# Outage Analysis of Cooperative NOMA in Millimeter Wave Vehicular Network at
Intersections

Baha Eddine Youcef Belmekki  Abdelkrim Hamza  LISIC Laboratory, Electronic and
Computer Faculty, USTHB, Algiers, Algeria,  Benoît Escrig  University of
Toulouse, IRIT Laboratory, School of ENSEEIHT, Institut National Polytechnique
de Toulouse, France, e-mail: $\\{$bahaeddine.belmekki,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study cooperative non-orthogonal multiple access scheme
(NOMA) for millimeter wave (mmWave) vehicular networks at intersection roads.
The intersection consists of two perpendicular roads. Transmissions occur
between a source, and two destinations nodes with a help of a relay. We assume
that the interference come from a set of vehicles that are distributed as a
one dimensional homogeneous Poisson point process (PPP). Our analysis includes
the effects of blockage from buildings at intersections. We derive closed form
outage probability expressions for cooperative NOMA, and compare them with
cooperative orthogonal multiple access (OMA). We show that cooperative NOMA
offers a significant improvement over cooperative OMA, especially for high
data rates. We also show that as the nodes reach the intersection, the outage
probability increases. Counter-intuitively, we show that the non line of sigh
(NLOS) scenario has a better performance than the line of sigh (LOS) scenario.
The analysis is verified with Monte Carlo simulations.

###### Index Terms:

5G, NOMA, mmWave, interference, outage probability, cooperative, vehicular
communications.

###  -A Motivation

Road traffic safety is a major issue, and more particularly at intersections
[1]. Vehicular communications provide helpful applications for road safety and
traffic management. These applications help to prevent accidents or alerting
vehicles of accidents happening in their surroundings. Hence, these
applications require high bandwidth and high spectral efficiency, to insure
high reliability and low latency communications. In this context, non-
orthogonal multiple access (NOMA) has been show to increase the data rate and
spectral efficiency [2]. Unlike orthogonal multiple access (OMA), NOMA allows
multiple users to share the same resource with different power allocation
levels. On the other hand, the needs of vehicular communications for the fifth
generation (5G) in terms of resources require a larger bandwidth. Since the
spectral efficiency of sub-6 GHz bands has already reached the theoretical
limits, millimeter wave (mmWave) frequency bands (20-100 GHz and beyond) offer
a very large bandwidth [3].

###  -B Related Works

####  -B1 Cooperative NOMA

NOMA is an efficient multiple access technique for spectrum use. It has been
shown that NOMA outperforms OMA [4]. However, few research investigates the
effect of co-channel interference and their impact on the performance
considering direct transmissions [5, 6, 7], and cooperative transmissions [8,
9].

####  -B2 Cooperative mmWave

In mmWave bands, few works studied cooperative communications using tools from
stochastic geometry [10, 11, 12, 13]. However, in [10, 11, 12], the effect of
small-scale fading is not taken into consideration. In [13], the authors
investigate the performance of mmWave relaying networks in terms of coverage
probability with best relay selection.

####  -B3 Vehicular communications at intersections

Several works studied the effect of the interference at intersections,
considering OMA. The performance in terms of success probability are derivated
considering direct transmission in [14, 15]. The performance of vehicle to
vehicle (V2V) communications are evaluated for multiple intersections schemes
considering direct transmission in [16]. In [17], the authors derive the
outage probability of a V2V communications with power control strategy of a
direct transmission. In [18], the authors investigate the impact of a line of
sight and non line of sight transmissions at intersections considering
Nakagami-$m$ fading channels. The authors in [19] study the effect of mobility
of vehicular communications at road junctions. In [20, 21, 22, 23, 24], the
authors respectively study the impact of non-orthogonal multiple access,
cooperative non-orthogonal multiple access, and maximum ratio combining with
NOMA at intersections. Following this line of research, we study the
performance of VCs at intersections in the presence of interference.

Following this line of research, we study the performance of vehicular
communications at intersections in the presence of interference. In this
paper, the authors extend their work [25] to cooperative transmissions using
NOMA considering mmWave networks. Our analysis includes the effects of
blockage from the building in intersections, and Nakagami-$m$ fading channels
between the transmitting nodes with difference values of $m$ for LOS and NLOS
are considered. Unlike other works that uses approximations, closed form
expressions are obtained for Nakagami-$m$ fading channel.

