# Cox Models for Vehicular Networks:
SIR Performance and Equivalence

Jeya Pradha Jeyaraj, Student Member, IEEE, and Martin Haenggi, Fellow, IEEE
The authors are with the Department of Electrical Engineering, University of
Notre Dame, Notre Dame, IN 46556, USA. (e-mail: $\\{$jjeyaraj,
mhaenggi$\\}$@nd.edu). This work was supported in part by Toyota Motor North
America R&D and by the U.S. National Science Foundation (grant CCF 2007498).

###### Abstract

We introduce a general framework for the modeling and analysis of vehicular
networks by defining street systems as random 1D subsets of $\mathbb{R}^{2}$.
The street system, in turn, specifies the random intensity measure of a Cox
process of vehicles, i.e., vehicles form independent 1D Poisson point
processes on each street. Models in this Coxian framework can characterize
streets of different lengths and orientations forming intersections or
T-junctions. The lengths of the streets can be infinite or finite and mutually
independent or dependent. We analyze the reliability of communication for
different models, where reliability is the probability that a vehicle at an
intersection, a T-junction, or a general location can receive a message
successfully from a transmitter at a certain distance. Further, we introduce a
notion of equivalence between vehicular models, which means that a
representative model can be used as a proxy for other models in terms of
reliability. Specifically, we prove that the Poisson stick process-based
vehicular network is equivalent to the Poisson line process-based and Poisson
lilypond model-based vehicular networks, and their rotational variants.

###### Index Terms:

Poisson line process, Poisson point process, stochastic geometry, and
vehicular networks.

##  I Introduction

###  I-A Motivation and Related Work

Through vehicle-to-vehicle (V2V) communication, vehicles can broadcast the
information required for safety such as their speed, brake status, position,
etc., to other vehicles. This can alert the vehicles about the events
happening in their vicinity that even the best sensors installed in the
vehicles may fail to anticipate. Further, infrastructure nodes such as
roadside units, smart traffic lights, etc., can inform the drivers of the
road/traffic conditions through vehicle-to-infrastructure (V2I) communication.
High reliability is a key requirement for safety-based V2V and V2I
applications due to their mission-critical nature. Reliability or success
probability is defined as the probability that a broadcast message is received
reliably at a certain distance. The factors that affect the reliability
include the locations of the transmitting and receiving vehicles and
interfering vehicles, and wireless medium.

An analysis of GPS traces of taxis in Beijing and Porto in [1] demonstrates
that (i) the taxi locations cannot be modeled as random points as in a 2D
Poisson point process (PPP) neglecting the street geometry as claimed in [2],
and (ii) it is desirable to model the intersections, which are critical for
vehicle safety. Limiting the reliability analysis to a single or a small
number of streets is certainly useful in its own right, albeit insufficient to
obtain insights into the network behavior at intersections and T-junctions [3,
4, 5] and in dense urban scenarios [6]. Hence, street geometry is integral to
vehicular network modeling.

The street systems in different geographical regions may differ in their
structure, street lengths, and degree of regularity. The authors in [7] have
studied the street systems of 18 cities in different parts of the world. They
divide the street systems into two classes—(i) self-organized patterns that
are formed historically without the control of any central agency, and (ii)
planned regular grid-like patterns. Cities such as Ahmedabad (India), Cairo
(Egypt), and Venice (Italy) are examples of self-organized patterns with
unimodal street length distributions. Los Angeles, Richmond, and San Francisco
in the United States are examples of grid-like patterns with multimodal street
length distributions. It is worth noting that that even in the cities
containing grid-like street patterns, the street lengths are finite and vary
significantly.

In the literature [8, 9, 10, 11, 12, 13, 14], street systems are most commonly
modeled as a random collection of lines with uniform orientations using
Poisson line processes (PLPs). While assuming that all streets are infinitely
long may lead to useful results, it either overestimates the total
interference or underestimates the local density of vehicles. For example, a
street in a city that is 2 km long may have a density of 50 cars per km. But
extending it to an infinite street would mean such a high car density extends
to very remote rural regions far outside the city, which is not realistic.
Further, in the PLP, each pair of streets forms an intersection and there are
no T-junctions. Fig. 1 shows a part of Rome as an example, where few streets
are long enough to be approximated by infinite streets, and there exist many
T-junctions.

Figure 1: Part of Rome’s city center. Visual inspection shows a mean street
length of about 250 m and many T-junctions.

Consequently, we need street models that can characterize finite variations in
the street lengths, and intersections and T-junctions. The edges of Poisson-
Voronoi tessellation, Poisson-Delaunay tessellation, and Poisson line
tessellation are considered to model streets of finite lengths in [15].
Estimators for the probability densities of the inter-node distances are
derived for these tessellations owing to their intractability. A simpler
alternative is to use line segments or sticks as we show later. In this work,
we introduce a general framework for the modeling and analysis of vehicular
networks. Under this framework, we develop models that represent street
systems varying from regular grid-like patterns to irregular hodgepodges and
characterize the uncertainty in the vehicle locations on the streets. In
particular, the street lengths can be infinitely long or varying finitely and
mutually independent or dependent resulting in intersections or T-junctions.

An important question is whether we need a different spatial model for each
region. Alternatively, does there exist an equivalence between the models such
that a single model is sufficient to analyze two different regions? If yes, a
representative subset can be used to analyze a larger set of models, reducing
the computation time and costs associated with network design and planning. We
show that some models developed within the framework are equivalent, in a
precise sense defined later. We use the mathematical toolsets from stochastic
geometry [16] to model and analyze random spatial patterns.

###  I-B Contributions

We present a general framework for the modeling and analysis of vehicular
networks, where the lengths of the streets can be infinite or finite, mutually
independent or dependent, and the street orientations can have different
levels of regularity. Lines or line segments (sticks) characterize the
streets. The street geometries of the spatial models that can be characterized
in this framework include but not limited to 1) the orthogonal grid with
exponential spacing (OG) whose lines are of infinite length with orthogonal
orientations, 2) the PLP whose lines are of infinite length with irregular
orientations, 3) the Poisson stick process (PSP) whose sticks can have random
lengths and orientations, and 4) the Poisson lilypond model (PLM) whose sticks
grow in a random direction until they touch another stick, mimicking the
growth of lilies in a pond.

The OG, PLP, and PSP inherently form intersections, whereas in the PLM, the
dependence between the sticks results in T-junctions. To the best of our
knowledge, this is the first work to model and analyze vehicular networks with
street systems formed by line segments involving intersections or T-junctions.

The vehicle locations on each street independently form 1D PPPs, i.e., the
spatial models in the presented framework are Cox vehicular networks. Our
contributions are:

  * •

We introduce a street system partitioning approach that decomposes a street
system into points of different orders, where the order quantifies the number
of street segments covering the point. This decomposition permits a unified
approach to the performance analysis of vehicles at general locations,
intersections, and T-junctions.

  * •

We derive the nearest-neighbor distance distribution in the PSP-based
vehicular network, and we show that the nearest-neighbor distance distribution
in the PLM-based vehicular network can be tightly approximated by that in the
PSP-based vehicular network.

  * •

We derive analytical expressions for the probability that a vehicle at a
general location, an intersection, or a T-junction successfully receives from
its transmitter located at a certain distance, in each of the considered
models.

  * •

We introduce a notion of equivalence between the models based on the success
probability. The expression for the success probability of the typical vehicle
in the PSP-based vehicular network can be used to evaluate that in the
vehicular networks formed by other models by mapping the parameters such as
street and vehicle intensities, and street length distribution. We show that
the equivalence also holds under random link distances.

  * •

We show that the Cox vehicular networks behave like a 2D PPP in the low-
reliability regime. In contrast, in the high-reliability regime, the success
probability of the typical vehicle in the Cox vehicular networks tends to that
in the network formed only by the street(s) that the typical vehicle belongs
to. This corroborates earlier findings [1] that the vehicles cannot be
generally modeled as 2D PPPs and that the street geometry plays an important
role.

##  II System Model

Throughout the paper, we will use the definitions and notations presented in
this section unless otherwise stated. Let $b(x,r)$ denote a disk of radius $r$
centered at $x$, and $o\triangleq(0,0)$ denote the origin. Let $|\cdot|_{d}$
denote the Lebesgue measure in $d$ dimensions.

###  II-A General Framework

######  Definition 1 (Street System).

A street system $\mathcal{S}$ is a stationary random closed subset of
$\mathbb{R}^{2}$ with $|\mathcal{S}|_{2}=0$ that contains no singletons or
isolated points. Due to the stationarity,

$\mathbb{E}|\mathcal{S}\cap\mathcal{B}|_{1}=\tau|\mathcal{B}|_{2}\hskip
8.53581pt\text{for Borel sets}\hskip
2.84526pt\mathcal{B}\subset\mathbb{R}^{2},$ (1)

where $\tau$ is the mean total street length per unit area.

$|\mathcal{S}|_{2}=0$ in Definition 1 implies that $\mathcal{S}$ is a random
1D subset of the plane consisting of lines, line segments or sticks, curved
segments or arcs, that characterize the streets. Let
$\Xi=\\{\xi_{1},\xi_{2},\ldots\\}$ be a collection of 1D subsets in
$\mathbb{R}^{2}$ such that for $i\neq j$, $|\xi_{i}\cap\xi_{j}|_{1}=0$ and
$\xi_{i}\cup\xi_{j}$ is not a 1D subset. The street system $\mathcal{S}$ is
the union of 1D subsets, i.e.,

$\mathcal{S}\triangleq\bigcup_{\xi\in\Xi}\xi.$ (2)

The elements of $\mathcal{S}$ are points in $\mathbb{R}^{2}$ but those of
$\Xi$ are 1D subsets, which are sets themselves, and hence
$\mathcal{S}\neq\Xi$. Further, $\mathcal{S}$ uniquely characterizes $\Xi$ and
vice-versa, i.e., there exists a one-to-one correspondence between
$\mathcal{S}$ and $\Xi$. Any street system $\mathcal{S}$ can be partitioned as
follows.

######  Definition 2 (Street System Decomposition).

Let $\mathcal{P}_{m}\triangleq\\{{z\in\mathbb{R}^{2}:|\mathcal{S}\cap
b(z,r)|_{1}\sim mr,r\to 0\\}}$ denote the set of points of order
$m\in\mathbb{N}$ in the street system $\mathcal{S}$.

The sets $\mathcal{P}_{m}$ are disjoint, and their union equals $\mathcal{S}$,
i.e., $\\{\mathcal{P}_{m}\\}_{m\in\mathbb{N}}$ is a partition of
$\mathcal{S}$. For $m\neq 2$, the sets $\mathcal{P}_{m}$ are countable and
form simple and stationary point processes. $\mathcal{P}_{2}$ is the only set
with $|\mathcal{P}_{2}|_{1}>0$, in fact, $|\mathcal{P}_{2}|_{1}=\infty$. We
have $\mathcal{P}_{2}=\mathcal{S}$ almost everywhere, i.e., $\mathcal{P}_{2}$
is an open set and $\mathcal{S}=\mathrm{cl}(\mathcal{P}_{2})$, where
$\mathrm{cl}$ denotes the closure. $\mathcal{P}_{1}$ are endpoints,
$\mathcal{P}_{3}$ are T-junctions, $\mathcal{P}_{4}$ are intersections,
$\mathcal{P}_{5}$ are intersections with a T-junction, $\mathcal{P}_{6}$ are
three-way intersections (three streets intersecting at one point), etc. Let

$\mathcal{M}\triangleq\\{m\in\mathbb{N}:\mathbb{P}(\mathcal{P}_{m}=\emptyset)=0\\}$
(3)

denote the index set of the non-empty components. Then
$\mathcal{S}=\bigcup_{m\in\mathcal{M}}\mathcal{P}_{m}$ is called an
$\mathcal{M}-$indexed street system. Using $\mathcal{M}$, we can categorize
different street systems. For example, a $(2,4)-$street system refers to a
street geometry without endpoints and T-junctions but with intersections.

