was analyzed in [26], in which the authors considered a $K$-tier HetNet
modeled by a combination of PPP and PCP. Interestingly, they used mapping
theorem [36, 37] to map the interference from $i$-th tiers onto one tier,
which forms a new and unknown distribution, and obtained the meta distribution
based on the new interference.

Unlike existing literature, this work aims to provide an alternative
approximation to compute the meta distribution in the case of downlink Poisson
networks with Rayleigh fading without requiring computing the moments of
success probability. Instead, the proposed approximation mainly requires
computing the joint distance distribution of the first and the second nearest
nodes in the network.

###  I-C Contribution

This paper derives a dominant interferer-based approximation of SINR meta
distribution in downlink networks. Different from existing literature which
mainly focused on beta distribution and approximated the moments of success
probability, we consider a novel method that approximates the interference,
consequently, the SINR meta distribution can be approximated by using a
Lambert W function. The resulting approximation shows a good matching in
different scenarios with both beta approximation and Monte-Carlo simulations
in a large range of BS densities.

The contributions of this paper are:

  * •

We first derive approximate analytical expressions of the downlink conditional
success probability for the PPP networks with a standard path loss model, and
all the channels are subject to Rayleigh fading, which is based on the exact
term of the dominant interferer(s) and the mean of the remaining interferers.

  * •

We obtain the expression of the dominant interferer-based approximation of the
SINR meta distribution, which can be written as a Lambert W function and
mainly requires computing the joint CDF and the PDF of the first and the
second nearest BS in the network. We show that the proposed approximation can
be extended to the $j$-th nearest interfering BSs, which is more general, but
the nearest interferer is already accurate enough, e.g., the gap between the
exact value and the proposed approximation in the case of PPP networks with
$\lambda=1$ km-2 and $\theta=26$ dB is about $0.02$, while at most be about
$0.03$ at other points. We also show that the proposed approximation performs
well at large path loss scenarios, say $\alpha=3.5,4$.

  * •

By applying the proposed approximation, we obtain the SINR meta distribution
in four different wireless networks modeled by Poisson bipolar network,
$K$-tier PPP, Matérn cluster process, and PLCP, and show that the proposed
approximation is good matching and simple to compute since it mainly requires
the joint distance distribution.

  * •

Additionally, we derive the SINR meta distribution of the PLCP network , which
has not been derived in existing literature, to the best of the authors’
knowledge. Our results reveal that the proposed approximation highly reduces
the complexity of computing its SINR meta distribution.

##  II The SINR Meta Distribution

In this section, we introduce the SINR meta distribution based on a standard
PPP wireless cellular network and focus on the downlink transmission in this
network where the base stations (BSs) are modeled by a PPP, denoted by $\Phi$,
with density $\lambda$. We simply assume that all the BSs are active. In
particular, we assume that the density of cellular users are much larger than
the density of BSs, which leads to all BSs being active, and the user connects
with the nearest BS since it provides the strongest average received power. We
are interested in the SINR meta distribution of the network and its
approximations.

Let $x_{i}$ be the locations of the BSs, where
$x_{i}\in\Phi,i\in\mathbb{N}\cup\\{0\\}$ and $x_{0}$ is location of the
nearest BS to the origin. Here, we condition a user to be at the origin and
this user becomes the typical user on averaging over the point process. We
focus on the SINR of the typical user which is equivalent to any other
arbitrary deterministic location owing to the stationarity of PPP. We use a
standard path-loss model with exponent $\alpha>2$ and $h_{{x_{i}}}$ models the
small scale Rayleigh fading of the channel between the typical user and the
$i$-th BS, which is i.i.d and follows the exponential distribution with mean
of unity.

Let $p_{t}$ be the transmit power of the BSs, the SINR at the typical user is

$\displaystyle{\rm
SINR}=\frac{p_{t}h_{x_{0}}||x_{0}||^{-\alpha}}{\sum_{x\in\Phi\setminus\\{x_{0}\\}}p_{t}h_{x}||x||^{-\alpha}+\sigma^{2}},$
(1)

where $\sigma^{2}$ is thermal noise. Consequently, the conditional success
probability of the typical link is given by

$\displaystyle P_{s}(\theta)=\mathbb{P}({\rm SINR}>\theta|\Phi).$ (2)

For an arbitrary realization of $\Phi$, we analyze the fraction of links that
exceed the reliability threshold $\theta$, and the target reliability is an
argument to the SINR meta distribution. With that being said, our goal is to
obtain the percentiles of users that achieve downlink coverage (SINR above
$\theta$) at an arbitrary realization of the PPP network.

######  Definition 1 (Meta Distribution).

Generally, the SINR meta distribution of downlink is defined in [5] as

$\displaystyle\bar{F}(\theta,\gamma)=\mathbb{P}(P_{s}(\theta)>\gamma),$ (3)

where $\gamma\in[0,1]$ and $P_{s}(\theta)$ is known as the success probability
given the realizations of $\Phi$, i.e. the locations of BSs.

##  III Mathematical Analysis

Before investigating the SINR meta distribution, we first introduce some
important distance distributions in PPP networks. In the following text, let
$R_{i}=||x_{i}||$ be the distance from the $i$-th interfering BS to the
typical user and recall that $x_{0}$ is location of the serving BS. To
simplify the notation, we use $x_{i}$ to present the locations of BSs which
are ordered by the distances to the origin. That is, $x_{j}$ presents the
$j$-th nearest interfering BS and $R_{j}$ denotes the distance to the $j$-th
nearest interferer BS.

######  Lemma 1 (Distance Distribution).

As mentioned above, $R_{0}$ is the distance between the user and the nearest
BS, which is the serving BSs, and $R_{j}$ is the distance to the $j$-th
closest BS. The joint and marginal distance distributions are respectively
given in [38] as

$\displaystyle f_{R_{0},R_{1},\cdots,R_{j}}(r_{0},r_{1},\cdots,r_{j})$
$\displaystyle=(2\lambda\pi)^{j+1}r_{0}r_{1}\cdots r_{j}\exp(-\lambda\pi
r_{j}^{2}),$ $\displaystyle f_{R_{0},R_{1}}(r_{0},r_{1})$
$\displaystyle=(2\pi\lambda)^{2}r_{0}r_{1}\exp(-\pi\lambda r_{1}^{2}),$
$\displaystyle f_{R_{1}}(r_{1})$
$\displaystyle=2(\pi\lambda)^{2}r_{1}^{3}\exp(-\pi\lambda r_{1}^{2}),$
$\displaystyle f_{R_{0}}(r_{0})$ $\displaystyle=2\pi\lambda
r_{0}\exp(-\pi\lambda r_{0}^{2}),$ (4)

consequently, the conditional PDF and CDF are

$\displaystyle f_{R_{0}\mid R_{1}}(r_{0}\mid
R_{1})=\frac{2r_{0}}{R_{1}^{2}},\quad(0<r_{0}<R_{1})$ (5) $\displaystyle
f_{R_{1},\cdots,R_{j}}(r_{1},\cdots,r_{j})=\frac{f_{R_{0},R_{1},\cdots,R_{j}}(r_{0},r_{1},\cdots,r_{j})}{f_{R_{0}\mid
R_{1}}(r_{0}\mid R_{1})},$ (6) $\displaystyle F_{R_{0}\mid R_{1}}(r_{0}\mid
R_{1})=\frac{r_{0}^{2}}{R_{1}^{2}},\quad(0<r_{0}<r_{1}).$ (7)

After obtaining the distance distributions, we are able to compute the SINR
meta distribution and its approximation, which are provided in the next two
subsections.

###  III-A Existing Methods for Computing SINR Meta Distribution

Clearly, SINR meta distribution is a two parameters complementary cumulative
distribution function (CCDF). We first give the exact expression of SINR meta
distribution. Typically, (3) is solved by using the Gil-Pelaez theorem [5,
12]. The exact expression of SIR meta distribution is given in [5] and noise
is added as an exponential factor to the moments.

######  Theorem 1 (Exact Expression of SINR Meta Distribution).

The exact expression is given by

$\displaystyle\bar{F}(\theta,\gamma)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{\Im(\exp(-jt\log\gamma)M_{jt}(\theta))}{t}{\rm
d}t,$ (8)

where $j$ is the imaginary unit, $M_{b}(\theta)$ is the $b$-th moment of
$P_{s}(\theta)$ and $\Im(\cdot)$ is the imaginary part of a complex number,
and

$\displaystyle M_{b}(\theta)=\int_{0}^{\infty}\exp\bigg{(}-\lambda
F_{b}\bigg{)}\exp\bigg{(}-\frac{b\theta}{p_{t}}r^{\alpha}\sigma^{2}\bigg{)}f_{R_{0}}(r){\rm
d}r,$ (9)

in which,

$\displaystyle F_{b}$
$\displaystyle=\pi\delta\sum_{k=1}^{\infty}\binom{b}{k}(-1)^{k+1}\frac{\alpha\theta^{k}R_{0}^{2}}{-2+k\alpha}{}_{2}F_{1}(k,-\delta+k,1-\delta+k,-\theta),$
(10)

where $\delta=2/\alpha$ and ${}_{2}F_{1}$ is the Gaussian hypergeometric
function.