###  -C Contributions

The contributions of this paper are as follows:

  * •

We study the impact and the improvement of using cooperative NOMA on a mmWave
vehicular network at intersection roads. Closed form expressions of the outage
probability are obtained.

  * •

Our analysis includes the effects of blockage from the building in
intersections, and Nakagami-$m$ fading channels with difference values of $m$
for LOS and NLOS are considered.

  * •

We evaluate the performance of NOMA for both intersection, and show that the
outage probability increases when the vehicles move toward the intersections.
We also show the effect of LOS and NLOS on the performance at the
intersection.

  * •

We compare all the results obtained with cooperative OMA, and show that
cooperative NOMA is superior in terms of outage probability than OMA.

##  I System Model

Figure 1: Cooperative NOMA system model for vehicular communications involving
one relay two receiving node. The receiving nodes can be vehicles or as part
of the communication infrastructure. For instance, $S$ and $D_{1}$ are
vehicles, and $R$ and $D_{2}$ are infrastructures.

###  I-A Scenario Model

In this paper, we consider a mm-Wave vehicular network using a cooperative
NOMA transmission between a source, denoted $S$, and two destinations denoted
$D_{1}$ and $D_{2}$ with the help of a relay denoted $R$. The set
$\\{S,R,D_{1},D_{2}\\}$ denotes the nodes and their locations as depicted in
Fig.1.

We consider, an intersection scenario involving two perpendicular roads, an
horizontal road denoted by $X$, and a vertical road denoted by $Y$. In this
paper, we consider both V2V and V2I communications111The Doppler shift and
time-varying effect of V2V and V2I channels is beyond the scope of this
paper., hence, any node of the set $\\{{S,R,D_{1},D_{2}}\\}$ can be on the
road or outside the roads. We denote by $M$ the receiving node, and by $m$ the
distance between the node $M$ and the intersection, where
$M\in\\{R,D_{1},D_{2}\\}$ and $m\in\\{r,d_{1},d_{2}\\}$, as shown in Fig.1.
The angle $\theta_{M}$ is the angle between the node ${M}$ and the X road (see
Fig.1). Note that the intersection is the point where the $X$ road and the $Y$
road intersect. The set $\\{{S,R,D_{1},D_{2}}\\}$ is subject to interference
that are originated from vehicles located on the roads.

The set of interfering vehicles located on the $X$ road that are in a LOS with
$\\{{S,R,D_{1},D_{2}}\\}$, denoted by $\Phi^{\textrm{LOS}}_{X}$ (resp. on axis
$Y$, denoted by $\Phi^{\textrm{LOS}}_{Y}$) are modeled as a One-Dimensional
Homogeneous Poisson Point Process (1D-HPPP), that is,
$\Phi^{\textrm{LOS}}_{X}\sim\textrm{1D-HPPP}(\lambda^{\textrm{LOS}}_{X},x)$
(resp.$\Phi^{\textrm{LOS}}_{Y}$
$\sim\textrm{1D-HPPP}(\lambda^{\textrm{LOS}}_{Y},y)$, where $x$ and
$\lambda^{\textrm{LOS}}_{X}$ (resp. $y$ and $\lambda^{\textrm{LOS}}_{Y}$) are
the position of the LOS interferer vehicles and their intensity on the $X$
road (resp. $Y$ road).

Similarly, the set of interfering vehicles located on the $X$ road that are in
a NLOS with $\\{{S,R,D_{1},D_{2}}\\}$, denoted by $\Phi^{\textrm{NLOS}}_{X}$
(resp. on axis $Y$, denoted by $\Phi^{\textrm{NLOS}}_{Y}$) are modeled as a
One-Dimensional Homogeneous Poisson Point Process (1D-HPPP), that is,
$\Phi^{\textrm{NLOS}}_{X}\sim\textrm{1D-HPPP}(\lambda^{\textrm{NLOS}}_{X},x)$
(resp.$\Phi^{\textrm{NLOS}}_{Y}$
$\sim\textrm{1D-HPPP}(\lambda^{\textrm{NLOS}}_{Y},y)$, where $x$ and
$\lambda^{\textrm{NLOS}}_{X}$ (resp. $y$ and $\lambda^{\textrm{NLOS}}_{Y}$)
are the position of the NLOS interferer vehicles and their intensity on the
$X$ road (resp. $Y$ road). The notation $x$ and $y$ denotes both the
interferer vehicles and their locations.