######  Definition 3 (Vehicular Point Process).

A vehicular point process $\mathcal{V}\subset\mathbb{R}^{2}$ is a Cox process
with random intensity measure
$\Upsilon(\mathcal{B})=\lambda|\mathcal{S}\cap\mathcal{B}|_{1}$.

Equivalently,
$\Upsilon(\mathcal{B})=\lambda|\mathcal{P}_{2}\cap\mathcal{B}|_{1}$ since a.s.
$\mathcal{V}\subset\mathcal{P}_{2}$. This implies that the vehicles form
independent 1D PPPs on each street. By Definition 1, the 2D intensity measure
of $\mathcal{V}$ is
$\mathbb{E}[\Upsilon(\mathcal{B})]=\lambda\mathbb{E}[|\mathcal{S}\cap\mathcal{B}|_{1}]=\lambda\tau|\mathcal{B}|_{2}$.

###  II-B Vehicular Network Models

Here, we present a few models that fall under our framework. A street system
may include curves or circles. In this work, we focus only on street systems
formed by lines or sticks.

A line (an infinitely long street) $L$ can be represented as

$L({x},{\varphi})=\\{{(a,b)\in\mathbb{R}^{2}:a\cos\varphi+b\sin\varphi=x\\}},$
(4)

where $(|x|,\varphi)$ are the polar coordinates of the foot of the
perpendicular from the origin $o$ to $L$.

######  Definition 4 (Line-based Poisson Street System).

Consider a marked point process on $\mathbb{R}\times\Omega$, whose ground
process is a PPP $\Phi_{1}$ of intensity $\mu$ on $\mathbb{R}$ and marks are
i.i.d. on $\Omega\subseteq[0,\pi)$. Let $\nu$ denote the distribution of
$\Omega$. The collection of lines
$\Xi_{\mathrm{L}}=\\{{L(x,\varphi):(x,\varphi)\in\Phi_{1}\times\Omega\\}}$
with the intensity measure $\Lambda_{\Xi}(\mathrm{d}x\hskip
0.85358pt\mathrm{d}\varphi)=\mu\hskip
0.85358pt\mathrm{d}x\times\mathrm{d}\nu(\varphi)$ forms

  * •

an orthogonal grid with exponential spacing (OG) if
$\mathrm{d}\nu(\varphi)=0.5\delta_{0}+0.5\delta_{\pi/2}$,

  * •

a Poisson line process (PLP) if
$\mathrm{d}\nu(\varphi)=\mathrm{d}\varphi/\pi$.

The line-based Poisson street system is
$\mathcal{S}=\bigcup_{L\in\Xi_{\mathrm{L}}}L$.

The OG and PLP inherently form intersections and hence are classified as
$(2,4)-$street systems. Streets of varying finite lengths can be represented
using sticks. A stick $S$ is defined by its midpoint $y$, orientation
$\varphi$, and half-length $h$, and represented as a closed set as

$\displaystyle S(y,\varphi,h)=[y-hu(\varphi),y+hu(\varphi)],$ (5)

where $u(\varphi)=(\cos\varphi,\sin\varphi)$. Alternatively,
$S(y,\varphi,h)=y+{\mathrm{rot}}_{\varphi}([-h,h])$, where
$\mathrm{rot}_{\varphi}$ denotes the rotation by $\varphi$ around $o$. Now, we
are ready to define the stick-based street system.

######  Definition 5 (Stick-based Poisson Street System).

Let $\mathcal{Q}=\\{(y_{i},\bm{t}_{i})\\}$, $i\in\mathbb{N}$, denote an i.i.d.
marked point process with the ground process $\\{y_{i}\\}$ forming a 2D PPP
$\Phi_{2}$ of intensity $\mu$. Associate with $y_{i}$ a 2D mark
$\bm{t}_{i}=(\varphi_{i},h_{i})$, where the orientation $\varphi_{i}$ is
i.i.d. on $[0,\pi)$, and $h_{i}$ is the half-length. The countable collection
of sticks $\Xi_{\mathrm{S}}=\\{S(y_{i},\varphi_{i},h_{i})\\}$ forms

  * •

a Poisson stick process (PSP) if the half-lengths are i.i.d. with some
distribution $F_{H}$.

  * •

a Poisson lilypond model (PLM)111The PLM is denoted as LM-I in [17]. We note
that the other model, LM-II in [17], has similar properties as the PLM. if
each stick grows from zero length at a constant rate on both sides until one
of its endpoints hits another stick, thereby forming a T-junction.

Then $\mathcal{S}=\bigcup_{S\in\Xi_{\mathrm{S}}}S$ defines the stick-based
Poisson street system.

Let $y_{i}\equiv(\gamma_{i},\phi_{i})$ in polar coordinates, where
$\gamma_{i}\in\mathbb{R}^{+}$, and $\phi_{i}\in[0,2\pi)$. The PSP has no
T-junctions a.s., thus forming a $(1,2,4)-$street system, whereas the PLM has
no intersections a.s. due to its touch-and-stop growth mechanism, thus forming
a $(1,2,3)-$street system.

We only consider the street systems containing points of order up to 4. The
analyses shown in this work can be extended to higher-order street systems as
well.

We refer to the vehicular point processes formed by the OG, PLP, PSP, and PLM
as the OG-PPP, PLP-PPP, PSP-PPP, and PLM-PPP, respectively. Fig. 2 depicts
sample realizations.

(a) $\mu=1$ and $\lambda=0.1$

(b) $\mu=1$ and $\lambda=0.1$

(c) $\mu=0.1$ and $\lambda=0.1$

(d) $\mu=0.1$ and $\lambda=0.1$

Figure 2:  Snapshots of vehicular networks: (a) OG-PPP (b) PLP-PPP (c) PSP-PPP
and (d) PLM-PPP. Lines or sticks denote the streets, and ‘$\circ$’ denote the
vehicles. The stick lengths in (c) are Rayleigh distributed with parameter 2.

###  II-C Performance Metric and Types of Vehicles

Each vehicle broadcasts with probability $p$ following the slotted ALOHA
protocol. Then the intensity of active transmitters on each street in each
time slot is $\lambda p$. Our metric of interest is the success probability or
reliability, which is the probability of a vehicle successfully receiving the
message from a transmitter at distance $D$. If a transmitter can communicate
to a receiver at a distance $D$, then the other receivers within distance $D$
are also highly likely to receive the message. The transmitter can be another
vehicle, a roadside unit, a pedestrian, or any other node.

To define a meaningful network-wide metric, we focus on the success
probability of a representative vehicle (receiver) whose performance
corresponds to the average of that of all vehicles. In point process theory,
this representative vehicle is called ‘the typical point.’ In our context, it
is ‘the typical vehicle.’ As vehicles are located on the street(s), having a
vehicle at the origin implies that at least one street passes through the
origin. Under expectation over the vehicular point process, a vehicle
conditioned to be at the origin becomes the typical vehicle. Note that we can
condition the typical vehicle to be at any location since the vehicular
network is stationary owing to the underlying stationary street system
(Definition 1) and the homogeneity of the PPP. The typical vehicle’s
transmitter is assumed to be at a distance $D$ from the origin.

The performance of vehicular communication at intersections and T-junctions is
crucial as they are more prone to accidents. In view of this, we evaluate the
success probabilities of three kinds of vehicles: (i) the typical general
vehicle whose order is 2; (ii) the typical intersection vehicle whose order is
4; and (iii) the typical T-junction vehicle whose order is 3. The term typical
vehicle refers to all the three types of typical vehicles unless otherwise
stated. Mathematically, the success probability of the typical vehicle of
order $m$ at $o$ is defined as

$p_{m}(\theta)=\mathbb{P}({\sf{SIR}}>\theta\mid o\in\mathcal{P}_{m}),\hskip
5.69054ptm=2,3,4,$ (6)

where SIR is the signal-to-interference ratio measured at $o$ and $\theta$
parametrizes the target rate. The SIR for the typical vehicle with its
transmitter at distance $D$ is given by

${\sf
SIR}=\frac{gD^{-\alpha}}{\sum_{z\in\mathcal{V}}g_{z}\|z\|^{-\alpha}\mathit{B}_{z}},$
(7)

where $I=\sum_{z\in\mathcal{V}}g_{z}\|z\|^{-\alpha}$ is the total interference
power at the origin. The channel power gains $g$ and $g_{z}$ are exponentially
distributed with mean 1 (Rayleigh fading), and $\alpha$ is the path-loss
exponent. $(\mathit{B}_{z})_{z\in\mathcal{V}}$ is an i.i.d. sequence of
Bernoulli random variables with mean $p$, the probability that the transmitter
$z$ is active. Equipped with the performance metric, we next define the
equivalence of spatial models.

###  II-D Equivalence

######  Definition 6 (Equivalence).

Two spatial models $\mathrm{A}$ and $\mathrm{B}$ are said to be
$\epsilon$-equivalent with respect to the typical vehicle of order $m$ if the
total variation distance of their SIR distributions is at most $\epsilon$,
i.e.,
$\max\limits_{\theta}|p_{m}^{\mathrm{A}}(\theta)-p_{m}^{\mathrm{B}}(\theta)|=\epsilon$,
$0\leq\epsilon\leq 1$. If this holds with $\epsilon=0$, we call them strictly
equivalent. Model $\mathrm{A}$ is said to be asymptotically equivalent to
model $\mathrm{B}$ in the lower regime of $\theta$ if
$(1-p_{m}^{\mathrm{A}}(\theta))/(1-p_{m}^{\mathrm{B}}(\theta))\to 1$ as
$\theta\to 0$, and in the higher regime of $\theta$ if
$p_{m}^{\mathrm{A}}(\theta)/p_{m}^{\mathrm{B}}(\theta)\to 1$ as
$\theta\to\infty$.222For all models, the success probability of the typical
vehicle $p_{m}(\theta)\to 1$ as $\theta\to 0$. Hence, considering ratios of
success probabilities is not meaningful in this regime as they would all be
equivalent. Similarly, the outage probability $1-p_{m}(\theta)\to 1$ as
$\theta\to\infty$, so in this regime, it does not make sense to consider the
ratio of outage probabilities. To obtain non-trivial equivalence results, the
quantities of interest need to go to zero.

If the street system $\mathrm{A}$ has index set $\mathcal{M}_{\mathrm{A}}$ and
the street system $\mathrm{B}$ has index set $\mathcal{M}_{\mathrm{B}}$, then
the equivalence of $\mathrm{A}$ and $\mathrm{B}$ is defined only for the
typical vehicles of orders
$\mathcal{M}_{\mathrm{A}}\cap\mathcal{M}_{\mathrm{B}}$. If $\mathrm{A}$ and
$\mathrm{B}$ are strictly equivalent, then either $\mathrm{A}$ or $\mathrm{B}$
is sufficient to capture all the geographical regions that can be
characterized by them. Else, substituting one model for the other would depend
on the complexity of the model and the value of $\epsilon$, which we will
discuss in detail in Section V-C.