###### Proof:

See Appendix -A. ∎

However, (8) is difficult to compute since it requires the imaginary moments
and three nested integrals. The beta distribution approximation is widely used
to approximate the SINR meta distribution since their higher moments are very
close [5], and it only requires the first and the second moments of the
success probability.

######  Remark 1 (Beta Approximation).

Using the beta approximation as mentioned in [5], the SINR meta distribution
can be approximated as

$\displaystyle\bar{F}^{\prime}(\theta,\gamma)\approx
1-I_{\gamma}\bigg{(}\frac{M_{1}(\theta)(M_{1}(\theta)-M_{2}(\theta))}{M_{2}(\theta)-M_{1}^{2}(\theta)},\frac{(M_{1}(\theta)-M_{2}(\theta))(1-M_{1}(\theta))}{M_{2}(\theta)-M_{1}^{2}(\theta)}\bigg{)},$
(11)

where,

$\displaystyle I_{x}(a,b)=\frac{\int_{0}^{x}t^{a-1}(1-t)^{b-1}{\rm
d}t}{B(a,b)},$ (12)

and $B(a,b)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}{\rm d}t$.

As shown in (11), beta approximation only requires to compute the first two
moments, which highly reduces the computing complexity.

###  III-B Proposed Approximation

Notice that the closer interference has the higher impact on the system
performance compared to the rest of the interferers. Here, we provide an
alternative approximation which approximates the interference by considering
the $j$-th nearest interfering BSs and the conditional expectation of the sum
of the remaining interfering BSs [39, 40].

We rewrite the aggregate interference based on the approximation policy
mentioned above.

######  Lemma 2 (Approximated Interference).

The interference at the typical user can be approximated as

$\displaystyle I_{j}$
$\displaystyle\approx\sum_{k=1}^{j}p_{t}h_{x_{k}}R_{k}^{-\alpha}+p_{t}G(R_{j}),$
(13)

where

$\displaystyle G(R_{j})=\frac{2\pi\lambda}{\alpha-2}R_{j}^{2-\alpha}.$ (14)

###### Proof:

As mentioned, the closer interfering BS has higher impact on the system
performance, hence, we consider the interference term composed of the exact
expression of the closest $j$ interferers and the conditional mean of the rest
of the terms,

$\displaystyle I_{j}$
$\displaystyle=\sum_{i\in\mathbb{N}}p_{t}h_{x_{i}}||x_{i}||^{-\alpha}=\sum_{k=1}^{j}p_{t}h_{x_{k}}R_{k}^{-\alpha}+\sum_{k=j+1}^{\infty}p_{t}h_{x_{k}}R_{k}^{-\alpha}$
$\displaystyle\approx\sum_{k=1}^{j}p_{t}h_{x_{k}}R_{k}^{-\alpha}+p_{t}\mathbb{E}\bigg{[}\sum_{i\in\mathbb{N}\setminus\\{1,\cdots,j\\}}h_{x_{i}}||x_{i}||^{-\alpha}\biggm{|}R_{j}\bigg{]},$
(15)

and let $G(R_{j})$ denotes the average interference (without $p_{t}$) from the
remaining interferers, which is a function of the location of the $j$-th
closest interferer, $R_{j}$, given by

$\displaystyle G(R_{j})$
$\displaystyle=\mathbb{E}\bigg{[}\sum_{i\in\mathbb{N}\setminus\\{1,\cdots,j\\}}h_{x_{i}}||x_{i}||^{-\alpha}\biggm{|}R_{j}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\bigg{[}\sum_{i\in\mathbb{N}\setminus\\{1,\cdots,j\\}}R_{i}^{-\alpha}\biggm{|}R_{j}\bigg{]}\stackrel{{\scriptstyle(b)}}{{=}}\frac{2\pi\lambda}{\alpha-2}R_{j}^{2-\alpha},$
(16)

where step $(a)$ follows from the assumption that all fading gains are
independent and exponentially distributed with mean of unity and step $(b)$
follows Campbell’s theorem [41] with conversion from Cartesian to polar
coordinates. ∎

Approximated success probability is the final requirement to derive the
approximated SINR meta distribution. Based on the approximation of the
aggregate interference, we rewrite the interference term in SINR and the
success probability is given in the following lemma.

######  Lemma 3 (Approximated Conditional Success Probability).

The conditional success probability is approximated by

$\displaystyle
P_{s,j}(\theta)\approx\exp\bigg{(}-\frac{\theta}{p_{t}}R_{0}^{\alpha}(p_{t}G(R_{j})+\sigma^{2})\bigg{)}\bigg{(}\frac{1}{1+\theta
R_{0}^{\alpha}\sum_{k=1}^{j}R_{k}^{-\alpha}}\bigg{)}.$ (17)

###### Proof:

Recall that $R_{i}=||x_{i}||$ is the distance from the $i$-th closest
interfering BS to the typical user and $R_{0}$ is the distance to the serving
BS. The conditional success probability is

$\displaystyle P_{s,j}(\theta)$
$\displaystyle=\mathbb{P}\bigg{(}\frac{p_{t}h_{x_{0}}R_{0}^{-\alpha}}{\sum_{i\in\mathbb{N}}p_{t}h_{x_{i}}R_{i}^{-\alpha}+\sigma^{2}}>\theta\biggm{|}\Phi\bigg{)}$
$\displaystyle=\mathbb{P}\bigg{(}h_{x_{0}}>\frac{\theta}{p_{t}}R_{0}^{\alpha}(\sum_{k=1}^{j}p_{t}h_{x_{k}}R_{k}^{-\alpha}+\sum_{i\in\mathbb{N}\setminus\\{1,\cdots,j\\}}p_{t}h_{x_{i}}R_{i}^{-\alpha}+\sigma^{2})\biggm{|}\Phi\bigg{)}$
$\displaystyle\approx\mathbb{E}_{h_{\\{x_{k}\\}}}\bigg{[}\exp\bigg{(}-\theta
R_{0}^{\alpha}\sum_{k=1}^{j}h_{x_{k}}R_{k}^{-\alpha}-\theta
R_{0}^{\alpha}G(R_{j})-\frac{\theta}{p_{t}}R_{0}^{\alpha}\sigma^{2}\bigg{)}\bigg{]}$
$\displaystyle=\exp\bigg{(}-\frac{\theta}{p_{t}}R_{0}^{\alpha}(p_{t}G(R_{j})+\sigma^{2})\bigg{)}\prod_{k=1}^{j}\bigg{(}\frac{1}{1+\theta
R_{0}^{\alpha}R_{k}^{-\alpha}}\bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\approx}}\exp\bigg{(}-\frac{\theta}{p_{t}}R_{0}^{\alpha}(p_{t}G(R_{j})+\sigma^{2})\bigg{)}\bigg{(}\frac{1}{1+\theta
R_{0}^{\alpha}\sum_{k=1}^{j}R_{k}^{-\alpha}}\bigg{)},$ (18)

in which step (a) follows by ignoring the higher order terms of
$\theta(\frac{R_{0}}{R_{j}})^{\alpha}$. ∎

Now we are able to proceed to the final expression of the proposed
approximation. Since there is less multiplication in conditional success
probability in (17) compared to (46), the SINR meta distribution can be easily
obtained.

######  Theorem 2 (Approximated Meta Distribution).

The approximated SINR meta distribution,
$\bar{F}_{j}^{\prime}(\theta,\gamma)$, is given by

$\displaystyle\bar{F}_{j}^{\prime}(\theta,\gamma)\approx\int_{(\mathbb{R}^{+})^{j}}F_{R_{0}|R_{1}}(K_{j}(r_{1},r_{2},...,r_{j}))f_{R_{1},R_{2},...R_{j}}(r_{1},r_{2},...,r_{j}){\rm
d}r_{1}{\rm d}r_{2},...{\rm d}r_{j},$ (19)

which is a $j$-dimensional integral, and

$\displaystyle
K_{j}(r_{1},r_{2},...,r_{j})=\bigg{(}-\frac{1}{\theta\sum_{k=1}^{j}r_{k}^{-\alpha}}+\frac{1}{s(r_{j})}W\bigg{(}0,\frac{s(r_{j})\exp(s(r_{j})\theta^{-1}(\sum_{k=1}^{j}r_{k}^{-\alpha})^{-1})}{\gamma\theta
r_{1}^{-\alpha}}\bigg{)}\bigg{)}^{\frac{1}{\alpha}},$ (20)

where $W(k,x)$ is the $k$-th branch of the Lambert W function,
$s(r_{j})=\frac{\theta}{p_{t}}(p_{t}G(r_{j})+\sigma^{2})$,
$F_{R_{0}|R_{1}}(r)$ and $f_{R_{1},R_{2},...R_{j}}(r_{1},r_{2},...,r_{j})$ are
given in (6) and (7).