###  I-B Blockage Model

At the intersection, the mmWave signals cannot penetrate the buildings and
other obstacles, which causes the link to be in LOS, or in NLOS. The event of
a link between a node $a$ and $b$ is in a LOS and NLOS, are respectively
defined as $\textrm{LOS}_{ab}$, and $\textrm{NLOS}_{ab}$. The LOS probability
function $\mathbb{P}(\textrm{LOS}_{ab})$ is used, where the link between $a$
and $b$ has a LOS probability $\mathbb{P}(\textrm{LOS}_{ab})=\exp(-\beta
r_{ab})$ and NLOS probability
$\mathbb{P}(\textrm{NLOS}_{ab})=1-\mathbb{P}(\textrm{LOS}_{ab})$, where the
constant rate $\beta$ depends on the building size, shape and density [26].

###  I-C Transmission and Decoding Model

The transmission is subject to a path loss, denoted by $r_{ab}^{-\alpha}$
between the nodes $a$ and $b$, where $r_{ab}=\|a-b\|$, and $\alpha$ is the
path loss exponent. The path exponent
$\alpha\in\\{\alpha_{\textrm{LOS}},\alpha_{\textrm{NLOS}}\\}$, where
$\alpha=\alpha_{\textrm{LOS}}$, when the transmission is in LOS, whereas
$\alpha=\alpha_{\textrm{NLOS}}$, when transmission is in NLOS.

We consider slotted ALOHA protocol with parameter $p$, i.e., every node
accesses the medium with a probability $p$.

We use a Decode and Forward (DF) decoding strategy, i.e., $R$ decodes the
message, re-encodes it, then forwards it to $D_{1}$ and $D_{2}$. We also use a
half-duplex transmission in which a transmission occurs during two phases.
Each phase lasts one time slot. During the first phase, $S$ broadcasts the
message to $R$ ($S\rightarrow R$). During the second phase, $R$ broadcasts the
message to $D_{1}$ and $D_{2}$ ($R\rightarrow D_{1}$ and $R\rightarrow
D_{2}$).

###  I-D NOMA Model

We consider, in this paper, that the receiving nodes, $D_{1}$ and $D_{2}$, are
ordered according to their quality of service (QoS) priorities [9, 27]. We
consider the case when node $D_{1}$ needs a low data rate but has to be served
immediately, whereas node $D_{2}$ requires a higher data rate but can be
served later. For instance, $D_{1}$ can be a vehicle that needs to receive
safety data information about an accident in its surrounding, whereas $D_{2}$
can be a user that accesses the internet connection.

###  I-E Directional Beamforming Model

We model the directivity similar to in [28], where the directional gain,
denoted $G(\omega)$, within the half power beamwidth ($\phi/2$) is $G_{max}$
and is $G_{min}$ in all other directions. The gain is then expressed as

$G(\omega)=\left\\{\begin{array}[]{ll}G_{max},&\textrm{if}\>\>|\omega|\leq\frac{\phi}{2};\\\
G_{min},&\textrm{otherwise}.\end{array}\right.$ (1)

In this paper, we consider a perfect beam alignment between the nodes, hence
$G_{eq}=G_{max}^{2}$. The impact of beam misalignment is beyond the scope of
this paper.

###  I-F Channel and Interference Model

We consider an interference limited scenario, that is, the power of noise is
set to zero ($\sigma^{2}=0$). Without loss of generality, we assume that all
nodes transmit with a unit power. The signal transmitted by $S$, denoted
$\chi_{S}$ is a mixture of the message intended to $D_{1}$ and $D_{2}$. This
can be expressed as

$\chi_{S}=\sqrt{a_{1}}\chi_{D1}+\sqrt{a_{2}}\chi_{D2},$

where $a_{i}$ is the power coefficients allocated to $D_{i}$, and $\chi_{Di}$
is the message intended to $D_{i}$, where $i\in\\{1,2\\}$. Since $D_{1}$ has
higher power than $D_{2}$, that is $a_{1}\geq a_{2}$, then $D_{1}$ comes first
in the decoding order. Note that, $a_{1}+a_{2}=1$.
The signal received at $R$ during the first time slot is expressed as