###  II-E Further Notation

Conditioning on $o\in\mathcal{P}_{2}$ implies that a street passes through the
origin, conditioning on $o\in\mathcal{P}_{3}$ implies that a street passes
through the origin while another street ends at the origin, and conditioning
on $o\in\mathcal{P}_{4}$ implies that two streets pass through the origin. We
denote by $\mathcal{V}^{m}=(\mathcal{V}\mid o\in\mathcal{P}_{m})$ the point
process of vehicles with the origin being on at least one street. Also, let
$\mathcal{V}_{o}^{m}$ denote the vehicles on the streets that pass through or
end at the origin. $\mathcal{V}_{!}$ denotes the vehicles on the rest of the
streets, i.e., $\mathcal{V}_{!}=\mathcal{V}^{m}\setminus\mathcal{V}_{o}^{m}$.
We index the streets based on the distances of the perpendiculars from the
streets to the origin. Based on the indexing, $V_{k}$ denotes the point
process of vehicles on the $k$th street. Then
$\displaystyle\mathcal{V}_{o}^{m}=\cup_{1\leq k\leq\lceil m/2\rceil}V_{k}$,
and $\displaystyle\mathcal{V}_{!}=\cup_{k>\lceil m/2\rceil}V_{k}$. Let
$I_{o}^{m}$ and $I_{!}$ denote the interference from the vehicles in
$\mathcal{V}_{o}^{m}$ and $\mathcal{V}_{!}$, respectively. $\delta\triangleq
2/\alpha$, where $\alpha$ is the path-loss exponent.

##  III Properties of Vehicular Networks

######  Lemma 1.

The mean total street length per unit area in the OG and PLP is $\tau=\mu$.

###### Proof:

The total expected length of the lines that intersect $b(o,r)$ is given by

$\displaystyle\mathbb{E}[|\mathcal{S}\cap b(o,r)|_{1}]$
$\displaystyle=\mu\int_{0}^{\pi}\int_{-r}^{r}2\sqrt{r^{2}-u^{2}}\hskip
1.42262pt\mathrm{d}u\hskip 1.42262pt\mathrm{d}v(\varphi)$
$\displaystyle=\mu\hskip 0.85358pt|b(o,r)|_{2}.$ (8)

By Definition 1 and (8), we get $\tau=\mu$. ∎

######  Lemma 2 ([18], Eq. 9).

The mean total stick length per unit area in the PSP is
$\tau=2\mu\mathbb{E}[H]$.

Let $f_{H}(h)$ denote the probability density function (PDF) of the half-
lengths of the sticks.

######  Lemma 3.

The half-lengths of the sticks that pass through the typical vehicle are
distributed with density $\tilde{f}_{H}(h)=hf_{H}(h)/\mathbb{E}[H]$.

Lemma 3 is a case of the inspection paradox [19]. The length of the stick that
passes through the typical vehicle is biased by the fact that the mean number
of points on the stick is proportional to its length. As a result, the half-
lengths of those sticks follow the density function $\tilde{f}_{H}(h)$. For
example, consider a case where half the streets are of length $10^{-2}$ and
the rest are of length $10^{2}$, and the intensity of vehicles on each street
is 1. Then the typical vehicle lies on a long street with a probability of
about $99.99\%$, which is very different from the $50\%$ probability of the
typical street to be a long one. On the other hand, in the PLM, a stick that
ends at a T-junction follows the inherent distribution $f_{H}(h)$ since each
stick has exactly one such endpoint a.s. If all the sticks are of the same
length $2h_{0}$, then $\tilde{f}_{H}(h)=f_{H}(h)=\delta(h-h_{0})$.

In the lilypond model, the growth of a stick is determined by the locations
and the orientations of the other sticks since each stick grows at a constant
rate until one of its endpoints hits another stick. This simultaneous touch-
and-stop growth process makes it difficult, most likely impossible, to derive
the exact distribution of the half-lengths. However, a suitable approximation
for the distribution of the half-lengths can be found, as stated in Lemma 4.

######  Lemma 4.

The half-lengths of the sticks in the PLM are approximately Rayleigh
distributed with PDF $\hat{f}_{H}(h)=2bh\exp(-bh^{2})$, $h\geq 0$. It follows
that ${\mathbb{E}}[H]\approx\sqrt{\pi/4b}$, where $b$ is proportional to the
street intensity $\mu$ and can be estimated from the empirical mean of the
half-lengths.

###### Proof:

See Appendix -A. ∎

(a)

(b)

Figure 3: (a) Fitting $h\hat{f}_{H}(h)/\mathbb{E}[H]$ to the half-lengths of
the streets that pass through the typical vehicle in the PLM. (b) Mean and
variance of number of neighbors to the typical general vehicle in the PLM-PPP
vs. PSP-PPP with $f_{H}(h)=\hat{f}_{H}(h)=2bh\exp(-bh^{2})$. $\mu=0.01$ and
$\lambda=0.3$. The value of $b$ corresponding to $\mu=0.01$ is 0.0103. The
intensity of 2D PPP is $2\lambda\mu\mathbb{E}[H]$, which is the 2D intensity
of the PLM-PPP/PSP-PPP.

$\displaystyle F^{\mathrm{PSP-PPP}}_{R}(r)=$ $\displaystyle
1-\bigg{[}\int\limits_{0}^{\infty}\frac{1}{h}\int\limits_{0}^{h}\exp(-\lambda\ell(\gamma,0,0))\tilde{f}_{H}(h)\hskip
0.85358pt\mathrm{d}\gamma\hskip 0.85358pt\mathrm{d}h\bigg{]}^{m/2}$
$\displaystyle\times\exp\bigg{(}-\frac{\mu}{\pi}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{r+h}\exp(-\lambda\ell(\gamma,\phi,\varphi))\gamma{f}_{H}(h)\hskip
0.85358pt\mathrm{d}\gamma\hskip 0.85358pt\mathrm{d}\phi\hskip
0.85358pt\mathrm{d}\varphi\hskip 0.85358pt\mathrm{d}h\bigg{)}.$ (10)

Combining Lemmas 3 and 4, we note that the PDF of half-lengths of the sticks
that pass through the typical vehicle in PLM can be approximated as
$\tilde{f}_{H}(h)\approx h\hat{f}_{H}(h)/\mathbb{E}[H]$. Fig. 3a illustrates
their histogram and the fitted density functions $\tilde{f}_{H}(h)$ and
$f_{H}(h)$. We observe that $h\hat{f}_{H}(h)/\mathbb{E}[H]$ provides a good
fit to the histogram, validating the approximation $\hat{f}_{H}(h)$.

To appreciate the differences between the PSP-PPP and PLM-PPP, we consider the
case where the half-lengths in the PSP are Rayleigh distributed as in the PLM.
Let $N_{o}(r)$ denote the number of neighbors to the typical vehicle at $o$
within a distance $r$. Fig. 3b compares the mean and variance of $N_{o}(r)$ in
the PLM-PPP and PSP-PPP. The correlation among the stick lengths resulting
from the touch-and-stop mechanism in the PLM-PPP leads to a smaller variance
of $N_{o}(r)$ than in the PSP-PPP, where the stick lengths are independent. On
the other hand, $\mathbb{E}[N_{o}]$ is the same for both the PSP-PPP and PLM-
PPP when both follow the same distribution for half-lengths as their first-
order statistics are the same. Also, the statistics of $N_{o}$ for the PLM-PPP
and PSP-PPP differ significantly from that for a 2D PPP of equivalent
intensity, highlighting the differences in having the street geometry and not.

(a)

(b)

Figure 4: Comparison of (a) mean distance from the typical general vehicle to
its $n$th-nearest neighbor and (b) nearest-neighbor distance distributions in
the PLM-PPP and PSP-PPP given by (10) with
$f_{H}(h)=\hat{f}_{H}(h)=2bh\exp(-bh^{2})$, where $b=1.04$ for $\mu=1$, and
$0.0103$ for $\mu=0.01$. $\lambda=0.3$.

######  Lemma 5.

The nearest-neighbor distance distribution for a vehicular network with the
street system characterized by Definition 1 can be decomposed as

$\displaystyle F_{R}(r)$
$\displaystyle=1-(1-F_{R,\mathcal{V}_{o}^{m}}(r))(1-F_{R,\mathcal{V}_{!}}(r)),$
(9)

where $F_{R,\mathcal{V}_{o}^{m}}(r)$ is the probability of finding a neighbor
in $\mathcal{V}_{o}^{m}$ within distance $r$, and $F_{R,\mathcal{V}_{!}}(r)$
is with respect to $\mathcal{V}_{!}$. $F_{R,\mathcal{V}_{!}}(r)$ also denotes
the contact distance distribution, the distribution of the distance from an
arbitrary location in the plane to the nearest vehicle in $\mathcal{V}$.

###### Proof:

See Appendix -B. ∎

######  Lemma 6.

The nearest-neighbor distance distribution for the OG-PPP/PLP-PPP is
$F_{R}(r)=1-\exp(-m\lambda
r-2\mu\textstyle\int_{0}^{r}(1-\exp(-2\lambda\sqrt{r^{2}-u^{2}}\hskip
0.85358pt))\mathrm{d}u)$, where $m\in\\{{2,4\\}}$.

###### Proof:

See Appendix -C. ∎

An alternative proof of the nearest-neighbor distance distribution for the
case $m=2$ in the PLP-PPP can be found in [13]. Lemma 6 presents a slightly
more general result that also applies to an intersection vehicle and the OG-
PPP, which can be obtained by rotating each line $L\in\Xi_{L}$ (Definition 4)
constituting the PLP-PPP such that they are orthogonal to each other.

######  Theorem 1.

The nearest-neighbor distance distribution for the PSP-PPP is given by (10),
where
$\ell(\gamma,\phi,\varphi)=\ell_{1}(\gamma,\phi,\varphi)\mathds{1}_{\gamma\leq
r}+\ell_{2}(\gamma,\phi,\varphi)\mathds{1}_{\gamma>r}$.
$\ell_{1}(\gamma,\phi,\varphi),\ell_{2}(\gamma,\phi,\varphi)=|\min(u_{1},h)\pm\min(u_{2},h)|$,
where
$u_{1},u_{2}=|-\gamma\cos(\phi-\varphi)\pm\sqrt{r^{2}-\gamma^{2}\sin^{2}(\phi-\varphi)}|$.
$\tilde{f}_{H}(h)=h{f}_{H}(h)/\mathbb{E}[H]$, and $m\in\\{{2,4\\}}$.

###### Proof:

See Appendix -D. ∎

Fig. 4a shows the normalized mean distance $\mathbb{E}[R_{n}]/n$ to the $n$th-
nearest neighbor in the PLM-PPP and PSP-PPP with Rayleigh distributed half-
lengths. The rate of change in $\mathbb{E}[R_{n}]/n$ of the PLM-PPP from that
of the PSP-PPP is at most 6% for $\mu=0.01$, and 5% for $\mu=1$. Fig. 4b
compares the nearest-neighbor distance distributions for different values of
$\mu$. We see that the nearest-neighbor distance distribution in the PLM-PPP
is tightly upper bounded by that in the PSP-PPP in accordance with Fig. 4a,
which shows that the mean distance to the nearest neighbor in the PLM-PPP is
tightly lower bounded by that in the PSP-PPP. We presume that the inference
obtained from Fig. 4b extends to the $n$th-nearest neighbor for $n>1$ as well.