###### Proof:

By using the definition of SINR meta distribution, the CCDF of the conditional
success probability is given by

$\displaystyle\bar{F}_{j}^{\prime}(\theta,\gamma)=\mathbb{P}(P_{s,j}(\theta)>\gamma)$
$\displaystyle=\mathbb{E}_{R_{1},\cdots,R_{j}}\bigg{[}\mathbb{P}\bigg{(}\exp\bigg{(}-\frac{\theta}{p_{t}}R_{0}^{\alpha}(p_{t}G(R_{j})+\sigma^{2})\bigg{)}\prod_{k=1}^{j}\bigg{(}\frac{1}{1+\theta
R_{0}^{\alpha}R_{k}^{-\alpha}}\bigg{)}>\gamma\mid
R_{1},\cdots,R_{j}\bigg{)}\bigg{]}$
$\displaystyle=\mathbb{E}_{R_{1},\cdots,R_{j}}\bigg{[}\mathbb{P}\bigg{(}\exp(-s(R_{j})R_{0}^{\alpha})>\gamma\prod_{k=1}^{j}\bigg{(}1+\theta
R_{0}^{\alpha}R_{k}^{-\alpha}\bigg{)}\mid R_{1},\cdots,R_{j}\bigg{)}\bigg{]}$
$\displaystyle\approx\mathbb{E}_{R_{1},\cdots,R_{j}}\bigg{[}\mathbb{P}\bigg{(}\exp(-s(R_{j})R_{0}^{\alpha})>\gamma\bigg{(}1+\theta
R_{0}^{\alpha}\sum_{k=0}^{j}R_{k}^{-\alpha}\bigg{)}\mid
R_{1},\cdots,R_{j}\bigg{)}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{R_{1},\cdots,R_{j}}\bigg{[}\mathbb{P}\bigg{(}R_{0}<\bigg{(}-\frac{1}{\theta\sum_{k=1}^{j}R_{k}^{-\alpha}}+$
$\displaystyle\qquad\qquad\quad\frac{1}{s(R_{j})}W\bigg{(}0,\frac{s(R_{j})\exp(s(R_{j})\theta^{-1}(\sum_{k=1}^{j}R_{k}^{-\alpha})^{-1})}{\gamma\theta\sum_{k=1}^{j}R_{k}^{-\alpha}}\bigg{)}\bigg{)}^{\frac{1}{\alpha}}\mid
R_{1},\cdots,R_{j}\bigg{)}\bigg{]},$ (21)

where step $(a)$ follows from that Lambert W function defined by:
$W(0,x)\exp(W(0,x))=x$, and this step can be solved by using MATLAB or
Mathematica, and the proof completes by using the conditional CDF of $R_{0}$
and the joint PDF of $R_{1},\cdots,R_{j}$ provided in Lemma 1. ∎

Observing that (19) requires to computing the joint PDF from $R_{1}$ to
$R_{j}$, we observe that it is still too complex and not practical in general.
Considering that the nearest interferer BS has the highest impact on the
system performance, we further simplify the proposed approximation which only
composed of the exact expression of the nearest interferer and the conditional
mean of the rest of the terms.

We would like to clarify that the approximation proposed in Cor. 1 is
considering the dominant interferer exactly while the rest are considered in
an average sense. However, we name it the dominant interferer-based
approximation to simplify the name.

######  Corollary 1 (The Dominant Interferer-based Meta distribution).

If we only consider the first nearest interferer BS exactly while the rest are
considered in an average sense, (19) can be further simplified,

$\displaystyle\bar{F}^{\prime}_{1}(\theta,\gamma)\approx\int_{0}^{\infty}F_{R_{0}|R_{1}}(K_{1}(r,\theta,\gamma))f_{R_{1}}(r){\rm
d}r,$ (22)

where,

$\displaystyle K_{1}(r,\theta,\gamma)=\bigg{(}-\frac{1}{\theta
r^{-\alpha}}+\frac{1}{s(r)}W\bigg{(}0,\frac{s(r)\exp(s(r)\theta^{-1}r^{\alpha})}{\gamma\theta
r^{-\alpha}}\bigg{)}\bigg{)}^{\frac{1}{\alpha}},$ (23)

in which $s(r)=\frac{\theta}{p_{t}}(p_{t}G(r)+\sigma^{2})$ and
$G(r)=\frac{2\pi\lambda}{\alpha-2}r^{2-\alpha}$.

###### Proof:

Proof completes by setting $j=1$ in (19). ∎

######  Remark 2.

In the case of $\sigma^{2}=0$, e.g., approximation of SIR meta distribution,
$s(r)=\theta G(r)$, (23) can be further simplified,

$\displaystyle K_{1}^{\prime}(r,\theta,\gamma)=\bigg{(}-\frac{1}{\theta
r^{-\alpha}}+\frac{1}{s(r)}W\bigg{(}0,\frac{s^{\prime}(r)}{\gamma}\exp(s^{\prime}(r))\bigg{)}\bigg{)}^{\frac{1}{\alpha}},$
(24)

where $s^{\prime}(r)=\frac{2\pi\lambda}{(\alpha-2)}r^{2}$. Moreover, if we
only consider the dominant interferer, that is $G(r)=0$,

$\displaystyle
K_{1}^{\prime\prime}(r,\theta,\gamma)=\bigg{(}\frac{1}{\gamma\theta
r^{-\alpha}}-\frac{1}{\theta
r^{-\alpha}}\bigg{)}^{\frac{1}{\alpha}}=r\bigg{(}\frac{1-\gamma}{\gamma\theta}\bigg{)}^{\frac{1}{\alpha}},$
(25)

consequently, (22) becomes

$\displaystyle\bar{F}^{\prime\prime}(\theta,\gamma)=\min\bigg{(}1,\bigg{(}\frac{1-\gamma}{\gamma\theta}\bigg{)}^{\frac{1}{\alpha}}\bigg{)},$
(26)

which is the same as the result of the nearest-interferer-only approximation
mentioned in [11, Cor. 3].

The difference and relation between the proposed approximation, dominant-
interferer approximation (22), and the nearest-interferer-only approximation,
[11, Cor. 3], are: (i) the nearest-interferer-only approximation is an upper
bound of the meta distribution while the proposed approximation is neither
upper bound nor lower bound and it is tighter (more details shown in Numerical
Results Section), (ii) the proposed approximation considered all the
interferer while the result in [11, Cor. 3] removed some interferers, and
(iii) the proposed approximated SINR meta distribution is an approximation of
exact SINR meta distribution and nearest-interferer-only meta distribution and
SIR meta distribution are special cases of the proposed approximation.

The SINR meta distribution is computed by the joint CDF and the PDF of the
first and the second nearest BSs, which are the fundamental distance
distributions of point processes. Hence, the proposed approximation has
different requirements compared with the beta approximation. In the point
processes which are able to obtain the joint distance distributions, the
proposed approximation can be a candidate approach to approximate the SINR
meta distribution. In the numerical results part, we show that the dominant
interferer-based approximation is already very accurate, e.g., the gap between
the exact value and the proposed approximation in the case of PPP networks
with $\lambda=1$ km-2 and $\theta=26$ dB is about $0.02$, while the gaps for
the remaining points are at most about $0.03$, in the downlink SINR meta
distribution of Poisson network with Rayleigh fading.

##  IV The Proposed Approximation in Poisson Networks

In this section, we use the proposed approximation in three wireless cellular
networks modeled by three commonly used point processes, respectively, and one
ad hoc network, to show its accuracy and tractability. The one ad hoc scenario
and three cellular scenarios are (i) Poisson bipolar networks, (ii) Matérn
cluster process (MCP), (iii) $K$-tier Poisson network, and (iv) Poisson line
Cox process (PLCP), respectively. In (i), (ii) and (iii), we give the
expressions of the SINR meta distribution for both beta approximation and the
proposed approximation. In (iv), we use the proposed approximation under PLCP-
distributed interference and PPP-approximated interference, respectively, and
compared the proposed approximation with a transdimensional PPP-based (TPPP-
based) approximation as mentioned in [28, (16)].

###  IV-A Poisson Bipolar Networks

In this part, we consider a Poisson bipolar model, where the transmitters form
a PPP, denoted by $\Phi$ with density $\lambda$, and each of the transmitter
has a dedicated receiver at distance $R$ in a random direction. Let $p_{t}$ be
the transmit power and all channels are subject to Rayleigh fading, and we use
the standard path loss model with exponent $\alpha$.

Before investigating the SINR meta distribution, we first introduce some
important distance distributions in Poisson bipolar networks. Note that in
Poisson bipolar network, each receiver has a dedicate transmitter at distance
$R$. Therefore, the distance to the serving BS is determined and independent
from the nearest interferer, while the distance distribution of the nearest
interferer follows the first contact distance distribution in PPP. The CDF of
$R_{0}$ given $R_{1}$ equals to the CDF of $R_{0}$, which is a shifted step
function, and $R_{1}$ is the the contact distance in PPP,

$\displaystyle f_{R_{0}}(r)$ $\displaystyle=\Delta_{R}(r),$ $\displaystyle
F_{R_{0}}(r)$ $\displaystyle=\left\\{\begin{aligned} 0,&\quad r<R,\\\ 1,&\quad
r\geq R,\\\ \end{aligned}\right.,$ $\displaystyle f_{R_{1}}(r)$
$\displaystyle=2\pi\lambda r\exp(-\pi\lambda r^{2}),$ (27)

in which $\Delta_{a}(x)$ is an impulse at $a$ and satisfies
$\int_{0}^{\infty}\Delta_{a}(r)dr=1$.