$\displaystyle\mathcal{Y}_{R}=$ $\displaystyle
h_{SR}\sqrt{r_{SR}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{S}\mathds{1}(\textrm{LOS}_{SR})+h_{SR}\sqrt{r_{SR}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{S}\mathds{1}(\textrm{NLOS}_{SR})$
$\displaystyle+\sum_{x\in\Phi^{\textrm{LOS}}_{X_{R}}}h_{Rx}\sqrt{r_{Rx}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{x}+\sum_{y\in\Phi^{\textrm{LOS}}_{Y_{R}}}h_{Ry}\sqrt{r_{Ry}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{y}$
$\displaystyle+\sum_{x\in\Phi^{\textrm{NLOS}}_{X_{R}}}h_{Rx}\sqrt{r_{Rx}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{x}+\sum_{y\in\Phi^{\textrm{NLOS}}_{Y_{R}}}h_{Ry}\sqrt{r_{Ry}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{y}.$

The signal received at $D_{i}$ during the second time slot is expressed as

$\displaystyle\mathcal{Y}_{D_{i}}=$ $\displaystyle
h_{RD_{i}}\sqrt{r_{RD_{i}}^{-\alpha}\Upsilon}\>\chi_{R}\mathds{1}(\textrm{LOS}_{RD_{i}})+h_{RD_{i}}\sqrt{r_{RD_{i}}^{-\alpha}\Upsilon}\>\chi_{R}\mathds{1}(\textrm{NLOS}_{RD_{i}})$
$\displaystyle+\sum_{x\in\Phi^{\textrm{LOS}}_{X_{D_{i}}}}h_{Dix}\sqrt{r_{D_{i}x}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{x}+\sum_{y\in\Phi^{\textrm{LOS}}_{Y_{D_{i}}}}h_{Diy}\sqrt{r_{D_{i}y}^{-\alpha_{\textrm{LOS}}}\Upsilon}\>\chi_{y}$
$\displaystyle+\sum_{x\in\Phi^{\textrm{NLOS}}_{X_{D_{i}}}}h_{Dix}\sqrt{r_{D_{i}x}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{x}+\sum_{y\in\Phi^{\textrm{NLOS}}_{Y_{D_{i}}}}h_{Diy}\sqrt{r_{D_{i}y}^{-\alpha_{\textrm{NLOS}}}\Upsilon}\>\chi_{y},$

where $\mathcal{Y}_{M}$ is the signal received by $M$, and $\chi_{R}$ is the
message transmitted by $R$. The messages transmitted by the interfere node $x$
and $y$, are denoted respectively by $\chi_{x}$ and $\chi_{y}$. The term
$\Upsilon=G_{eq}\eta^{2}/(4\pi)^{2}$ models the directional gain, the
reference path loss at one meter, and $\eta$ is the wavelength of the
operating frequency.

The coefficients $h_{SR}$, and $h_{RD_{i}}$ denote the fading of the link
$S-R$, and $R-D_{i}$. The fading coefficients are distributed according to a
Nakagami-$m$ distribution with parameter $m$ [13], that is

$f_{h_{u}}(x)=2\Big{(}\frac{m}{\mu}\Big{)}^{m}\frac{x^{2m-1}}{\Gamma(m)}e^{-\frac{m}{\mu}x^{2}},$
(2)

where $u\in\\{SR,RD_{i}\\}$. The parameter
$m\in\\{m_{\textrm{LOS}},m_{\textrm{NLOS}}\\}$, where $m=m_{\textrm{LOS}}$
when $u$ is in a LOS, whereas $m=m_{\textrm{NLOS}}$, when $u$ is in a NLOS.
The parameter $\mu$ is the average received power.

Hence, the power fading coefficients $|h_{SR}|^{2}$, and $|h_{RD_{i}}|^{2}$
are distributed according to a gamma distribution, that is,

$f_{|h_{u}|^{2}}(x)=\Big{(}\frac{m}{\mu}\Big{)}^{m}\frac{x^{m-1}}{\Gamma(m)}e^{-\frac{m}{\mu}x}.$
(3)

The fading coefficients $h_{Rx}$,$h_{Ry}$,$h_{D_{i}x}$ and $h_{D_{i}y}$ denote
the fading of the link $R-x$, $R-y$, $D_{i}-x$, and $D_{i}-y$. The fading
coefficients are modeled as Rayleigh fading [29]. Thus, the power fading
coefficients $|h_{Rx}|^{2}$, $|h_{Ry}|^{2}$ $|h_{D_{i}x}|^{2}$ and
$|h_{D_{i}y}|^{2}$, are distributed according to an exponential distribution
with unit mean.