$\displaystyle\mathcal{L}_{I_{o}^{m}}^{\mathrm{PSP-
PPP}}(s)=\bigg{(}\int\limits_{0}^{\infty}\bigg{(}\frac{1}{2h}\int\limits_{-h}^{h}\exp\bigg{(}-\lambda
ps^{\delta/2}\int\limits_{(-w-h)s^{-\delta/2}}^{(-w+h)s^{-\delta/2}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\bigg{)}\mathrm{d}w\bigg{)}\tilde{f}_{H}(h)\hskip
0.85358pt\mathrm{d}h\bigg{)}^{m/2}.$ (16)
$\displaystyle\mathcal{L}_{I_{!}}^{\mathrm{PSP-
PPP}}(s)=\exp\bigg{(}-\frac{\mu}{\pi}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}\bigg{(}1-\exp\bigg{(}-\lambda
p\int\limits_{-h}^{h}\bigg{(}{1+\bigg{(}\frac{\mathfrak{F}(\gamma,u,\phi,\varphi)}{s^{\delta}}\bigg{)}^{1/\delta}}\bigg{)}^{-1}\mathrm{d}u\bigg{)}\bigg{)}\gamma{f}_{H}(h)\hskip
0.85358pt\mathrm{d}\gamma\hskip 0.85358pt\mathrm{d}\phi\hskip
0.85358pt\mathrm{d}\varphi\hskip 0.85358pt\mathrm{d}h\bigg{)}.$ (17)
$\displaystyle p_{2}^{\mathrm{PLM-PPP}}\approx{p}_{2}^{\mathrm{PSP-
PPP}}=\mathcal{L}_{I_{o}^{2}}^{\mathrm{PSP-PPP}}(\theta
D^{\alpha})\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(\theta D^{\alpha}).$ (19)
$\displaystyle p_{3}^{\mathrm{PLM-PPP}}\approx{p}_{2}^{\mathrm{PSP-
PPP}}\times\int\limits_{0}^{\infty}\exp\bigg{(}-\lambda
pD\theta^{\delta/2}\int\limits_{0}^{\frac{2h}{D\theta^{\delta/2}}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\bigg{)}\hat{f}_{H}(h)\hskip
0.85358pt\mathrm{d}h.$ (20)

######  Conjecture 1.

The distance from the typical general vehicle to the $n$th-nearest neighbor in
the PLM-PPP stochastically dominates that distance in the PSP-PPP with
Rayleigh distributed half-lengths.

To facilitate the comparison of the success probabilities in the general
street-based Cox models and the homogeneous PPP, we recall the success
probability of the typical vehicle in a PPP. Let $\Phi_{d}$ denote a
stationary $d-$dimensional PPP of intensity $\lambda_{d}$ and $c_{d}$ denote
the volume of a unit $d-$dimensional ball. In particular, $c_{1}=2$ and
$c_{2}=\pi$.

######  Lemma 7 ([16], Sec. 5.2).

The success probability $p_{\mathrm{s}}$ of the typical vehicle in $\Phi_{d}$
is

$\displaystyle
p^{\Phi_{d}}_{\mathrm{s}}=\exp(-c_{d}\lambda_{d}D^{d}\theta^{\delta}\Gamma(1+\delta^{\prime})\Gamma(1-\delta^{\prime})),$
(11)

where $\delta^{\prime}=d/\alpha$.

Next, we analyze the success probabilities of the typical vehicle in Cox
vehicular networks.

##  IV Success Probabilities

Using (7) and the notations in Section II-E, we express the success
probability $p_{m}$ of the typical vehicle of order $m$ as

$\displaystyle p_{m}$ $\displaystyle=\mathbb{P}(g>\theta D^{\alpha}I)$
$\displaystyle=\mathbb{E}[\exp(-\theta D^{\alpha}(I_{o}^{m}+I_{!})].$ (12)

We can simplify (12) for all the Cox vehicular networks considered but the
PLM-PPP as

$\displaystyle p_{m}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}[\exp(-\theta
D^{\alpha}I_{o}^{m})]\mathbb{E}[\exp(-\theta D^{\alpha}I_{!})]$ (13)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathcal{L}_{I_{o}^{m}}(s)\mathcal{L}_{I_{!}}(s)|_{s=\theta
D^{\alpha}},$ (14)

where $(a)$ is due to the independence of the PPPs on the streets, and $(b)$
applies the definition of the Laplace transform. For the PLM-PPP, (13) does
not hold as the length of each street is dependent on that of other streets.

(a)  (b)  (c)

Figure 5:  Success probabilities of the typical general and
intersection/T-junction vehicles in the (a) PLP-PPP (b) PSP-PPP and (c) PLM-
PPP. $\lambda p=0.3,D=0.25$, $\alpha=4$, $\mu=2$, $0.1$, and $0.3$ for the
PLP, PSP, and PLM, respectively. $f^{\mathrm{PSP}}_{H}(h)=\delta(h-10)$. The
equation numbers are given in parentheses in the legends.

######  Proposition 1.

The success probability of the typical vehicle in the OG-PPP/PLP-PPP is

$\displaystyle p_{m}=\exp\bigg{(}$ $\displaystyle-m\lambda
pD\theta^{\delta/2}\Gamma(1+\delta/2)\Gamma(1-\delta/2)$
$\displaystyle-2\mu\int_{0}^{\infty}(1-\mathcal{L}_{I_{x}}(\theta
D^{\alpha}))\hskip 0.85358pt\mathrm{d}x\bigg{)},$ (15)

where $\mathcal{L}_{I_{x}}(s)=\exp\big{(}-\lambda
ps^{\delta/2}\int_{v_{x}}^{\infty}\frac{1}{\left(1+v^{{1/\delta}}\right)\sqrt{v-v_{x}}}\mathrm{d}v\big{)}$
with $v_{x}=\frac{x^{2}}{s^{\delta}}$, and $m\in\\{2,4\\}$.

###### Proof:

See Appendix -E. ∎

The success probability of the typical vehicle of order 2 in the PLP-PPP is
derived in [12]. The success probability (15) depends only on the distances of
the interferers to the typical vehicle, not their orientations. In Appendix
-E, we give a general proof that shows the effect of the order of the vehicle
and the irrelevance of the orientations on the success probability.

######  Proposition 2.

The Laplace transform of the interference from the vehicles on streets that
pass through the typical vehicle of order $m\in\\{2,4\\}$ in the PSP-PPP with
half-length density function $f_{H}(h)$ is given by (16), where
$\tilde{f}_{H}(h)=hf_{H}(h)/\mathbb{E}[H]$.

###### Proof:

See Appendix -F. ∎

######  Proposition 3.

The Laplace transform of the interference from the vehicles on all but the
streets that pass through the typical vehicle in the PSP-PPP with half-length
density function $f_{H}(h)$ is given by (17), where
$\mathfrak{F}(\gamma,u,\phi,\varphi)=\gamma^{2}+u^{2}+2\gamma
u\cos(\phi-\varphi)$.

###### Proof:

See Appendix -G. ∎

Following (14), the success probability of the typical vehicle in the PSP-PPP
is

$\displaystyle p_{m}^{\mathrm{PSP-PPP}}=\mathcal{L}_{I_{o}^{m}}^{\mathrm{PSP-
PPP}}(\theta D^{\alpha})\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(\theta
D^{\alpha}),$ (18)

where $\mathcal{L}_{I_{o}^{m}}^{\mathrm{PSP-PPP}}(s)$ and
$\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(s)$ are given by (16) and (17),
respectively.

######  Proposition 4.

The success probabilities of the typical general vehicle (order 2) and the
typical T-junction vehicle (order 3) in the PLM-PPP can be approximated as in
(19) and (20), respectively. $\mathcal{L}_{I_{o}^{2}}^{\mathrm{PSP-PPP}}(s)$
and $\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(s)$ in (19) are given by (16) and
(17), respectively, with $f_{H}(h)=\hat{f}_{H}(h)=2bh\exp(-bh^{2})$.

###### Proof:

See Appendix -H. ∎

Fig. 5 compares the success probabilities of the typical general and
intersection/T-junction vehicles. We omit the plot for the success probability
of the typical vehicle in the OG-PPP as it is the same as that in the PLP-PPP
by Proposition 1. We see that the success probability of the typical general
vehicle is higher than that of the typical intersection/T-junction vehicle in
all the Cox vehicular networks. The reason is that as two streets pass through
or end at the typical vehicle at an intersection or a T-junction, the received
interference is higher compared to the typical general vehicle through which
only one street passes.

######  Remark 1.

Fig. 5c indicates that the success probability of the typical general vehicle
in the PLM-PPP is tightly lower bounded by that in the PSP-PPP with Rayleigh
distributed half-lengths.

######  Remark 2.

The approximation (20) to the success probability of the typical T-junction
vehicle serves as a tight lower bound (see Fig. 5c). The integral term on the
right-hand side of (20) is the Laplace transform of the interference from the
vehicles on the street that has its endpoint at the origin. We can rephrase
(20) as follows. The success probability of the typical T-junction vehicle in
the PLM-PPP is tightly lower bounded by the success probability of the typical
vehicle (with one street passing through) in the network formed by
conditioning a street in the PSP-PPP (with Rayleigh distributed half-lengths)
such that one of its endpoints is at the origin.

Next, we analyze the equivalence between the spatial models using their
properties and the success probabilities of the typical vehicle in those
models.

##  V Equivalence of Spatial Models

To differentiate the street intensities of the models OG, PLP, PSP, and PLM,
we denote them as $\mu_{{\mathrm{OG}}}$, $\mu_{\mathrm{PLP}}$,
$\mu_{\mathrm{PSP}}$, and $\mu_{\mathrm{PLM}}$, respectively.

###  V-A OG-PPP and PLP-PPP

######  Theorem 2.

The PLP-PPP of street intensity $\mu_{\mathrm{PLP}}$ is strictly equivalent to
the OG-PPP with street intensity $\mu_{\mathrm{OG}}=\mu_{\mathrm{PLP}}$.

###### Proof:

The equivalence is a direct consequence of the success probabilities of the
typical vehicle in the OG-PPP and PLP-PPP given in Proposition 1. ∎

###  V-B PSP-PPP and PLP-PPP

######  Theorem 3.

Let $H\triangleq cH_{1}$, where $c$ is a constant and $H_{1}$ is a random
variable with mean 1. The PLP is the limiting process of the PSP as
$c\to\infty$ and $\mu_{{\mathrm{PSP}}}\to 0$ such that
$2\mu_{{\mathrm{PSP}}}\mathbb{E}[H]=2c\mu_{{\mathrm{PSP}}}=\mu_{{\mathrm{PLP}}}$.

###### Proof:

Figure 6: Reparametrization of the PSP. The stick
$S(y,\varphi,h)\in\Xi_{\mathrm{S}}$ is extended to form a line
$S^{\prime}(r,\phi,q)$. The perpendicular from the extended stick is at a
distance $r$ from $o$ and forms an angle $\phi$ with the $x-$axis.