The SIR success probability and $b$-th moment of the conditional success
probability of a Poisson bipolar network are well known [41, 5] and given by

$\displaystyle P_{s,{\rm SIR}}(\theta)$
$\displaystyle=\exp(-C\theta^{\delta}),$ $\displaystyle M_{b,{\rm
SIR}}(\theta)$
$\displaystyle=\exp\bigg{(}-C\theta^{\delta}\frac{\Gamma(b+\delta)}{\Gamma(1+\delta)\Gamma(b)}\bigg{)},$
(28)

where $\delta=2/\alpha$ and $C$ is a coefficient that does not depend on
$\theta$: $C=\lambda\pi R^{2}\Gamma(1-\delta)\Gamma(1+\delta)$. Therefore, the
SINR success probability and the $b$-th moment are obtained by adding the
exponential term and given by, respectively,

$\displaystyle P_{s}(\theta)=$ $\displaystyle P_{s,{\rm
SIR}}(\theta)\exp\bigg{(}-\frac{\theta
R^{\alpha}}{{p_{t}}}\sigma^{2}\bigg{)},$ $\displaystyle M_{b}(\theta)=$
$\displaystyle\exp\bigg{(}-C\theta^{\delta}\frac{\Gamma(b+\delta)}{\Gamma(1+\delta)\Gamma(b)}\bigg{)}\exp\bigg{(}-b\frac{\theta
R^{\alpha}}{{p_{t}}}\sigma^{2}\bigg{)}.$ (29)

Therefore, the beta approximation of the SINR meta distribution of a Poisson
bipolar network is obtained by substituting (29) into (11), and the proposed
approximation of the SINR meta distribution is obtained by substituting (IV-A)
into (22).

###  IV-B Matérn Cluster Process

In this part, we consider a Matérn cluster point process to model the
locations of users and BSs [42]. In MCP, the clusters are modeled as disks
with radii $r_{c}$ whose centers are modeled as a PPP, $\Phi$ with density
$\lambda$, while the users are uniformly distributed within the disk. Assume
that the user associates with its cluster BS. Let $p_{t}$ be the transmit
power of BSs and all channels are subject to Rayleigh fading, and we use the
standard path loss model with exponent $\alpha$.

In order to use the proposed approximation, some distance distributions are
given below. Since we assume that the user associates with its cluster BS, the
distance to the serving BS is independent from the nearest interferer. The CDF
of $R_{0}$ given $R_{1}$ equals to the CDF of $R_{0}$ and $R_{1}$ is the the
first contact distance in PPP,

$\displaystyle f_{R_{0}}(r)$ $\displaystyle=\frac{2r}{r_{c}^{2}},\quad
r<r_{c},$ $\displaystyle f_{R_{1}}(r)$ $\displaystyle=2\pi\lambda
r\exp(-\pi\lambda r^{2}),$ $\displaystyle F_{R_{0}}(r)$
$\displaystyle=\frac{r^{2}}{r_{c}^{2}},\quad r<r_{c},$ (30)

where $r_{c}$ is the user cluster radius.

The SINR success probability and the $b$-th moment are, respectively, given by

$\displaystyle P_{s}(\theta)$ $\displaystyle=\mathbb{P}({\rm
SINR}>\theta)=\mathbb{P}\bigg{(}\frac{p_{t}hR_{0}^{-\alpha}}{I+\sigma^{2}}>\theta\bigg{)}=\mathbb{P}\bigg{(}h>\frac{\theta
R_{0}^{\alpha}}{p_{t}}(I+\sigma^{2})\bigg{)}=\exp\bigg{(}-g(R_{0})(I+\sigma^{2})\bigg{)}$
$\displaystyle=\prod_{i\in\mathbb{N}}\bigg{(}\frac{1}{1+g(R_{0})p_{t}R_{i}^{-\alpha}}\bigg{)}\exp(-g(R_{0})\sigma^{2}),$
$\displaystyle M_{b}(\theta)$
$\displaystyle=\mathbb{E}\bigg{[}\prod_{i\in\mathbb{N}}\bigg{(}\frac{1}{1+g(R_{0})p_{t}R_{i}^{-\alpha}}\bigg{)}^{b}\exp(-bg(R_{0})\sigma^{2})\bigg{]}$
$\displaystyle=\int_{0}^{r_{c}}\exp\bigg{(}-2\pi\lambda\int_{0}^{\infty}\bigg{[}1-\bigg{(}\frac{1}{1+g(r)p_{t}z^{-\alpha}}\bigg{)}^{b}\bigg{]}z{\rm
d}z\bigg{)}\exp(-bg(r)\sigma^{2})f_{R_{0}}(r){\rm d}r,$ (31)

where $g(r)=\frac{\theta r^{\alpha}}{p_{t}}$.

Therefore, the beta approximation of the SINR meta distribution of a MCP
network is obtained by substituting (31) into (11), and the proposed
approximation of the SINR meta distribution is obtained by substituting (30)
into (22).

###  IV-C $K$-tier Poisson Point Process

In this part, we consider a general $K$-tier cellular network model, where BSs
of each tier follow a homogeneous independent PPP $\Phi_{i}$ with density
$\lambda_{i}$. For the BSs in the $i$-th tier, the transmit power is
$p_{t,i}$. Assume that the user associates with the BS that provides the
strongest average received power.

In order to use the proposed approximation, which is a function of the first
and the second nearest BSs, we map the BSs on the $i$-th tier to the $1$-st
tier to obtain the distance distributions of the first and the second nearest
BSs. After mapping, the equivalent density in the $1$-st tier is

$\displaystyle\lambda_{i}^{\prime}=(\frac{p_{t,1}}{p_{t,i}})^{-\delta}\lambda_{i}.$
(32)

By the results of linear mapping and the superposition property of the point
process, the new $1$-tier network is still a homogeneous PPP $\Phi^{\prime}$
with density

$\displaystyle\lambda^{\prime}=\sum_{i=1}(\frac{p_{t,1}}{p_{t,i}})^{-\delta}\lambda_{i}^{\prime}.$
(33)

We then use the result from the single tier PPP network. The $b$-th moment is

$\displaystyle
M_{b}(\theta)=\mathbb{E}\bigg{[}\exp\bigg{(}-\lambda^{\prime}F_{b}\bigg{)}\exp\bigg{(}-\frac{b\theta}{p_{t}}R_{0}^{\alpha}\sigma^{2}\bigg{)}\bigg{]},$
(34)

where $F_{b}$ is given in (10).

Therefore, the beta approximation of the SINR meta distribution of a $K$-tier
PPP network is obtained by substituting (34) into (11), and the proposed
approximation of the SINR meta distribution is obtained by substituting the
equivalent BS density in (33) and the distance distribution in Lemma 1 into
(22).

###  IV-D Poisson Line Cox Process

Figure 1: Illustration of the conditional SINR success probability of a
Poisson line Cox process for $\lambda=1.6$ BS/km2, where $\lambda_{l}=0.4/\pi$
km/km2 and $\lambda_{p}=0.4$ /km. The markers denote the BSs, users are
indicated by crosses, blue segments denote the links and gray lines denote the
Poisson line process. The number next to each link is its success probability.
Note that all the links have a number denoting the success probability, but
due to the limitation of the space, some numbers may not show.

In this section, we consider a general Poisson line Cox process (PLCP)
cellular network model, as shown in Fig. 1, where the locations of BSs and
users are modeled by the same Poisson line process, with line density
$\mu_{l}=\pi\lambda_{l}$, where $\lambda_{l}$ is the point density of the
corresponding point process in representation space, and two independent 1D
PPPs, with densities $\lambda_{p}$ and $\lambda_{p,u}$, respectively. Let
$\Phi$ be the point set of the locations of BSs and the corresponding density
is $\lambda=\lambda_{p}\lambda_{l}\pi$. Assume that the user associates with
the nearest BS. Noting that in PLCP network, we define the user point process
since it influence the distance distribution, e.g., a typical line passes
though the user.

To obtain the distance distribution, we need to compute the probability of the
number of the points falling in a unit region. In PLCP, the PMF of the number
of points in $B(0,r)$ is given in [43, 44] and obtained by

$\displaystyle\mathbb{P}(\mathcal{N}_{p}(B(0,r)\cap\Phi_{0}^{!})=k)=(-\lambda_{p})^{k}\frac{\partial^{k}\mathcal{L}^{!}_{(0,r)}(\lambda_{p})}{\partial\lambda_{p}^{k}},$
(35)

where $\mathcal{N}_{p}(A\cap\Phi)$ denotes the counting measure, which counts
the number of points in point set $\Phi$ falling in the A, $\Phi_{0}^{!}$
denotes the reduced Palm distribution [44, Cor. 4.2], in which a typical line
passes though the origin, $B(0,r)$ denotes the ball centered at the origin
with radius $r$ and $\mathcal{L}^{!}_{(0,r)}(s)$ is Laplace transform of the
total chord length in $B(0,r)$ under the Palm distribution,

$\displaystyle\mathcal{L}_{(0,r)}^{!}(s)$
$\displaystyle=\exp\left[-2sr-2\pi\lambda_{l}\int_{0}^{r}1-\exp\left(-2s\sqrt{r^{2}-\rho^{2}}\right)\mathrm{d}\rho\right].$
(36)