The aggregate interference is defined as from the $X$ road at $M$, denoted
$I_{X_{M}}$, is expressed as

$I_{X_{M}}=I^{\textrm{LOS}}_{X_{M}}+I^{\textrm{NLOS}}_{X_{M}}=\sum_{x\in\Phi^{\textrm{LOS}}_{X_{M}}}|h_{Mx}|^{2}r_{Mx}^{-\alpha_{\textrm{LOS}}}\Upsilon+\sum_{y\in\Phi^{\textrm{NLOS}}_{X_{M}}}|h_{Mx}|^{2}r_{Mx}^{-\alpha_{\textrm{NLOS}}}\Upsilon,$
(4)

where $I^{\textrm{LOS}}_{X_{M}}$ denotes the aggregate interference from the
$X$ road that are in a LOS with $M$, and $I^{\textrm{NLOS}}_{X_{M}}$ denotes
the aggregate interference from the $X$ road that are in a NLOS with $M$.
Similarly, $\Phi^{\textrm{LOS}}_{X_{M}}$ and $\Phi^{\textrm{NLOS}}_{X_{M}}$,
denote respectively, the set of the interferers from the $X$ road at $M$ in a
LOS, and in NLOS.

In the same way, the aggregate interference is defined as from the $Y$ road at
$M$, denoted $I_{Y_{M}}$, is expressed as

$I_{Y_{M}}=I^{\textrm{LOS}}_{Y_{M}}+I^{\textrm{NLOS}}_{Y_{M}}=\sum_{y\in\Phi^{\textrm{LOS}}_{Y_{M}}}|h_{My}|^{2}r_{My}^{-\alpha_{\textrm{LOS}}}\Upsilon+\sum_{y\in\Phi^{\textrm{NLOS}}_{Y_{M}}}|h_{My}|^{2}r_{My}^{-\alpha_{\textrm{NLOS}}}\Upsilon,$
(5)

where $I^{\textrm{LOS}}_{Y_{M}}$ denotes the aggregate interference from the
$X$ road that are in a LOS with $M$, and $I^{\textrm{NLOS}}_{Y_{M}}$ denotes
the aggregate interference from the $Y$ road that are in a NLOS with $M$.
Similarly, $\Phi^{\textrm{LOS}}_{Y_{M}}$ and $\Phi^{\textrm{NLOS}}_{Y_{M}}$,
denote respectively, the set of the interferers from the $Y$ road at $M$ in a
LOS, and in NLOS.

##  II Cooperative NOMA Outage Expressions

###  II-A Signal-to-Interference Ratio (SIR) Expressions

We define the outage probability as the probability that the signal-to-
interference ratio (SIR) at the receiver is below a given threshold. According
to successive interference cancellation (SIC) [30], $D_{1}$ will be decoded
first at the receiver since it has the higher power allocation, and $D_{2}$
message will be considered as interference. The SIR at $R$ to decode $D_{1}$,
denoted $\textrm{SIR}^{(\alpha)}_{R_{1}}$, is expressed as

$\textrm{SIR}^{(\alpha)}_{R_{1}}=\frac{|h_{SR}|^{2}r_{SR}^{-\alpha}\Upsilon\,a_{1}}{|h_{SR}|^{2}r_{SR}^{-\alpha}\Upsilon
a_{2}+I_{X_{R}}+I_{Y_{R}}}.$ (6)

Since $D_{2}$ has a lower power allocation, $R$ has to decode $D_{1}$ message,
then decode $D_{2}$ message. The SIR at $R$ to decode $D_{2}$ message, denoted
$\textrm{SIR}^{(\alpha)}_{R_{2}}$, is expressed as 222Perfect SIC is
considered in this work, that is, no fraction of power remains after the SIC
process.

$\textrm{SIR}^{(\alpha)}_{R_{2}}=\frac{|h_{SR}|^{2}r_{SR}^{-\alpha}\Upsilon\,a_{2}}{I_{X_{R}}+I_{Y_{R}}}.$
(7)

The SIR at $D_{1}$ to decode its intended message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{1}}$, is given by

$\textrm{SIR}^{(\alpha)}_{D_{1}}=\frac{|h_{RD1}|^{2}r_{RD1}^{-\alpha}\Upsilon\,a_{1}}{|h_{RD1}|^{2}r_{RD1}^{-\alpha}\Upsilon
a_{2}+I_{X_{D1}}+I_{Y_{D1}}}.$ (8)