We formalize the heuristic arguments on the relation between the PLP and PSP
given in [18]. From Definitions 4 and 5, we learn that the parameter spaces of
line and stick are different. To compare the line process with the stick
process, we first establish compatible parametrizations. Let
$S^{\prime}(r,\phi,q)$ denote the infinitely extended stick $S(y,\varphi,h)$,
where $q=\overline{yz}$ is the distance between the midpoint of the stick
$y=(u,v)$ and $z=(r\cos\phi,r\sin\phi)$, the closest point to the origin from
$S^{\prime}(r,\phi,q)$. Fig. 6 illustrates $S^{\prime}(r,\phi,q)$ and
$S(y,\varphi,h)$ using overlaid dashed and solid lines, respectively. We can
express $y=(u,v)$ and $\varphi$ in terms of $(r,\phi)$ and $q$ as
$u=r\cos\phi-q\sin\phi,\hskip 0.85358ptv=r\sin\phi+q\cos\phi,\hskip
0.85358pt\text{and}\hskip 2.84526pt\varphi=\phi-\pi/2$. The differential
element $\mathrm{d}u\hskip 0.85358pt\mathrm{d}v\hskip
0.85358pt\mathrm{d}\varphi$ equals $\mathrm{d}r\hskip
0.85358pt\mathrm{d}\phi\hskip 0.85358pt\mathrm{d}q$, i.e., the alternate
parametrizations are equivalent.

By (4), the line is just the projection of the stick from the parameter space
$(r,\phi,q)$ to $(r,\phi)$ when the latter is extended to infinity. However,
we learn from Lemmas 1 and 2 that the properties of the PLP and PSP differ.
Then, to obtain the PLP from PSP, we need to equate their statistical
properties. Equating the mean total street length per unit area in the PLP and
PSP given in Lemmas 1 and 2, we obtain
$2\mu_{{\mathrm{PSP}}}\mathbb{E}[H]=\mu_{{\mathrm{PLP}}}$. For
$\mu_{{\mathrm{PSP}}}\mathbb{E}[H]=c\mu_{{\mathrm{PSP}}}$ to remain finite,
$\mu_{{\mathrm{PSP}}}$ should go to zero as $c\to\infty$. ∎

######  Corollary 1.

The PSP-PPP, and its limiting case, the PLP-PPP, are strictly equivalent. As
the PLP and PSP are identically distributed as $c\to\infty$ and
$\mu_{\mathrm{PSP}}\to 0$, equivalence is not restricted to PPPs on the
streets but also holds for general point processes of vehicles.

###  V-C PLM-PPP and PSP-PPP

We learned from Proposition 4 that the success probability of the typical
general vehicle in the PLM-PPP is approximated by that in the PSP-PPP with the
same half-length density function as the PLM-PPP. Furthermore, from Figs. 4a
and 4b, we infer that the $n$th-nearest-neighbor distance distribution in the
PLM-PPP is tightly upper bounded by that in the PSP-PPP. Consequently, we can
deduce that the approximation to the success probability in the PLM-PPP is
tight. Fig. 7 compares the simulated success probability of the typical
general vehicle in the PLM-PPP with its lower bound. The maximum difference
between the success probabilities in the PSP-PPP and PLM-PPP is
$\epsilon=0.0297$ for $\mu=0.01$ and 0.0219 for $\mu=1$. Though the PLM can
characterize T-junctions, it is too complex to permit an exact analytical
expression.

Figure 7: Success probability of the typical general vehicle in the PLM-PPP
vs. that in the PSP-PPP-based approximation given by $(19)$ with
$f_{H}(h)=2bh\exp(-bh^{2})$, where $b=1.04$ for $\mu=1$, and $0.0103$ for
$\mu=0.01$. $\lambda p=0.3$, $D=0.25$, and $\alpha=4$.

######  Remark 3.

The PLM-PPP is $\epsilon-$equivalent with $\epsilon\ll 1$ to the PSP-PPP with
the same half-length distribution and street intensity as the PLM-PPP.
Consequently, the PSP-PPP serves as a good substitute for the PLM-PPP.

We see from Fig. 7 that the success probabilities of the typical general
vehicle in the PLM-PPP and PSP-PPP with Rayleigh distributed half-lengths are
even closer in the asymptotic regions of $\theta$ than in the middle regions
of $\theta$. Theorem 4 proves this observation formally.

######  Theorem 4.

The PLM-PPP is asymptotically equivalent in both the lower and upper regimes
of $\theta$ to the PSP-PPP with the same half-length density as the PLM-PPP.

###### Proof:

First, we study the asymptotic behavior of the PSP-PPP as $\theta\to 0$ and
$\infty$. Then, we compare them with that of the PLM-PPP.

######  Lemma 8.

As $\theta\to 0$, the PSP-PPP behaves like a vehicular point process formed
only by the typical vehicle’s streets, i.e.,

$\displaystyle 1-p_{m}^{\mathrm{PSP-PPP}}(\theta)=\Theta(\theta^{\delta
m/4}).$ (21)

###### Proof:

See Appendix -I. ∎

######  Lemma 9.

As $\theta\to\infty$, the success probability of the typical vehicle in the
PSP-PPP tends to that in a 2D PPP, i.e.,

$p_{m}^{\mathrm{PSP-
PPP}}(\theta)\sim\exp(-\pi\lambda_{2}pD^{2}\theta^{\delta}\Gamma(1+\delta)\Gamma(1-\delta)),$
(22)

where $\lambda_{2}=2\mu\lambda\mathbb{E}[H]$ is the 2D intensity of the PSP-
PPP.

###### Proof:

See Appendix -J. ∎

Equipped with the two lemmas, we continue with the proof of the theorem. As
$\theta\to 0$, for SIR $>\theta$ to hold, it suffices not to have any
interferers within a small disk around the typical vehicle. With high
probability, the small disk intersects only the street(s) passing through the
typical vehicle. Consequently, the system performance converges to that of
only the typical vehicle’s streets as $\theta\to 0$. The outage probability of
the typical vehicle due to its streets alone is proportional to $\lambda
p\theta^{\delta m/4}$ as $\theta\to 0$. On the other hand, as
$\theta\to\infty$, for SIR $>\theta$, a large disk around the typical vehicle
must be devoid of interferers. It follows from Lemma 9 that the street
geometry outside a large disk does not matter as $\theta\to\infty$ since the
PSP-PPP is similar to a 2D PPP at a large scale.

Lemma 8 extends to the PLM-PPP as the interaction between the sticks is
irrelevant when $\theta\to 0$. Also, the success probability of the typical
general vehicle with respect to its street alone in the PLM-PPP is the same as
that in the PSP-PPP with the same half-length density as the PLM-PPP. In
Appendix -J, we reasoned that the PSP-PPP behaves like a 2D PPP as
$\theta\to\infty$ through mapping all the points on the sticks to their
respective midpoints. The same logic holds for the PLM-PPP as it is also
formed by sticks whose midpoints form a PPP. As the PLM-PPP and PSP-PPP with
the same half-length density as the PLM-PPP behave like 2D PPPs of the same
intensities as $\theta\to\infty$, they are equivalent as $\theta\to\infty$. ∎

(a)  (b)  (c)

Figure 8:  Success probability of the typical general vehicle in the (a) PSP-PPP (b) PLM-PPP (c) PLP-PPP vs. that of the typical vehicle in 1D and 2D PPPs. $\lambda p=0.3,D=0.25$, $\alpha=4$, $\mu=0.1$, $0.3$, and $2$ for the PSP, PLM, and PLP, respectively. $f^{\mathrm{PSP}}_{H}(h)=\delta(h-10)$. The equation numbers are given in parentheses in the legends. Table I: Conditions for equivalence between the spatial models.  Model A | Model B | Conditions for Equivalence
---|---|---
OG-PPP | PLP-PPP | $\mu_{\mathrm{OG}}=\mu_{\mathrm{PLP}}$
PLP-PPP | PSP-PPP |  $\mu_{{\mathrm{PSP}}}\to 0$ and $c\to\infty$ s.t. $2\mu_{{\mathrm{PSP}}}\mathbb{E}[H]=2c\mu_{{\mathrm{PSP}}}=\mu_{{\mathrm{PLP}}}$
PSP-PPP | PLM-PPP |  $\mu_{\mathrm{PSP}}=\mu_{\mathrm{PLM}}$, $f_{H}^{\mathrm{PSP}}(h)=f_{H}^{\mathrm{PLM}}(h)$

Fig. 8 compares the success probabilities of the typical general vehicle in
the street-based Cox vehicular networks with that of the typical vehicle in 1D
and 2D PPPs. The success probability of the typical vehicle in the PSP-PPP and
PLM-PPP tends to that in the network formed only by the typical vehicle’s
street(s) as $\theta\to 0$ and to that of the 2D PPP as $\theta\to\infty$, as
given in Theorem 4 and the lemmas therein. The same holds for the PLP-PPP, as
established in [20, Th. 4 and 5]. As vehicles on each street in the PLP-PPP
form a 1D PPP, the PLP-PPP behaves like a 1D PPP as $\theta\to 0$.

Until now, we have assumed that the link distance $D$ is fixed. Next, we
discuss the equivalence between the spatial models when the link distances are
random. Here, the success probability is obtained by averaging the Laplace
transform of the interference over the link distances.

###  V-D Equivalence Under Random Link Distances

####  V-D1 PLP-PPP vs. OG-PPP

The OG-PPP is just a rotational variant of the PLP-PPP. Both of them have the
same statistical properties such as the mean total street length per unit area
(Lemma 1), distribution of the distance to the nearest-neighbor (Lemma 6), and
the Laplace transform of the interference (Proposition 1). Hence, the OG-PPP
and PLP-PPP are strictly equivalent even if the link distances are random.

####  V-D2 PSP-PPP vs. PLP-PPP

The PLP is the limiting process of the PSP as the lengths of the sticks extend
to infinity and street intensity tends to zero (Theorem 3). By the inherent
nature of the PSP, it is equivalent to the PLP irrespective of the mode of
communication.

####  V-D3 PLM-PPP vs. PSP-PPP

We deduced from Figs. 4a and 4b that the $n$th-nearest-neighbor (or
interferer) distance distribution in the PLM-PPP is tightly upper bounded by
that in the PSP-PPP with Rayleigh distributed half-lengths. It follows that
irrespective of the distribution of the distance to the intended transmitter,
the PLM-PPP and PSP-PPP with Rayleigh distributed half-lengths are
$\epsilon-$equivalent. Fig. 9 validates the above inference for the case where
the typical general vehicle receives a message from its nearest neighbor
(transmitter) through simulations.

Figure 9: Success probabilities of the typical general vehicle that receives a
message from its nearest neighbor. $\lambda p=0.5$ and $\alpha=4$.
$f_{H}(h)=\hat{f}_{H}(h)=2bh\exp(-bh^{2})$, where $b=1.04$ for $\mu=1$, and
$0.0103$ for $\mu=0.01$.

######  Remark 4.

The notion of equivalence enables us to consider only a representative set of
spatial models to obtain insights on the effect of the network parameters,
thereby reducing the computational costs and time associated with large-scale
experiments and system-level simulations. The success probabilities of the
typical vehicle in the OG-PPP, PLP-PPP, and PLM-PPP can be obtained from that
in the PSP-PPP by a suitable mapping between the parameters as summarized in
Table I.

##  VI Conclusions and Future Work

We developed a Coxian framework for the modeling and analysis of vehicular
networks. The spatial models in this framework can characterize different
street geometries involving intersections and T-junctions, and street lengths
that can be independent or dependent on each other. Streets of infinite
lengths and different orientations forming intersections can be characterized
by PLPs and their rotational variants. PSPs and PLMs can model streets of
varying finite lengths forming intersections and T-junctions, respectively. We
evaluated the reliability of a vehicle-to-vehicle link in the Cox vehicular
networks when the receiving vehicle is at an intersection, a T-junction, or a
general location, and its transmitter is at a certain fixed distance. Our
approach to defining the street system as a union of points of different
orders facilitates general analytical results for different types of typical
vehicles. The expressions for the reliability can be used to investigate the
interplay among the network parameters such as data rate, street intensity,
vehicle density, and the type of vehicle.