######  Lemma 4 (Distance Distributions of PLCP).

The PDF of $R_{0}$ and the PDF of $R_{1}$ are, respectively, given by

$\displaystyle f_{R_{0}}(r_{0})$
$\displaystyle=2\bigg{(}\lambda_{p}+\pi\lambda_{l}\int_{0}^{r_{0}}\frac{2r_{0}\lambda_{p}}{\sqrt{r_{0}^{2}-x^{2}}}\exp(-2\lambda_{p}\sqrt{r_{0}^{2}-x^{2}}){\rm
d}x\bigg{)}$
$\displaystyle\quad\times\exp\left[-2\lambda_{p}r_{0}-2\pi\lambda_{l}\int_{0}^{r_{0}}1-\exp\left(-2\lambda_{p}\sqrt{r_{0}^{2}-\rho^{2}}\right)\mathrm{d}\rho\right],$
(37) $\displaystyle f_{\rm R_{1}}(r_{1})$
$\displaystyle=\exp\left[-2\lambda_{p}r_{1}-2\pi\lambda_{l}\int_{0}^{r_{1}}1-\exp\left(-2\lambda_{p}\sqrt{r_{1}^{2}-\rho^{2}}\right)\mathrm{d}\rho\right]$
$\displaystyle\quad\times\bigg{[}\lambda_{p}\bigg{(}-2+4r_{1}\lambda_{p}+4\lambda_{p}\lambda_{l}\pi\int_{0}^{r_{1}}2\sqrt{r_{1}^{2}-x^{2}}\exp(-2\lambda_{p}\sqrt{r_{1}^{2}-x^{2}}){\rm
d}x$
$\displaystyle\qquad-2\pi\lambda_{l}\int_{0}^{r_{1}}\frac{r_{1}(2-4\lambda_{p}\sqrt{r_{1}^{2}-x^{2}})}{\sqrt{r_{1}^{2}-x^{2}}}\exp(-2\lambda_{p}\sqrt{r_{1}^{2}-x^{2}}){\rm
d}x$
$\displaystyle\qquad+4\pi\lambda_{l}\int_{0}^{r_{1}}\frac{r_{1}\lambda_{p}}{\sqrt{r_{1}^{2}-x^{2}}}\exp(-2\lambda_{p}\sqrt{r_{1}^{2}-x^{2}}){\rm
d}x$
$\displaystyle\quad\qquad\times(r_{1}+\pi\lambda_{l}\int_{0}^{r_{1}}2\sqrt{r_{1}^{2}-x^{2}}\exp(-2\lambda_{p}\sqrt{r_{1}^{2}-x^{2}}){\rm
d}x)\bigg{)}$
$\displaystyle\quad+2(\lambda_{p}+\pi\lambda_{l}\int_{0}^{r_{1}}\frac{2r_{1}\lambda_{p}}{\sqrt{r_{1}^{2}-x^{2}}}\exp(-2\lambda_{p}\sqrt{r_{0}^{2}-x^{2}}){\rm
d}x)\bigg{]}.$ (38)

###### Proof:

The first contact distance distribution is derived by computing the void
probability, which is defined as the probability of no points in $B(0,r)$.
From the definition of PLCP, the void probability, which is also the CCDF of
$R_{0}$, immediately follows,

$\displaystyle\mathbb{P}(R_{0}>r_{0})$
$\displaystyle=\mathbb{P}(\mathcal{N}_{p}(B(0,r_{0})\cap\Phi_{0}^{!})=0)=\mathcal{L}^{!}_{(0,r_{0})}(\lambda_{p})$
$\displaystyle=\exp\left[-2\lambda_{p}r_{0}-2\pi\lambda_{l}\int_{0}^{r_{0}}1-\exp\left(-2\lambda_{p}\sqrt{r_{0}^{2}-\rho^{2}}\right)\mathrm{d}\rho\right],$
(39)

$f_{\rm R_{0}}(r_{0})$ then obtain by one minus the derivative of above
equation over $r_{0}$. Similarly, the CCDF of the distance to the second
neighbor in PLCP is derived by

$\displaystyle\mathbb{P}(R_{1}>r_{1})$
$\displaystyle=\mathbb{P}(\mathcal{N}_{p}(B(0,r_{1})\cap\Phi_{0}^{!})=0)+\mathbb{P}(\mathcal{N}_{p}(B(0,r_{1})\cap\Phi_{0}^{!})=1)$
$\displaystyle=\mathcal{L}^{!}_{(0,r_{1})}(\lambda_{p})+(-\lambda_{p})\frac{\partial\mathcal{L}^{!}_{(0,r_{1})}(\lambda_{p})}{\partial\lambda_{p}},$
(40)

proof completes by taking the derivative over $r_{1}$,

$\displaystyle f_{R_{1}}(r_{1})$
$\displaystyle=-\frac{\partial\mathcal{L}_{(0,r_{1})}^{!}(\lambda_{p})}{\partial
r_{1}}-(-\lambda_{p})\frac{\partial^{2}\mathcal{L}_{(0,r_{1})}^{!}(\lambda_{p})}{\partial\lambda_{p}\partial
r_{1}}.$ (41)

∎

In what follows, we compute the conditional distance distribution in PLCP
networks.

######  Lemma 5 (Conditional Distance Distribution of PLCP).

The conditional CDF $F_{R_{0}|R_{1}}(r_{0}|R_{1})$ is the final requirement to
approach the proposed approximation of the SINR meta distribution,

$\displaystyle F_{R_{0}|R_{1}}(r_{0}|R_{1})=$
$\displaystyle\frac{f_{R_{0}}(r_{0})}{f_{R_{1}}(R_{1})}-\frac{f_{R_{0}}(r_{0})}{f_{R_{1}}(R_{1})}\exp(-2\lambda_{p}(R_{1}-r_{0}))$
$\displaystyle\times\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{R_{1}}1-\exp\bigg{(}-2\lambda_{p}\bigg{(}\sqrt{R_{1}^{2}-\rho^{2}}-\sqrt{\max(0,r_{0}^{2}-\rho^{2})}\bigg{)}\bigg{)}{\rm
d}\rho\bigg{)}.$ (42)

###### Proof:

We first obtain $F_{R_{1}|R_{0}}(r_{1}|r_{0})$, which denotes the CDF of the
distance to the second nearest neighbor (which is the probability that the
distance to the second nearest neighbor is less than $r_{1}$) given the
nearest one located at distance $r_{0}$. To do so, we start with computing the
CCDF,

$\displaystyle\bar{F}_{R_{1}|R_{0}}(r_{1}|R_{0})$
$\displaystyle=\mathbb{P}(R_{1}>r_{1}|R_{0})$
$\displaystyle=\mathbb{P}(\mathcal{N}_{p}(B(0,r_{1})\cap\Phi_{0}^{!}-B(0,R_{0})\cap\Phi_{0}^{!})=0)=\mathcal{L}_{(0,r_{1})-(0,R_{0})}^{!}(\lambda_{p})$
$\displaystyle=\exp\bigg{(}-2\pi\lambda_{l}\int_{0}^{r_{1}}1-\exp\bigg{(}-2\lambda_{p}\bigg{(}\sqrt{r_{1}^{2}-\rho^{2}}-\sqrt{\max(0,r_{0}^{2}-\rho^{2})}\bigg{)}\bigg{)}{\rm
d}\rho\bigg{)}$ $\displaystyle\quad\times\exp(-2\lambda_{p}(r_{1}-R_{0})),$
(43)

in which the subscript $(0,r_{1})-(0,R_{0})$ denotes the area
$B(o,r_{1})\setminus B(o,R_{0})$, then the proof completes by applying the
Bayes’ theorem. ∎

Note that the interference distribution of PLCP networks is slightly different
from PPP networks. Hence, the approximation of the interference term
$G(R_{1})$ in (22) should be recomputed, which is shown in the following
lemma.

######  Lemma 6 (Interference Approximation of PLCP).

In PLCP, the conditional mean of the rest interference (without $p_{t}$), is
given by

$\displaystyle G_{1}(R_{1})=$ $\displaystyle
2\pi\lambda_{l}\lambda_{p}\bigg{(}\int_{R_{1}}^{\infty}\int_{0}^{\infty}(\rho^{2}+r^{2})^{\frac{-\alpha}{2}}{\rm
d}r{\rm
d}\rho+\int_{0}^{R_{1}}\int_{\sqrt{\rho^{2}-r^{2}}}^{\infty}(\rho^{2}+r^{2})^{\frac{-\alpha}{2}}{\rm
d}r{\rm d}\rho\bigg{)}$
$\displaystyle+2\lambda_{p}\frac{R_{1}^{1-\alpha}}{\alpha-1}.$ (44)

###### Proof:

In PLCP, the remaining interference is composed of three parts: (i)
interference from the typical line (the line passes though the origin,
$L_{0}$), (ii) interference from the remaining lines, which do not intersect
with $B(0,R_{1})$ and (iii) interference from the remaining lines, which
intersect with $B(0,R_{1})$:

$\displaystyle
G_{1}(R_{1})=\mathbb{E}\bigg{[}\sum_{i\in\mathbb{N}\setminus\\{1\\}}h_{x_{i}}||x_{i}||^{-\alpha}|R_{1}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sum_{x\in
L_{0}\setminus\\{B(0,R_{1})\\}}||x||^{-\alpha}|R_{1}\bigg{]}+\mathbb{E}\bigg{[}\sum_{x\in\Phi_{0}\setminus\\{B(0,R_{1})\\},\rho<R_{1}}||x||^{-\alpha}|R_{1}\bigg{]}+\mathbb{E}\bigg{[}\sum_{x\in\Phi_{0}\setminus\\{B(0,R_{1})\\},\rho>R_{1}}||x||^{-\alpha}|R_{1}\bigg{]},$
(45)

in which $\Phi_{0}=\Phi_{0}^{!}\setminus L_{0}$, $\rho$ denotes the distance
from the line to the origin, and the proof completes by using Campbell’s
theorem [41] with conversion from Cartesian to polar coordinates. ∎

Since PPP is a good approximation of PLCP [44], we compute the proposed
approximation based on (14) and (6), respectively, in the numerical result
section and compare these two approximations.