In order for $D_{2}$ to decode its intended message, it has to decode $D_{1}$
message. The SIR at $D_{2}$ to decode $D_{1}$ message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{2-1}}$, is expressed as

$\textrm{SIR}^{(\alpha)}_{D_{2-1}}=\frac{|h_{RD2}|^{2}r_{RD2}^{-\alpha}\Upsilon\,a_{1}}{|h_{RD2}|^{2}r_{RD2}^{-\alpha}\Upsilon
a_{2}+I_{X_{D2}}+I_{Y_{D2}}}.$ (9)

The SIR at $D_{2}$ to decode its intended message, denoted
$\textrm{SIR}^{(\alpha)}_{D_{2}}$, is expressed as

$\textrm{SIR}^{(\alpha)}_{D_{2}}=\frac{|h_{RD2}|^{2}r_{RD2}^{-\alpha}\Upsilon\,a_{2}}{I_{X_{D2}}+I_{Y_{D2}}}.$
(10)

###  II-B Outage Event Expressions

The outage event that $R$ does not decode $D_{1}$ message, denoted
$\textit{O}_{R_{1}}$, is given by

$\textit{O}_{R_{1}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\big{\\{}\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{1}}<\Theta_{1})\big{\\}},$
(11)

where $\Theta_{1}=2^{2\mathcal{R}_{1}}-1$, and $\mathcal{R}_{1}$ is the target
data rate of $D_{1}$.
Also, the outage event that $D_{1}$ does not decode its intended message,
denoted $\textit{O}_{D_{1}}$, is given by

$\textit{O}_{D_{1}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\big{\\{}\textrm{Z}_{RD_{1}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{1}}<\Theta_{1})\big{\\}},$
(12)

Then, the overall outage event related to $D_{1}$, denoted $\textit{O}_{(1)}$,
is given by

$\textit{O}_{(1)}\triangleq\left[\textit{O}_{R_{1}}\cup\textit{O}_{D_{1}}\right],$
(13)

The outage event that $R$ does not decode $D_{2}$ message, denoted
$\textit{O}_{R_{2}}$, is given by

$\textit{O}_{R_{2}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\bigcup_{i=1}^{2}\big{\\{}\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{i}}<\Theta_{i})\big{\\}},$
(14)

where $\Theta_{2}=2^{2\mathcal{R}_{2}}-1$ ($i=2)$, and $\mathcal{R}_{2}$ is
the target data rate of $D_{2}$. Also, the outage event that $D_{2}$ does not
decode its intended message, denoted $\textit{O}_{D_{2}}$, is given by

$\textit{O}_{D_{2}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\bigcup_{i=1}^{2}\big{\\{}\textrm{Z}_{RD_{2}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-i}}<\Theta_{i})\big{\\}},$
(15)

Finally, the overall outage event related to $D_{2}$, denoted
$\textit{O}_{(2)}$, is given by

$\textit{O}_{(2)}\triangleq\left[\textit{O}_{R_{2}}\cup\textit{O}_{D_{2}}\right].$
(16)

###  II-C Outage Probability Expressions

In the following, we will express the outage probability related to
$\textit{O}_{(1)}$ and $\textit{O}_{(2)}$. The probability
$\mathbb{P}(\textit{O}_{(1)})$ is given, when
$\Theta_{1}<\frac{a_{1}}{a_{2}}$, by (17)

$\mathbb{P}(\textit{O}_{(1)})=1-\Bigg{\\{}\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\Lambda\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}\times\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{RD_{1}})\Lambda\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{RD_{1}}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}\Bigg{\\}},$
(17)

where $\Psi_{1}=\Theta_{1}/(a_{1}-\Theta_{1}a_{2})$. The expression of
$\Lambda\Big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\Big{)}$ is given
by

$\Lambda\Big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\Big{)}=\\\
\prod_{\textrm{K}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\sum_{k=0}^{m-1}\frac{1}{k!}\>\big{(}-\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{b}}}\big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha_{L}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{b}}}\big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m\>\Psi}{\mu\>r_{ab}^{-\alpha}\Upsilon}\big{)}}.$
(18)

The probability $\mathbb{P}(\textit{O}_{(2)})$ is given, when
$\Theta_{1}<\frac{a_{1}}{a_{2}}$, by (19)

$\mathbb{P}(\textit{O}_{(2)})=1-\Bigg{\\{}\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\Lambda\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}\times\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{RD_{2}})\Lambda\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{RD_{2}}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}\Bigg{\\}},$
(19)

where $\Psi_{\mathrm{max}}=\mathrm{max}(\Psi_{1},\Psi_{2})$, and
$\Psi_{2}=\Theta_{2}/a_{2}$.
Proof: See Appendix A. $\blacksquare$

Figure 2: Outage probability as function of $\lambda$ considering cooperative
NOMA, for LOS transmission, NLOS, and LOS/NLOS (the equation (17) and (19)).