The concept of equivalence demonstrates that one does not need different
spatial models to analyze the reliability of a vehicle-to-vehicle link in
different geographical regions. The models considered in the Coxian framework
are equivalent in terms of reliability. This implies that the expression for
the reliability of the typical vehicle in the PSP-PPP is sufficient to
evaluate that in the OG-PPP, PLP-PPP, and PLM-PPP by appropriately mapping the
system parameters. Also, the vehicular networks behave like PPPs only in the
asymptotic regimes of the reliability or data rate. Hence, the street geometry
is relevant to understand vehicular network behavior.

An interesting future extension would be to understand how two or more models
developed in the Coxian framework can be superimposed to represent an
intricate geographic region with intersections and T-junctions, and streets
and highways. Also, one may look for tractable models that characterize curved
streets and examine whether they are equivalent to the line/stick-based
vehicular networks.

###  -A Proof of Lemma 4

To prove Lemma 4, we make use of [17, Prop. 6.2], which asserts the following:
There exist $a,b\geq 0$, such that the complementary cumulative distribution
function (CCDF) of the half-lengths $H$ in the lilypond model can be bounded
as

$\displaystyle 1-F_{H}(h)\leq a\exp(-bh^{2}),\hskip 8.53581pth\geq 0.$ (23)

Our goal is to find a tight approximation. By the properties of the cumulative
distribution function, $a\geq 1$. However, $a>1$ makes the bound (23) loose
for small $h$, which implies that $a=1$ is the natural choice. This reduces
the bound (23) to the CCDF of the Rayleigh distribution. Fig. 10 fits the CCDF
and PDF of the half-lengths to that of the Rayleigh distribution. We observe
that the upper bound on the CCDF is loose for small values of $h$. Instead of
choosing a value for $b$ that provides an upper bound, we approximate the CCDF
of the half-lengths with an appropriate value for $b$ that minimizes the gap
between the empirical and theoretical CCDFs. It follows from the Rayleigh
approximation that the mean half-length is
$\mathbb{E}[H]\approx\sqrt{\pi/4b}$. The value of $b$ is found by equating
$\mathbb{E}[H]$ to the empirical mean of the half-lengths.

(a)

(b)

Figure 10: Fitting the (a) CCDF and (b) PDF of the half-lengths to that of the
Rayleigh distribution. $\mu=0.01$.

Next, we study the relation between $b$ and $\mu$. Suppose we scale the PLM by
an arbitrary factor $\nu>0$. Then the street intensity $\mu$ is scaled by
$1/\nu^{2}$ and the half-lengths of the sticks are scaled by $\nu$, inversely
proportional to $\sqrt{\mu}$. As the PLM retains its lilypond nature with
scaling, the mean of the scaled half-lengths $H^{\prime}$ is scale-invariant,
i.e., $\sqrt{\pi/4b^{\prime}}\approx\mathbb{E}[H^{\prime}]=\mathbb{E}[\nu
H]\approx\sqrt{\nu^{2}\pi/4b}$. This implies that $b^{\prime}=b/\nu^{2}$, and
thus $b$ scales with $\mu$.

$\displaystyle{1-F_{R,\mathcal{V}_{o}^{m}}(r)}$
$\displaystyle={\mathbb{E}\bigg{[}\prod\limits_{k=1}^{m/2}\mathbb{E}\bigg{[}\prod\limits_{z\in{V}_{k}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}\bigg{]}}\stackrel{{\scriptstyle(a)}}{{=}}\bigg{[}\int\limits_{0}^{\infty}\frac{1}{h}\int\limits_{0}^{h}\exp(-\lambda\ell(\gamma,0,0))\tilde{f}_{H}(h)\hskip
0.28453pt\mathrm{d}\gamma\hskip 0.28453pt\mathrm{d}h\bigg{]}^{m/2}.$ (27)
$\displaystyle{1-F_{R,\mathcal{V}_{!}}(r)}$
$\displaystyle=\exp\bigg{(}-\frac{\mu}{\pi}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{r+h}\exp(-\lambda\ell(\gamma,\phi,\varphi))\gamma{f}_{H}(h)\hskip
0.28453pt\mathrm{d}\gamma\hskip 0.28453pt\mathrm{d}\phi\hskip
0.28453pt\mathrm{d}\varphi\hskip 0.28453pt\mathrm{d}h\bigg{)}.$ (28)

###  -B Proof of Lemma 5

For the typical vehicle at the origin, the probability that its nearest
neighbor is at a distance greater than $r$, $1-F_{R}(r)$, is given by

$\displaystyle 1-F_{R}(r)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in\mathcal{V}^{m}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\bigg{[}\prod\limits_{z\in\mathcal{V}_{o}^{m}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}\mathbb{E}\bigg{[}\prod\limits_{z\in\mathcal{V}_{!}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{k=1}^{\lceil
m/2\rceil}\prod\limits_{z\in{V}_{k}}\mathsf{1}\\{{z\notin b(o,r)\\}}\bigg{]}$
$\displaystyle\times\mathbb{E}\bigg{[}\prod\limits_{k>\lceil
m/2\rceil}\prod\limits_{z\in{V}_{k}}\mathsf{1}\\{{z\notin b(o,r)\\}}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\underbrace{\mathbb{E}\bigg{[}\prod\limits_{k=1}^{\lceil
m/2\rceil}\mathbb{E}\bigg{[}\prod\limits_{z\in{V}_{k}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}\bigg{]}}_{1-F_{R,\mathcal{V}_{o}^{m}}(r)}$
$\displaystyle\times\underbrace{\mathbb{E}\bigg{[}\prod\limits_{k>\lceil
m/2\rceil}\mathbb{E}\bigg{[}\prod\limits_{z\in{V}_{k}}\mathsf{1}\\{{z\notin
b(o,r)\\}}\bigg{]}\bigg{]}}_{1-F_{R,\mathcal{V}_{!}}(r)},$ (24)

where $(a)$ and $(b)$ exploit the independence of the 1D PPPs.

The contact distance distribution $G_{R}(r)$ is the probability of finding a
vehicle within a distance $r$ from an arbitrary location in the plane, say the
origin. No streets pass through or contain the origin a.s. This implies that
$G_{R}(r)$ is defined with respect to $\mathcal{V}$, which is equivalent in
distribution to $\mathcal{V}^{m}\setminus\mathcal{V}_{o}^{m}=\mathcal{V}_{!}$.
Hence $G_{R}(r)=F_{R,\mathcal{V}_{!}}(r)$.

###  -C Proof of Lemma 6

Using the probability generating functional (PGFL) of the PPP, we can express
$1-F_{R,\mathcal{V}_{!}}(r)$ in (24) as

$\displaystyle 1-{F}_{R,\mathcal{V}_{!}}(r)$
$\displaystyle=\exp\bigg{(}-2\mu\int\limits_{0}^{\pi}\int\limits_{0}^{r}(1-\exp(-2\lambda\sqrt{r^{2}-u^{2}}))\mathrm{d}u\hskip
0.85358pt\mathrm{d}v(\varphi)\bigg{)},$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\bigg{(}-2\mu\int\limits_{0}^{r}(1-e^{-2\lambda\sqrt{r^{2}-u^{2}}})\mathrm{d}u\bigg{)},$
(25)

where $(a)$ follows from the independence between $u$ and $\varphi$ and the
fact that $\textstyle\int_{\mathbb{R}}\mathrm{d}v(\varphi)=1$ for the OG-
PPP/PLP-PPP. Since the vehicle locations follow a 1D PPP, with respect to the
streets that pass through the origin, we have
$1-F_{R,\mathcal{V}_{o}^{m}}(r)=(e^{-2\lambda r})^{m/2}$.

###  -D Proof of Theorem 1

The probability that there are no vehicles on a stick $S(y,\varphi,h)$ within
a distance $r$ from the origin is $\exp(-\lambda|S(y,\varphi,h)\cap
b(o,r)|_{1})$. We derive $|S(y,\varphi,h)\cap b(o,r)|_{1}$ as follows: The
midpoint $y$ is denoted as $(\gamma,\phi)$ in polar coordinates. Using (5), we
can express the points on a stick $S(y,\phi,h)$ of length $2h$ as
$(\gamma\cos\phi+u\cos\varphi,\gamma\sin\phi+u\sin\varphi)$, $u\in(-h,h)$.
Then the squared distance between a point on $S(y,\phi,h)$ and the origin is
$\gamma^{2}+u^{2}+2\gamma u\cos(\phi-\varphi)$. The points of intersection of
$S(y,\varphi,h)$ on $b(o,r)$ are at distances
$u_{1},u_{2}=|-\gamma\cos(\phi-\varphi)\pm\sqrt{r^{2}-\gamma^{2}\sin^{2}(\phi-\varphi)}|$
from the midpoint of the stick. They follow from solving

$\gamma^{2}+u^{2}+2\gamma u\cos(\phi-\varphi)=r^{2}$ (26)

with respect to $u$. For a stick $S(y,\varphi,h)$ to intersect $b(o,r)$, $y$
must be within $b(o,r+h)$. We consider two cases—$y\in b(o,r)$, and $y\in
b(r,r+h)$:

Figure 11: Realizations corresponding to cases (1a), (1b), (2a), and (2b). The
midpoints of the streets are highlighted using filled ‘$\circ$’. The points of
intersection of the street on $b(o,r)$ are at distances $u_{i}$,
$i\in\\{{1,2\\}}$, from the midpoint of the street. $u_{i}$ in the cases (1b)
and (2b) refers to $u_{1}$ or $u_{2}$.

1\. $y\in b(o,r)$: Let $|S(y,\varphi,h)\cap b(o,r)|\triangleq\ell_{1}$ for
$y\in b(o,r)$. When

  * a.

$S(y,\varphi,h)\cap b(o,r)=S(y,\varphi,h)$: The stick lies within $b(o,r)$.
Then $\ell_{1}(\gamma,\phi,\varphi)=2h$.

  * b.

$S(y,\varphi,h)\cap b(o,r)\subset S(y,\varphi,h)$: The stick is not fully
contained in $b(o,r)$. For a stick that passes through $b(o,r)$,
$\ell_{1}(\gamma,\phi,\varphi)=u_{1}+u_{2}$. For a stick with only one
endpoint in $b(o,r)$, $\ell_{1}(\gamma,\phi,\varphi)=u_{1}+h$ or $h+u_{2}$.

Summarizing the above cases, we write
$\ell_{1}(\gamma,\phi,\varphi)=\min(u_{1},h)+\min(u_{2},h)$.

2\. $y\in b(r,r+h)$: Let $|S(y,\varphi,h)\cap b(o,r)|\triangleq\ell_{2}$ for
$y\in b(r,r+h)$. When

  * a.

$S(y,\varphi,h)\cap b(o,r)=\emptyset$: The stick does not intersect $b(o,r)$.
Then $\ell_{2}(\gamma,\phi,\varphi)=0$.

  * b.