Consequently, the proposed approximation of the SINR meta distribution is
obtained by substituting the distance distribution in (42) and (41) into (22).
Besides, authors in [28] provides an interesting approximation to the SIR meta
distribution of PLP with fixed transmission distance and we compare the
proposed approximation to their approximation (by adding the noise and taking
the expectation over $R_{0}$).

Here, we provide some advantages and limitations of the proposed
approximation. The proposed approximation has different requirements, which
are joint distance distributions and a spatial average of the interference
except for the nearest interferer, compared to the traditional approximation.
Therefore, for the point processes where the joint distance distributions are
available or can be approximated, the proposed approximation is simple since
it only needs one integral which is about the distance to the nearest
interferer (the expectation over $R_{1}$). Besides, computing moments is not
always trivial, it depends on the system model, e.g., the point process. The
moments of PPP, MCP, and Poisson bipolar networks are trivial, for PLCP,
however, the moments are not straightforward owing to the locations of
interferers. For instance in [44, (7.46)] is the Laplace transform of the
interference in PLCP networks, which is the PGFL part of the moments.
Therefore, the proposed dominant interferer-based approximation does provide
some simplicity in computing SINR meta distribution. However, the proposed
approximation is restricted by the ability to solve $R_{0}$ (step (a) in
(III-B)), e.g., it can not be solved under Nakagami-m fading, and for the
point processes which has an unknown joint distance distribution, the proposed
approximation may not work.

##  V Numerical Results

In this section, we validate the proposed approximation via Monte-Carlo
simulations with a large number of iterations ($5\times 10^{5}$ iterations, in
which $100$ for location realizations and $5000$ for fading realizations) to
ensure the accuracy, and compare it with the beta approximation and the exact
value of the SINR meta distribution. We start with the standard PPP network.
We first generate two realizations of PPPs for the locations of users and BSs.
Users are located in the Voronoi cells formed by BSs. While the realizations
of PPPs are fixed, the fading realizations change in each iteration and we
compute the success probability for each link. We then obtain the CCDF of the
success probability of each realization. Unless stated otherwise, we use the
system parameters listed herein. The transmit power of BS is $p_{t}=10$ W,
path-loss exponent is $\alpha=4$, noise power is $\sigma^{2}=10^{-9}$ W and
the density of BS changes from $0.1$ to $10$ BS/km2. In the single-tier case
either $p_{t}$ or $\sigma^{2}$ can be set to $1$ since only their ratio
matters, however, we set the values of $p_{t}$ or $\sigma^{2}$ separately
since we have a $K$-tier PPP network and we would like to be consistent in
parameters.

Note that in (22), Lambert W function $W(\cdot,\cdot)$ is used. When we
compute this function in MATLAB, we need to write the code by using
logarithmic input which allows for much larger arguments than the built-in
function to avoid infinity output, and related codes can be found in [45].

Besides, to analyze the accuracy of the proposed approximation, we verify it
through KL divergence [46] numerically which compares the probability
distribution of the proposed approximation with the exact distribution:

$\displaystyle D_{\rm
KL,prop|exact}=\sum_{\gamma^{\prime}\in\chi}f_{1}^{\prime}(\gamma^{\prime},\theta)\log\bigg{(}\frac{f_{1}^{\prime}(\gamma^{\prime},\theta)}{f(\gamma^{\prime},\theta)}\bigg{)},$

in which $f_{1}^{\prime}(\gamma^{\prime},\theta)$ is obtained by discretizing
$\bar{F}_{1}^{\prime}(\gamma,\theta)$:
$f_{1}^{\prime}(\gamma^{\prime}(i),\theta)=\bar{F}_{1}^{\prime}(\gamma(i),\theta)-\bar{F}_{1}^{\prime}(\gamma(i+1),\theta)$,
$\gamma^{\prime}(i)=\gamma(i)$, $\chi$ denotes the probability space:
$\chi=0.01,0.02,\cdots,0.99$, and similar steps for obtaining
$f(\gamma^{\prime},\theta)$ from $F(\gamma^{\prime},\theta)$. In addition, we
compared the KL divergence of the proposed approximation with the beta
approximation and the KL divergence of beta approximation is

$\displaystyle D_{\rm
KL,beta|exact}=\sum_{\gamma^{\prime}\in\chi}f^{\prime}(\gamma^{\prime},\theta)\log\bigg{(}\frac{f^{\prime}(\gamma^{\prime},\theta)}{f(\gamma^{\prime},\theta)}\bigg{)},$

in which $f^{\prime}(\gamma^{\prime},\theta)$ is obtained by discretizing
$\bar{F}^{\prime}(\gamma,\theta)$. Note that the above analysis is based on
the numerical results, and the lower absolute value implies better
approximation performance.

Figure 2: SINR meta distribution of PPP networks at $\lambda=1$ BS/km2,
$\alpha=4$ and $p_{t}=10$ W. The solid lines are exact values based on
simulations, dash lines are for beta approximation, and markers for the
proposed approximation, respectively.  Figure 3: SINR meta distribution of PPP
networks at different density values from $\lambda=0.1$ to $\lambda=10$
BS/km2, $\alpha=4$ and $p_{t}=10$ W. The solid lines are exact values based on
simulations, dash lines are for beta approximation and markers are for the
proposed approximation, respectively. $\gamma=0.3,0.6,0.9$ in (a), (b), (c),
respectively.

Figure 4: SINR meta distribution of PPP networks at different path-loss values
(a) $\alpha=2.5$, (b) $\alpha=3$, (c) $\alpha=3.5$, respectively, while
$\lambda=1$ BS/km2 and $p_{t}=10$ W. The solid lines are exact values based on
simulations, dash lines are for beta approximation and markers are for the
proposed approximation, respectively. (d) The nearest-interferer-only
approximation and the proposed dominant interferer-based approximation of SINR
meta distribution at $\alpha=3.5$.

In Fig. 2, we plot the exact value, beta approximation and the proposed
approximation of SINR meta distribution against $\theta$ or $\gamma$ at
different values of $\gamma$ or $\theta$ under $\lambda=1$ km-2. As shown, the
proposed approximation shows good matching at low values of $\theta$ and
$\gamma$. However, when it comes to very high values, especially $\theta$, a
small gap exists. The reason for the gap at high values of $\theta$ is that we
ignore the high order terms of $\theta$ in both (III-B) and (III-B). Such gaps
exist even if we increase the values of $j$ in (19) ($j=2$ shows almost the
same curve as $j=1$, in which the difference is less than $0.1$, hence,
omitted in Fig. 5), hence the dominant interferer-based approximation (Cor. 1)
is a more efficient way (comparing (22) to (19) ) since no need of computing
joint distribution of $R_{1},\cdots,R_{j}$ (only $f_{R_{1}}(r)$ and
$F_{R_{0}|R_{1}}(r)$ are required) and a simple integral completes the
analysis. Notice that, the logarithmic scale is used for the y-axis in Fig. 2
since the gap is negligible in linear scale (e.g., the gap between the
proposed approximation and exact value at $\theta=26$ dB is actually 0.02).

Fig. 3 shows the gap between the proposed approximation and beta and exact
value of SINR meta distribution for a large range of BS densities. The
proposed approximation shows very good performance at a large range of density
of BS and low values of $\theta$, and its performance is even better than beta
distribution, while saving a large amount of computing time.

Notice that the accuracy of dominant interferer-based approximation strongly
depends on the path loss exponent. While $\alpha=4$ is an important case, user
connections in mm-Wave band will be line-of-sight, say UAV networks, hence in
Fig. 4 (a),(b),(c) we plot $\alpha=2.5,3,3.5$, respectively. Clearly, the
proposed approximation becomes more accurate with the increase of $\alpha$ and
in small values of $\alpha$ a gap exists. In Fig. 4 (d) we plot the nearest-
interferer-only approximation, which is an upper bound of the system
performance, and as mentioned in Remark 2, the proposed approximation is
tighter by considering the mean of the remaining interferers (all the
interferers except the nearest one) but it is neither upper bound nor lower
bound: as can be observed from Fig. 2 that the proposed approximation does
indeed become higher than the exact value specially at high values of
$\theta$. On the other hand, it can be observed from Fig. 4 that the proposed
approximation is lower than the exact value.