##  III Laplace Transform Expressions

We present the Laplace transform expressions of the interference from the X
road at the receiving node denoted by $M$, denoted
$\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}$, and from the Y road at the receiving
node denoted by $M$, denoted $\mathcal{L}_{I^{\textrm{K}}_{Y_{M}}}$. We only
present the case when $\alpha_{\textrm{K}}=2$ due to the lack of space. The
Laplace transform expressions of the interference at the node $M$ for an
intersection scenario, when $\alpha_{\textrm{K}}=2$ are given by

$\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}(s)=\exp\Bigg{(}\dfrac{-\emph{p}\lambda^{\textrm{K}}_{X}s\pi}{\sqrt{\big{[}m\sin(\theta_{{M}})\big{]}^{2}+s}}\Bigg{)},$
(20)

and

$\mathcal{L}_{I^{\textrm{K}}_{Y_{M}}}(s)=\exp\Bigg{(}\dfrac{-\emph{p}\lambda^{\textrm{K}}_{Y}s\pi}{\sqrt{\big{[}m\cos(\theta_{{M}})\big{]}^{2}+s}}\Bigg{)}.$
(21)

Proof: See Appendix B. $\blacksquare$

##  IV Simulations and Discussions

In this section, we evaluate the performance of cooperative NOMA at road
intersections. In order to verify the accuracy of the theoretical results,
Monte Carlo simulations are carried out by averaging over 10,000 realizations
of the PPPs and fading parameters. In all figures, Monte Carlo simulations are
presented by marks, and they match perfectly the theoretical results, which
validates the correctness of our analysis. We set, without loss of generality,
$\lambda^{\textrm{LOS}}_{X}=\lambda^{\textrm{LOS}}_{Y}=\lambda^{\textrm{NLOS}}_{X}=\lambda^{\textrm{NLOS}}_{Y}=\lambda$.
$S=(0,0),R=(50,0),D_{1}=(100,10),D_{2}=(100,-10)$, $\beta=9.5\times 10^{3}$
[26], $\mu=1$. We set $\alpha_{\textrm{LOS}}=2$, $\alpha_{\textrm{NLOS}}=4$,
$m_{\textrm{LOS}}=2$, and $m_{\textrm{NLOS}}=1$. Finally, we set $G_{max}=18$
dBi, $\eta=30$ GHz.

Figure 3: Outage probability as a function of $\|S-D_{1}\|=\|S-D_{2}\|$. The
relay $R$ is always at mid distance between the source and the destination.

Fig. 2 plots the outage probability as function of $\lambda$ considering
cooperative NOMA, for LOS transmission, NLOS, and LOS/NLOS. We can see that
LOS scenario has the highest outage probability. This is because, when the
interference are in direct line of sight with the set $\\{S,R,D_{1},D_{2}\\}$,
the power of aggregate interference increases, hence reducing the SIR and
increasing the outage. on the other hand, the NLOS scenario has the smallest
outage, since the interference are in non line of sight with the transmitting
nodes. The model for this paper include a blockage model that includes both
LOS and NLOS. Therefore, we wan see that the performance are between the LOS
scenario and NLOS scenario, which are two extreme cases.

Fig.3 plots the outage probability as a function of the distance between the
source and the destinations. Without loss of generality, we set $R$ at mid
distance between $S$ and the two destinations $D_{1}$ and $D_{2}$. We can see
that cooperative NOMA outperforms cooperative OMA when $a_{1}=0.8$ for both
$D_{1}$ and $D_{2}$. However, this is not the case for $a_{1}=0.6$, when NOMA
outperforms OMA only for $D_{2}$. This is because when $a_{1}$ decreases, less
power is allocated to $D_{1}$, hence it increases the outage probability. We
can also see from Fig.3 that the outage probability increases until 200 m for
$D_{1}$ (100 m for $D_{2}$). This because, as the distance between the
transmitting and the receiving nodes increases, the LOS probability decreases,
and the NLOS probability increases, hence decreasing the outage probability.