$S(y,\varphi,h)\cap b(o,r)\subset S(y,\varphi,h)$: For a stick that passes
through $b(o,r)$, $\ell_{2}(\gamma,\phi,\varphi)=|u_{1}-u_{2}|$. For a stick
with only one endpoint in $b(o,r)$, $\ell_{2}(\gamma,\phi,\varphi)=|u_{1}-h|$
or $|h-u_{2}|$.

Thus, $\ell_{2}(\gamma,\phi,\varphi)=|\min(u_{1},h)-\min(u_{2},h)|$. Then the
probability of not finding a vehicle on $S(y,\varphi,h)$ within $b(o,r)$ is
$\exp(-\lambda\ell(\gamma,\phi,\varphi))$, where
$\ell(\gamma,\phi,\varphi)=\ell_{1}(\gamma,\phi,\varphi)\mathds{1}_{\gamma\leq
r}+\ell_{2}(\gamma,\phi,\varphi)\mathds{1}_{r<\gamma\leq r+h}$. Fig. 11
illustrates the above cases.

For the typical vehicle’s street to pass through or end at the origin, its
midpoint must belong to $b(o,h)$. The midpoint of the typical vehicle’s street
of half-length $h$ is at a distance $\gamma\in[0,h]$ from the origin. Using
the above results and the decomposition of the nearest-neighbor distance
distribution in Lemma 5, we can write ${1-F_{R,\mathcal{V}_{o}^{m}}(r)}$ and
${1-F_{R,\mathcal{V}_{!}}(r)}$ as in (27) and (28), respectively, by applying
the PGFL of the PPP.

Note that $(a)$ in (27) follows from the facts that (i) the rotation of the
typical vehicle’s streets around the typical vehicle does not change
$u_{1},u_{2}$, and hence we can set $\phi=\varphi=0$, and (ii) $m/2$ streets
are independent. Substituting (27) and (28) in (9), we obtain (10).

###  -E Proof of Proposition 1

By (4), we can express the points on $L(x,\varphi)$ as
$(x\cos\phi-u\sin\phi,x\sin\phi+u\cos\phi)$, $u\in\mathbb{R}$. Let $x$ refer
to the $k$th-nearest point in $\Phi_{1}$ from the origin. Then $V_{k}$ denotes
the point process of vehicles on $L(x,\phi)$ (Section II-E). The Laplace
transform of the interference $\mathcal{L}_{I_{x}}(s)$ from the vehicles on
$L(x,\varphi)$ to the typical vehicle at the origin is given by

$\displaystyle\mathcal{L}_{I_{x}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in
V_{k}}\mathbb{E}_{g_{z}}\left[\exp\left(-sg_{z}\|z\|^{-\alpha}\mathit{B}_{z}\right)\right]\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in
V_{k}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}$ (29)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\bigg{(}-\lambda
p\int_{-\infty}^{\infty}\frac{1}{1+\left(\frac{x^{2}+u^{2}}{s^{\delta}}\right)^{1/\delta}}{\mathrm{d}u}\bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\exp\bigg{(}-\lambda
ps^{\delta/2}\int_{v_{x}}^{\infty}\frac{1}{\left(1+v^{{1/\delta}}\right)\sqrt{v-v_{x}}}\mathrm{d}v\bigg{)},$
(30)

where $\delta=2/\alpha$, $(a)$ substitutes
$\|z\|_{2}=\|(x\cos\phi-u\sin\phi,x\sin\phi+u\cos\phi)\|_{2}$ and the PGFL of
the PPP. $\lambda p$ is the intensity of the active transmitters for which
$\mathit{B}_{z}=1$, $z\in V_{k}$. $(b)$ is due to the change of variable
$v=\frac{x^{2}+u^{2}}{s^{\delta}}$, and $v_{x}=\frac{x^{2}}{s^{\delta}}$.

We learned from (14) that the success probability is the product of the
Laplace transforms of the interference $I_{o}^{m}$ and $I_{!}$, which we
derive below.

$\displaystyle\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(\theta D^{\alpha})\sim
1-\frac{\theta\mu\lambda
p}{\pi}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}\int\limits_{-h}^{h}\bigg{(}\frac{D^{2}}{{\gamma^{2}+u^{2}+2\gamma
u\cos(\phi-\varphi)}}\bigg{)}^{1/\delta}\gamma{f}_{H}(h)\mathrm{d}u\hskip
0.28453pt\mathrm{d}\gamma\hskip 0.28453pt\mathrm{d}\phi\hskip
0.28453pt\mathrm{d}\varphi\hskip 0.28453pt\mathrm{d}h=1-\Theta(\theta).$ (41)
$\displaystyle\lim_{\theta\to 0}\hskip 2.84526pt1-p_{m}^{\mathrm{PSP-
PPP}}=1-\lim_{\theta\to 0}\hskip
2.84526pt\mathbb{E}\bigg{[}\exp\Bigg{(}-\lambda
ps^{\delta/2}\int\limits_{(-W-H)s^{-\delta/2}}^{(-W+H)s^{-\delta/2}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\bigg{)}\Bigg{]}^{m/2}$
(42) $\displaystyle\hskip 79.66771pt=1-\mathbb{E}\Bigg{[}1-\lambda
ps^{\delta/2}\int\limits_{(-W-H)s^{-\delta/2}}^{(-W+H)s^{-\delta/2}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\Bigg{]}^{m/2}$
(43) $\displaystyle\hskip 79.66771pt=1-\mathbb{E}\Bigg{[}1-\frac{\lambda
ps^{\delta/2}}{2}\int\limits_{(-W-H)^{2}s^{-\delta/2}}^{(-W+H)^{2}s^{-\delta/2}}\frac{1}{(1+u^{{1/\delta}})\sqrt{u}}\hskip
0.85358pt\mathrm{d}u\Bigg{]}^{m/2}\sim K_{\alpha}\lambda ps^{\delta m/4}.$
(44)

####  -E1 Laplace Transform of $I_{o}^{m}$

The Laplace transform of the interference from $V_{0}$ to the typical general
vehicle (order 2) can be obtained by setting $x=0$ in (30), i.e.,

$\displaystyle\mathcal{L}_{I_{o}^{2}}(s)$ $\displaystyle=\exp(-2\lambda
ps^{\delta/2}\Gamma(1+\delta/2)\Gamma(1-\delta/2)).$ (31)

For the typical intersection vehicle (order 4), as two streets pass through
the origin, $\mathcal{V}_{o}^{4}=V_{0}\cup V_{1}$ (Sec. II-E). Then
${\mathcal{L}}_{I_{o}^{4}}(s)$ can be written as in (29) as follows:

$\displaystyle{\mathcal{L}}_{I_{o}^{4}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in V_{0}\cup
V_{1}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\bigg{(}\mathbb{E}\Bigg{[}\prod\limits_{z\in
V_{0}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}\bigg{)}^{2},$ (32)

where $(c)$ results from $V_{0}$ being identically distributed as $V_{1}$. It
follows that $\mathcal{L}_{I_{o}^{4}}(s)=\mathcal{L}^{2}_{I_{o}^{2}}(s)$.

####  -E2 Laplace Transform of $I_{!}$

The aggregate Laplace transform of the interference from the vehicles on all
the streets that do not pass through the typical vehicle is given by

$\displaystyle\mathcal{L}_{I_{!}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in\mathcal{V}_{!}}\mathbb{E}_{g_{z}}\left[\exp\left(-sg_{z}\|z\|^{-\alpha}\mathit{B}_{z}\right)\right]\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{=}}\mathbb{E}\bigg{[}\prod\limits_{k>m/2}\mathbb{E}\bigg{[}\prod\limits_{z\in
V_{k}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}\bigg{]}$ (33)
$\displaystyle\stackrel{{\scriptstyle(f)}}{{=}}\exp\bigg{(}-\mu\int_{0}^{\pi}\int_{-\infty}^{\infty}(1-\mathcal{L}_{I_{x}}(s))\hskip
1.42262pt\mathrm{d}x\hskip 1.42262pt\mathrm{d}\nu(\varphi)\bigg{)}$
$\displaystyle\stackrel{{\scriptstyle(g)}}{{=}}\exp\bigg{(}-\mu\int_{-\infty}^{\infty}(1-\mathcal{L}_{I_{x}}(s))\hskip
1.42262pt\mathrm{d}x\bigg{)},$ (34)

where $(e)$ follows from $\mathcal{V}_{!}=\cup_{k>m/2}V_{k}$ and the fact that
the $V_{k}$’s are independent 1D PPPs, $(f)$ uses (30) and the PGFL of the PPP
with respect to $x$ and $\varphi$, and $(g)$ results from
$\mathcal{L}_{I_{x}}(s)$ being independent of $\varphi$, and
$\int_{\mathbb{R}}\mathrm{d}\nu(\varphi)=1$. Substituting (31), (32), and (34)
in (14), we obtain (15).

###  -F Proof of Proposition 2

Using (5), we can denote the points on a stick $S(y,\phi,h)$ of length $2h$ as
$(\gamma\cos\phi+u\cos\varphi,\gamma\sin\phi$ $+u\sin\varphi)$, $u\in(-h,h)$.
The midpoint of the street that passes through the typical vehicle is at a
signed distance $W\sim\mathcal{U}(-h,h)$ from the origin. Then the endpoints
of the street are at signed distances $-W-h$, and $-W+h$ to the origin. As in
(29), the Laplace transform of the interference $I_{o}^{2}$ for the typical
general vehicle can be written as

$\displaystyle\mathcal{L}_{I_{o}^{2}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in\mathcal{V}_{o}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\bigg{[}\exp\bigg{(}-\lambda
p\int\limits_{-W-H}^{-W+H}\frac{1}{1+\left(\frac{u}{s^{\delta/2}}\right)^{2/\delta}}{\mathrm{d}u}\bigg{)}\bigg{]}$
(35)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathbb{E}\bigg{[}\exp\bigg{(}-\lambda
ps^{\delta/2}\int\limits_{\frac{(-W-H)}{s^{\delta/2}}}^{\frac{(-W+H)}{s^{\delta/2}}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\bigg{)}\bigg{]},$
(36)

where $(a)$ substitutes $\|z\|_{2}=\|(u\cos\varphi,u\sin\varphi)\|_{2}$ and
applies the PGFL of the PPP, and $(b)$ results from the change of variable
$v=\frac{u}{s^{\delta/2}}$. $(c)$ evaluates the expectation with respect to
$H$ using $\tilde{f}_{H}(h)$ rather than ${f}_{H}(h)$ based on Lemma 3. For
the typical intersection vehicle, as two streets pass through the origin and
the point processes on them are independent, the corresponding Laplace
transform of the interference is the square of that given in (36) similar to
(32).

###  -G Proof of Proposition 3

This proof uses the same notation as in Appendix -F. Let
$\mathcal{A}\triangleq\mathbb{R}^{+}\times[0,2\pi)\times[0,\pi)\times\mathbb{R}^{+}$
and $a=(\gamma,\phi,\varphi,h)\in\mathcal{A}$. From Section II-E, we have
$V_{k}$, $k>m/2$, denoting the point process of vehicles on the $k$th street
$S(\gamma,\phi,\varphi,h)$ that does not pass through the origin. The Laplace
transform of the interference $\mathcal{L}_{I_{a}}(s)$ from the vehicles on
$S(\gamma,\phi,\varphi,h)$ is

$\displaystyle\mathcal{L}_{I_{a}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{z\in
V_{k}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\bigg{(}-\lambda
p\int\limits_{-h}^{h}\frac{1}{1+\frac{{\mathfrak{F}(\gamma,u,\phi,\varphi)}^{1/\delta}}{s}}\mathrm{d}u\bigg{)},$
(37)

where $(a)$ applies the PGFL of the PPP,
$\|z\|_{2}=\|(\gamma\cos\phi+u\cos\varphi,\gamma\sin\phi+u\sin\varphi)\|_{2}$,
and $\mathfrak{F}(\gamma,u,\phi,\varphi)=\gamma^{2}+u^{2}+2\gamma
u\cos(\phi-\varphi)$.