TABLE I: KL divergence analysis of PPP networks Parameters | $\alpha=2.5$ | $\alpha=3$ | $\alpha=3.5$ | $\alpha=4$ |  $\alpha=4,\lambda=0.1,0.5,5,10$ km-2
---|---|---|---|---|---
$D_{\rm KL,prop|exact}$ ($\theta=-10$ dB) | 0.0417 | 0.0095 | 0.0082 | 0.0037 | 0.0097,0.0050,0.0085,0.0119
$D_{\rm KL,beta|exact}$ ($\theta=-10$ dB) | 0.0122 | 0.0284 | 0.0101 | -0.0045 | 0.0111,0.0025,0.0042,0.0335
$D_{\rm KL,prop|exact}$ ($\theta=0$ dB) | 0.0109 | 0.0105 | 0.0037 | 0.0020 | 0.0007,0.0123,0.0039,0.0180
$D_{\rm KL,beta|exact}$ ($\theta=0$ dB) | 0.0142 | 0.0126 | 0.0079 | 0.0064 | 0.0013,0.0167,0.0138,0.0322
$D_{\rm KL,prop|exact}$ ($\theta=12$ dB) | 0.0012 | 0.006 | 0.0042 | 0.0011 | -0.0030,0.0033,0.0064,0.0146
$D_{\rm KL,beta|exact}$ ($\theta=12$ dB) | 0.0031 | 0.0025 | 0.0017 | 0.0016 | -0.0025,0.0039,0.0033,0.0094

The KL divergence analysis of the proposed approximation is provided in Table
I. As shown in Table I, the proposed approximation at high values of $\alpha$
shows a competitive results compared with the beta approximation.

In the following part of this section, we follow the same simulation steps by
conditioning on each realization of point processes, generating a large
iterations of channel fading and computing the conditional success probability
of each link. We plot the simulation results of the proposed approximation in
the four different scenarios, Poisson bipolar network, MCP, $K$-tier PPP and
PLCP, and compare the results with the traditional beta approximation and
simulation based exact value of SINR meta distribution.

###  V-A Numerical Results of Poisson Bipolar Networks

Figure 5: Meta distribution of a Poisson bipolar network at $\lambda=10$
BS/km2, $\alpha=4$, $p_{t}=10$ W and $R=50$ m. The solid lines are exact
values based on simulations and markers for the proposed approximation,
respectively.  Figure 6: Meta distribution of a Poisson bipolar network at
$\lambda=10$ BS/km2, $\alpha=4$, $p_{t}=10$ W. The solid lines are exact
values based on simulations, dash lines are for beta approximation and markers
are for the proposed approximation, respectively. $R$ is in meter and
$\gamma=0.3,0.6,0.9$ in (a,b,c), respectively.

Fig. 5 and Fig. 6 show the SINR meta distribution of Poisson bipolar networks.
The proposed approximation shows good matching at all values of $\theta$ and
$\gamma$. In Fig. 6, we plot the SINR meta distribution under $10$ different
values of $R$. With the increase of the distance, system’s reliability drops
sharply and approaches 0, which is owing to the fixed transmission distance
and the noise power.

TABLE II: KL divergence analysis of Poisson bipolar networks Parameters | $R=15$ | $R=30$ | $R=45$ | $R=60$ | $R=75$ | $R=100$ | $R=125$ | $R=150$
---|---|---|---|---|---|---|---|---
$D_{\rm KL,prop|exact}$ ($\theta=-10$ dB) | 0.0013 | 0.0068 | 0.0241 | -0.0030 | 0.0367 | 0.0255 | 0.0289 | 0.0127
$D_{\rm KL,beta|exact}$ ($\theta=-10$ dB) | -0.0065 | -0.0226 | -0.0285 | -0.0646 | -0.0814 | -0.1464 | -0.1239 | -0.0165
$D_{\rm KL,prop|exact}$ ($\theta=0$ dB) | -0.0034 | 0.0006 | 0.0386 | 0.0269 | 0.0153 | 0.0098 | 0.0096 | 0.0230
$D_{\rm KL,beta|exact}$ ($\theta=0$ dB) | -0.0267 | -0.0552 | -0.1085 | -0.1574 | -0.0740 | 0.0173 | 0.0900 | 0.0164
$D_{\rm KL,prop|exact}$ ($\theta=12$ dB) | 0.0142 | 0.0440 | 0.0097 | 0.0148 | 0.0186 | 0.0207 | 0.0204 | 0.0050
$D_{\rm KL,beta|exact}$ ($\theta=12$ dB) | -0.0410 | -0.1482 | -0.0015 | 0.0543 | 0.0348 | 0.0090 | 0.0379 | 0.0143

The KL divergence analysis of the proposed approximation under Poisson bipolar
networks is provided in Table II, in which the unit of $R$ is meter. As shown
in Table II, the proposed approximation shows a competitive results compared
with the beta approximation.

###  V-B Numerical Results of MCP

Figure 7: Meta distribution of MCP networks at $\lambda=10$ BS/km2, $\alpha=4$
and $p_{t}=10$ W. The solid lines are exact values based on simulations, dash
lines are for beta approximation and markers for the proposed approximation,
respectively.  Figure 8: Meta distribution of a MCP network at $\lambda=1$
BS/km2, $\alpha=4$ and $p_{t}=10$ W. The solid lines are exact values based on
simulations, dash lines are for beta approximation and markers are for the
proposed approximation, respectively. $r_{c}$ is in meter and
$\gamma=0.3,0.6,0.9$ in (a,b,c), respectively.

Fig. 7 and Fig. 8 show the SINR meta distribution of MCP networks. In Fig. 8,
we plot the SINR meta distribution under different values of user cluster
radii. As expected, SINR meta distribution decrease with the increase of the
user cluster radii. Besides, compared to the Poisson bipolar networks, the
reliability of MCP networks drops slower and does not approach 0. Clearly, the
proposed approximation shows good performance, especially at large values of
$\gamma$.

TABLE III: KL divergence analysis of MCP networks Parameters | $r_{c}=100$ | $r_{c}=150$ | $r_{c}=200$ | $r_{c}=250$ | $r_{c}=300$ | $r_{c}=350$ | $r_{c}=400$
---|---|---|---|---|---|---|---
$D_{\rm KL,prop|exact}$ ($\theta=-10$ dB) | -0.0031 | 0.0059 | -0.0015 | 0.0173 | -0.0041 | 0.0218 | -0.0012
$D_{\rm KL,beta|exact}$ ($\theta=-10$ dB) | 0.0008 | 0.0220 | 0.0293 | 0.0249 | -0.0345 | -0.0161 | -0.0116
$D_{\rm KL,prop|exact}$ ($\theta=0$ dB) | -0.0067 | 0.0117 | 0.0001 | 0.0063 | -0.0027 | 0.0043 | 0.0006
$D_{\rm KL,beta|exact}$ ($\theta=0$ dB) | 0.0206 | 0.0069 | -0.0237 | 0.0332 | 0.0343 | 0.0332 | 0.0136
$D_{\rm KL,prop|exact}$ ($\theta=12$ dB) | 0.0007 | 0.0052 | 0.0004 | 0.0011 | 0.0002 | 0.0004 | 0.0002
$D_{\rm KL,beta|exact}$ ($\theta=12$ dB) | -0.0378 | 0.0435 | 0.0296 | 0.0029 | -0.0011 | -0.0018 | -0.0115

The KL divergence analysis of the proposed approximation under MCP networks is
provided in Table III, in which the unit of $r_{c}$ is meter. As shown in
Table III, the proposed approximation shows a competitive results compared
with the beta approximation.

###  V-C Numerical Results of $K$-tier Networks

Figure 9: Meta distribution of a $K$-tier PPP network, where $K=2$,
$\lambda_{1}=1$ BS/km2, $\lambda_{2}=3$ BS/km2, $\alpha=4$, $p_{t,1}=10$ W and
$p_{t,2}=5$. The solid lines are exact values based on simulations, dash lines
are for beta approximation and markers for the proposed approximation,
respectively.  Figure 10: Meta distribution of a $K$-tier PPP network, where
$K=3$, $\lambda_{1}=0.1$ BS/km2, $\lambda_{2}=0.5$ BS/km2, $\lambda_{3}=5$
BS/km2, $\alpha=4$, $p_{t,1}=10$ W, $p_{t,2}=2$ W and $p_{t,3}=0.2$ W. The
solid lines are exact values based on simulations, dash lines are for beta
approximation and markers for the proposed approximation, respectively.

Fig. 9 and Fig. 10 show the SINR meta distributions for a $2$-tier and
$3$-tier network, respectively. Here we can see that the proposed
approximation also works well for multi-tier networks by mapping the $i$-th
tier to the first tier and obtaining the equivalent distance distributions and
the density of the new point process. In this way, we can avoid computing the
association probability and the summation of the aggregate interference of
different tiers.

Since the analysis of $K$-tier PPP is similar to PPP, the KL divergence
analysis is omitted.

###  V-D Numerical Results of PLCP

Figure 11: The SINR meta distribution of a PLCP network, where
$\lambda_{l}=8/\pi$ km/km2, $\lambda_{p}=0.2$ /km (the density of BSs is $1.6$
BS/km2), $\alpha=4$, $p_{t}=10$ W. The solid lines are exact values based on
simulations and markers for the proposed approximation, respectively.  Figure
12: The SINR meta distribution of PLCP networks, while the density of the BS
is fixed at $1.6$ BS/km2, the line and point densities are different,
$\lambda_{l,1}=5\lambda_{l,2}$, $\alpha=4$, $p_{t}=10$ W. The unit of
$\lambda_{l}$ is in km/km2. The solid lines are exact values based on
simulations and markers for the proposed approximation, respectively. The
green curves with diamond markers are the SINR meta distribution of a 2D PPP
network with density $\lambda=\pi\lambda_{l,1}\lambda_{p,1}$ and the gray
curves with square markers are the SINR meta distribution of a 1D PPP network
with density $\lambda=2\lambda_{p,2}$.

Fig. 11 and Fig. 12 display the SINR meta distribution of PLCP networks. In
Fig. 12, we plot the SINR meta distribution in two different line and point
densities. While we fix the overall density of the BSs, with the increasing of
the line density, the fraction of links exceeding the reliability threshold
increases. The reason is that as the line density increases
($\lambda_{l}\to\infty$) while the overall average number of points remains
unchanged, the PLCP converges to a homogeneous 2D PPP with point density
$\lambda=\pi\lambda_{l}\lambda_{p}$, as shown in the green curves with diamond
markers in Fig. 12; and as the line density decreases ($\lambda_{l}\to 0$)
while the overall average number of points remains unchanged, the PLCP reduces
to a homogeneous 1D PPP with point density $\lambda=\lambda_{p}$. However,
since the analysis of 1D PPP with $\lambda_{p}=\infty$ is challenging, we plot
the SINR meta distribution of 1D PPP with density $\lambda=2\lambda_{p,2}$, as
shown in the gray curves with square markers in Fig. 12. In these two
asymptotical scenarios, the first contact distance in the case of 2D PPP is
further than that of 1D PPP. Hence, system is more reliable with the line
density decreasing since shorter serving distance and the system performance
is bounded by 2D PPP and 1D PPP with corresponding point densities.

Figure 13: The SINR meta distribution of PLCP networks under three
approximations, (i) TPPP approximation [28, (16)]: green and crossing markers,
(ii) proposed approximation with PPP-approximated interference: orange and
diamond markers, and (iii) proposed approximation with PLCP interference: red
dash lines, and (iv) the exact value: blue and solid lines. The curves are
plotted under $\lambda_{l}=8/\pi$ km/km2 and $\lambda_{p}=0.2$ /km.

Besides, in Fig. 13, we plot the proposed approximation with PLCP
interference, PPP-approximated interference, respectively, and the
approximation mentioned in [28, (16)], which is based on the transdimensional
PPP (TPPP). As shown, PPP is a good approximation of PLCP and all
approximations provide good matching results.

TABLE IV: KL divergence analysis of PLCP networks Parameters |  $\lambda_{l}=1.6/\pi$ km/km2, $\lambda_{p}=1$ /km |  $\lambda_{l}=8/\pi$ km/km2,$\lambda_{p}=0.2$ /km
---|---|---
$D_{\rm KL,prop|exact}$ ($\theta=-10$ dB) | -0.0234 | -0.0031
$D_{\rm KL,prop(PPP)|exact}$ ($\theta=-10$ dB) | -0.0280 | -0.0103
$D_{\rm KL,TPPP(16)|exact}$ ($\theta=-10$ dB) | 0.1796 | -0.0606
$D_{\rm KL,prop|exact}$ ($\theta=0$ dB) | -0.0079 | -0.0028
$D_{\rm KL,prop(PPP)|exact}$ ($\theta=0$ dB) | -0.0094 | -0.0052
$D_{\rm KL,TPPP(16)|exact}$ ($\theta=0$ dB) | 0.0262 | -0.0404
$D_{\rm KL,prop|exact}$ ($\theta=12$ dB) | -0.0045 | -0.0013
$D_{\rm KL,prop(PPP)|exact}$ ($\theta=12$ dB) | -0.0052 | -0.0020
$D_{\rm KL,TPPP(16)|exact}$ ($\theta=12$ dB) | -0.0115 | -0.0143

The KL divergence analysis of the proposed approximation under PLCP networks
is provided in Table IV. As shown in Table IV, the proposed approximation
shows a good matching result, and the PPP interference also shows a good
matching result, which is because PPP is a good approximation of PLCP.

##  VI Conclusion

This paper analyzes a dominant interferer-based approximation, considering the
dominate interferers exactly while the rest interferers in an average sense,
for SINR meta distribution in the downlink Poisson cellular networks. We first
obtain the proposed approximation in a standard PPP networks with all the
channel links subject to the Rayleigh fading. The applied approximation shows
good matching with the exact value of SINR meta distribution at large range of
BS densities, SINR thresholds and various system parameters. Compared with the
traditional method, the proposed approximation does not require computing the
moments of conditional success probability. It can be derived in a simple form
based on the Lambert W function and the joint CDF and PDF of the first and the
second nearest BSs’ distance distribution, which highly reduces calculation
complexity and is highly time-efficient. To illustrate the accuracy and
operability, we extend the proposed approximation in four different scenarios
and compare the results with the popular used beta approximation as well as
the Monte-Carlo simulations. Meanwhile, we derive the SINR meta distribution
for the PLCP networks for the first time based on the proposed approximation
since it avoids computing the $b$-th moments and highly reduce the complexity
of the calculation process.

Throughout this paper, we focus on validating the proposed approximation with
simulations and beta approximation in downlink scenarios. For the uplink
cases, system models are more complex, e.g., transmit power may be a function
of the serving distance (inverse power control), the locations of interferers
can be closer to the BSs, and maybe more than one active user within one
Voronoi cell (depends on the transmission policy). Therefore, the distance
distributions, as well as system model, are way more complicated than downlink
scenarios. While we only compute the SINR meta distribution in the downlink in
this paper, uplink analysis, as well as some more complex system models, such
as cluster distributed BSs [26] and [47], can be an interesting future
research direction.

###  -A Proof of Theorem 1

Given $\Phi$, the success probability is given by

$\displaystyle
P_{s}(\theta)=\prod_{i\in\mathbb{N}}\bigg{(}\frac{1}{1+\theta||x_{i}||^{-\alpha}||x_{0}||^{\alpha}}\bigg{)}\exp\bigg{(}-\frac{\theta}{p_{t}}||x_{0}||^{\alpha}\sigma^{2}\bigg{)}.$
(46)

Hence, following the Campbell’s theorem [41] with conversion from Cartesian to
polar coordinates, we have the $b$-th moment,

$\displaystyle
M_{b}(\theta)=\mathbb{E}_{||x_{0}||}\bigg{[}\mathbb{E}\bigg{[}\prod_{i\in\mathbb{N}}\bigg{(}\frac{1}{1+\theta||x_{i}||^{-\alpha}||x_{0}||^{\alpha}}\bigg{)}^{b}\biggm{|}||x_{0}||\bigg{]}\exp\bigg{(}-\frac{b\theta}{p_{t}}||x_{0}||^{\alpha}\sigma^{2}\bigg{)}\bigg{]}$
$\displaystyle=\mathbb{E}_{R_{0}}\bigg{[}\exp\bigg{(}-2\pi\lambda\int_{r_{0}}^{\infty}\bigg{[}1-\bigg{(}\frac{1}{1+\theta
r^{-\alpha}R_{0}^{\alpha}}\bigg{)}^{b}\bigg{]}r{\rm
d}r\bigg{)}\exp\bigg{(}-\frac{b\theta}{p_{t}}R_{0}^{\alpha}\sigma^{2}\bigg{)}\bigg{]}$
$\displaystyle=\mathbb{E}_{R_{0}}\bigg{[}\exp\bigg{(}-\lambda
F_{b}\bigg{)}\exp\bigg{(}-\frac{b\theta}{p_{t}}R_{0}^{\alpha}\sigma^{2}\bigg{)}\bigg{]},$
(47)

where,

$\displaystyle
F_{b}=2\pi\int_{R_{0}}^{\infty}\bigg{[}1-\bigg{(}\frac{1}{1+\theta
r^{-\alpha}R_{0}^{\alpha}}\bigg{)}^{b}\bigg{]}r{\rm
d}r\stackrel{{\scriptstyle(a)}}{{=}}\pi\delta\int_{R_{0}^{\alpha}}^{\infty}\bigg{[}1-\bigg{(}\frac{u}{u+\theta
R_{0}^{\alpha}}\bigg{)}^{b}\bigg{]}u^{\delta-1}{\rm d}u$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\pi\delta\int_{R_{0}^{\alpha}}^{\infty}\sum_{k=1}^{\infty}\binom{b}{k}(-1)^{k+1}\bigg{(}\frac{\theta
R_{0}^{\alpha}}{u+\theta R_{0}^{\alpha}}\bigg{)}^{k}u^{\delta-1}{\rm
d}u=\pi\delta\sum_{k=1}^{\infty}\binom{b}{k}(-1)^{k+1}(\theta
R_{0}^{\alpha})^{k}\int_{R_{0}^{\alpha}}^{\infty}\bigg{(}\frac{u^{\delta-1}}{(u+\theta
R_{0}^{\alpha})^{k}}\bigg{)}{\rm d}u$
$\displaystyle=\pi\delta\sum_{k=1}^{\infty}\binom{b}{k}(-1)^{k+1}\frac{\alpha\theta^{k}R_{0}^{2}}{-2+k\alpha}{}_{2}F_{1}(k,-\delta+k,1-\delta+k,-\theta),$
(48)

where step $(a)$ follows from using the replacement $u=r^{\alpha}$ and step
$(b)$ follows from the binomial expansion. Notice that the above expression is
different from the one derived in [19], since the association policies are
different: the typical user in this section associated with the nearest BS,
while the user in the Poisson bipolar network has a dedicated serving BS.
Therefore, the locations of interfering BSs in this section are different from
[19]: lower bound of the integration is not zero. Besides, since we included
the noise here, the relative distance process (RDP) approach mentioned in [5]
and [48] cannot be used.

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