Using (33), (37), and applying PGFL with respect to midpoints, orientations
and half-lengths, we write the Laplace transform of the interference from
$\mathcal{V}_{!}$ as

$\displaystyle\mathcal{L}_{I_{!}}(s)$
$\displaystyle=\mathbb{E}\bigg{[}\prod\limits_{k>m/2}\mathbb{E}\bigg{[}\prod\limits_{z\in
V_{k}:\mathit{B}_{z}=1}\frac{1}{1+s\|z\|^{-\alpha}}\bigg{]}\bigg{]}$
$\displaystyle=\exp\bigg{(}-\frac{\mu}{\pi}\int_{\mathcal{A}}(1-\mathcal{L}_{I_{a}}(s))\gamma{f}_{H}(h)\mathrm{d}\mathcal{A}\bigg{)},$
(38)

where $\mathrm{d}\mathcal{A}=\mathrm{d}\gamma\hskip
0.85358pt\mathrm{d}\phi\hskip 0.85358pt\mathrm{d}\varphi\hskip
0.85358pt\mathrm{d}h$.

###  -H Proof of Proposition 4

####  -H1 Success probability of the typical general vehicle in the PLM-PPP

We learned from Conjecture 1 that the probability of finding the $n$th-nearest
neighbor closer is higher in the PSP-PPP than in the PLM-PPP. Consequently,
the success probability of the typical general vehicle in the PLM-PPP is lower
bounded by that in the PSP-PPP. As this inference is based on a conjecture, we
present the result for the success probability of the typical general vehicle
in the PLM-PPP as an approximation rather than a bound. By (12) and (14), we
have

$\displaystyle p_{2}^{\mathrm{PLM-PPP}}$
$\displaystyle=\mathcal{L}_{I_{o}^{2}+I_{!}}^{\mathrm{PLM-PPP}}(s)$
$\displaystyle\approx\mathcal{L}_{I_{o}^{2}}^{\mathrm{PSP-
PPP}}(s)\mathcal{L}_{I_{!}}^{\mathrm{PSP-
PPP}}(s)\mid_{f_{H}(h)=\hat{f}_{H}(h)}.$ (39)

####  -H2 Success probability of the typical T-junction vehicle in the PLM-PPP

We have $\mathcal{V}_{o}^{3}=V_{0}\cup V_{1}$, where $V_{0}$ denote the
vehicles on the street that passes through the origin, and $V_{1}$ denote the
vehicles on the street with one endpoint at the T-junction. The success
probability of the typical T-junction vehicle (order 3), $p_{3}^{\mathrm{PLM-
PPP}}$, is given by

$\displaystyle p_{3}$
$\displaystyle=\mathbb{E}\Bigg{[}\prod\limits_{\begin{subarray}{c}z\in
V_{0}\cup V_{1}:\\\
\mathit{B}_{z}=1\end{subarray}}\frac{1}{1+s\|z\|^{-\alpha}}\prod\limits_{\begin{subarray}{c}z\in\mathcal{V}_{!}:\\\
\mathit{B}_{z}=1\end{subarray}}\frac{1}{1+s\|z\|^{-\alpha}}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\approx}}\mathbb{E}\Bigg{[}\prod\limits_{\begin{subarray}{c}z\in
V_{0}\cup\mathcal{V}_{!}:\\\
\mathit{B}_{z}=1\end{subarray}}\frac{1}{1+s\|z\|^{-\alpha}}\Bigg{]}\mathbb{E}\Bigg{[}\prod\limits_{\begin{subarray}{c}z\in
V_{1}:\\\ \mathit{B}_{z}=1\end{subarray}}\frac{1}{1+s\|z\|^{-\alpha}}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\approx}}\mathcal{L}_{I_{o}^{2}}^{\mathrm{PSP-
PPP}}(s)\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(s)\hskip
0.85358pt\mathbb{E}\Bigg{[}\prod\limits_{\begin{subarray}{c}z\in V_{1}:\\\
\mathit{B}_{z}=1\end{subarray}}\frac{1}{1+s\|z\|^{-\alpha}}\Bigg{]}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathcal{L}_{I_{o}^{2}}^{\mathrm{PSP-
PPP}}(s)\mathcal{L}_{I_{!}}^{\mathrm{PSP-PPP}}(s)$
$\displaystyle\times\int\limits_{0}^{\infty}\exp\bigg{(}-\lambda
ps^{\delta/2}\int\limits_{0}^{2hs^{-\delta/2}}\frac{1}{1+v^{{2/\delta}}}\mathrm{d}v\bigg{)}\hat{f}_{H}(h)\mathrm{d}h,$
(40)

where $(a)$ assumes independence between $V_{1}$ and
$V_{0}\cup\mathcal{V}_{!}$, and $(b)$ follows from (39). The integral
expression in $(c)$ can be derived similarly to (36) using the detail that for
the street whose one endpoint is a T-junction at the origin, its other
endpoint is at a distance $2H$ from the origin. $\hat{f}_{H}(h)$ is the
approximated density of the half-length of the street that ends at a
T-junction.

###  -I Proof of Lemma 8

We can approximate $\mathcal{L}_{I_{!}}(\theta D^{\alpha})$ given by (38)
using Taylor’s series as $\theta\to 0$ as in (41). Using (35) and (41), the
outage probability as $\theta\to 0$, can be written as in (42), where
$s=\theta D^{\alpha}$. By interchanging the limit and expectation (using the
monotone convergence theorem), and applying Taylor’s theorem inside the
expectation in (42), we obtain (43). It transforms to (44) by the change of
variable $u=v^{2}$, whose integral term evaluates to a constant $K_{\alpha}$
as $\theta\to 0$. Note that $K_{\alpha}$ depends on $\alpha$.

###  -J Proof of Lemma 9

Consider a model $\mathcal{V}^{\prime}$ formed by mapping all the points on
each stick to its midpoint. The expected number of active transmitters on each
stick is $2\lambda p\mathbb{E}[H]$. Since the midpoints follow a 2D PPP of
intensity $\mu$, $\mathcal{V}^{\prime}$ forms a non-simple 2D PPP with density
$2\mu\lambda p\mathbb{E}[H]$. It can be seen as a marked point process whose
ground process is a 2D PPP of intensity $\mu$ and the marks $2\lambda
p\mathbb{E}[H]$ refer to the multiplicity of the points. Let $\mathbb{M}$
define the mapping from $\mathcal{V}$ to $\mathcal{V}^{\prime}$. Using (6) and
(7), the success probability of the typical vehicle in the PSP-PPP can be
expressed as

$p_{m}^{\mathrm{PSP-
PPP}}=\mathbb{P}\bigg{(}g>D^{\alpha}\sum\limits_{z\in\mathcal{V}}g_{z}\|\theta^{-1/\alpha}z\|^{-\alpha}B_{z}\bigg{)}.$

Similarly, the success probability of the typical vehicle in
$\mathcal{V}^{\prime}$ is written as

${p_{m}^{\mathcal{V}^{\prime}}}=\mathbb{P}\bigg{(}g>D^{\alpha}\sum\limits_{z\in\mathcal{V}}g_{z}\|\theta^{-1/\alpha}\mathbb{M}(z)\|^{-\alpha}B_{z}\bigg{)}.\\\
$

For sticks of half-length $h$,
$|\|z\|-\|\mathbb{M}(z)\||\leq\|z-\mathbb{M}(z)\|\leq h<\infty$. Then

$|\|\theta^{-1/\alpha}z\|-\|\theta^{-1/\alpha}\mathbb{M}(z)\||\to 0\hskip
8.53581pt\text{as}\hskip 8.53581pt\theta\to\infty.$ (45)

Then $p_{m}^{\mathrm{PSP-PPP}}\to p_{m}^{\mathcal{V}^{\prime}}$ as
$\theta\to\infty$. The order $m$ of the typical vehicle is irrelevant as the
mapping $\mathbb{M}$ does not distinguish an intersection from a non-
intersection. The success probability of the typical vehicle
$p_{m}^{\mathcal{V}^{\prime}}$ can be obtained by substituting
$\lambda_{2}=2\mu\lambda p\mathbb{E}[H]$ in (11).

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|  Jeya Pradha Jeyaraj (S’17) received the M.Tech degree in electrical
engineering from the Indian Institute of Technology Kanpur, India, in 2013.
She is currently pursuing the Ph.D. degree in electrical engineering at the
University of Notre Dame, Indiana, USA, since 2016. She is also working
towards the M.S. degree in applied and computational mathematics and
statistics. Before joining her Ph.D., she worked as a Software Development
Engineer in the 3G cellular technologies group at Cisco Systems, India. She
held internship positions in the research and development divisions at Toyota
Motor North America and Qualcomm in the USA. Her research interests include
vehicular networks, stochastic geometry, spectrum sharing, and data mining.
---|---
|  Martin Haenggi (S‘95-M‘99-SM‘04-F‘14) received the Dipl.-Ing. (M.Sc.) and
Dr.sc.techn. (Ph.D.) degrees in electrical engineering from the Swiss Federal
Institute of Technology in Zurich (ETH) in 1995 and 1999, respectively.
Currently he is the Freimann Professor of Electrical Engineering and a
Concurrent Professor of Applied and Computational Mathematics and Statistics
at the University of Notre Dame, Indiana, USA. In 2007-2008, he was a visiting
professor at the University of California at San Diego, and in 2014-2015 he
was an Invited Professor at EPFL, Switzerland. He is a co-author of the
monographs "Interference in Large Wireless Networks" (NOW Publishers, 2009)
and ‘Stochastic Geometry Analysis of Cellular Networks’ (Cambridge University
Press, 2018) and the author of the textbook "Stochastic Geometry for Wireless
Networks" (Cambridge, 2012), and he published 16 single-author journal
articles. His scientific interests lie in networking and wireless
communications, with an emphasis on cellular, amorphous, ad hoc (including D2D
and M2M), cognitive, and vehicular networks. He served as an Associate Editor
of the Elsevier Journal of Ad Hoc Networks, the IEEE Transactions on Mobile
Computing (TMC), the ACM Transactions on Sensor Networks, as a Guest Editor
for the IEEE Journal on Selected Areas in Communications, the IEEE
Transactions on Vehicular Technology, and the EURASIP Journal on Wireless
Communications and Networking, as a Steering Committee member of the TMC, and
as the Chair of the Executive Editorial Committee of the IEEE Transactions on
Wireless Communications (TWC). From 2017 to 2018, he was the Editor-in-Chief
of the TWC. Currently he is an editor for MDPI Information. For both his M.Sc.
and Ph.D. theses, he was awarded the ETH medal. He also received a CAREER
award from the U.S. National Science Foundation in 2005 and three awards from
the IEEE Communications Society, the 2010 Best Tutorial Paper award, the 2017
Stephen O. Rice Prize paper award, and the 2017 Best Survey paper award, and
he is a Clarivate Analytics Highly Cited Researcher.
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