Figure 4: Outage probability as a function of $\lambda$ considering
cooperative NOMA and cooperative OMA. Figure 5: Outage probability as a
function of the distance form the intersection considering cooperative NOMA
and cooperative OMA, for LOS scenarion and NLOS scenario.

Fig.4 plots the outage probability as a function of $\lambda$ considering
cooperative NOMA and cooperative OMA for several values of data rates. We can
see that NOMA outperforms OMA. We can also see that $D_{1}$ has a better
performance than $D_{2}$. This is because $D_{1}$ has a smaller target data
rate, since $D_{1}$ need to be served quickly (e.g., alert message). We can
also see that, as the data rates increases ($R_{1}=1.2$bits/s and
$R_{2}=4$bits/s), the gap of performance between NOMA and OMA increases. This
is because, as the data rates increases, the decoding threshold of OMA
increases dramatically ($\Theta_{\textrm{OMA}}=2^{4\mathcal{R}}-1$). The
increase of the threshold becomes larger for $D_{2}$, since it has a higher
data rate that $D_{1}$.

Fig.5 plots the outage probability of the distance from the intersection
considering cooperative NOMA and cooperative OMA, for LOS scenario and NLOS
scenario. Without loss of generality, we set $R$ at mid distance between $S$
and the two destinations $D_{1}$ and $D_{2}$. We notice from Fig.5 that as
nodes approach the intersection, the outage probability increases. This
because when the nodes are far from the intersection, only the interferes in
the same road segment contribute to the aggregate interference, but as the
node approach the intersection, both road segments contribute to the aggregate
interference. However, we can see that $D_{2}$ has a severe outage in LOS
scenario compared to NLOS, and that the increases of the outage for $D_{2}$ in
LOS, when the nodes move toward the intersection is negligible. This is
because, in a LOS scenario, the interferers from both road segment contributes
the aggregate interference, whether the nodes are close or far away from the
intersection.

##  V Conclusion

In this paper, we studied cooperative NOMA for mmWave vehicular networks at
intersection roads. The analysis was conducted using tools from stochastic
geometry and was verified with Monte Carlo simulations. We derived closed form
outage probability expressions for cooperative NOMA, and compared them with
cooperative OMA. We showed that cooperative NOMA exhibited a significant
improvement compared to cooperative OMA, especially for high data rates.
However, data rates have to respect a given condition, if not, the performance
of cooperative NOMA will decreases drastically. We also showed that as the
nodes reach the intersection, the outage probability increased. Counter-
intuitively, we showed that NLOS scenario has a better performance than LOS
scenario.

## Appendix A

To calculate $\mathbb{P}(\textit{O}_{(1)})$, we express it as a function of a
success probability $\mathbb{P}(\textit{O}_{(1)}^{C})$, where
$\mathbb{P}(\textit{O}_{D_{1}}^{C})$ is expressed as

$\mathbb{P}(\textit{O}_{(1)})=1-\mathbb{P}(\textit{O}^{C}_{(1)}),$ (22)

The probability $\mathbb{P}(\textit{O}^{C}_{(1)})$ is expressed as

$\mathbb{P}(\textit{O}^{C}_{(1)})=1-\mathbb{P}(\textit{O}^{C}_{R_{1}}\cap\textit{O}^{C}_{D_{1}})\\\
=\mathbb{P}(\textit{O}^{C}_{R_{1}})\mathbb{P}(\textit{O}^{C}_{D_{1}}),$ (23)

where

$\textit{O}^{C}_{R_{1}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\big{\\{}\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{1}}\geq\Theta_{1})\big{\\}}$
(24)
$\textit{O}^{C}_{D_{1}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\big{\\{}\textrm{Z}_{RD_{1}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{1}}\geq\Theta_{1})\big{\\}}.$
(25)

We calculate The probability $\mathbb{P}(\textit{O}^{C}_{R_{1}})$ as

$\displaystyle\mathbb{P}(\textit{O}^{C}_{R_{1}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{1}}\geq\Theta_{1})\Bigg{\\}}\Bigg{]}$
(26) $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{1}}\geq\Theta_{1}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\frac{|h_{SR}|^{2}r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{1}}{|h_{SR}|^{2}r_{SR}^{-\alpha_{{\textrm{Z}}}}\Upsilon
a_{2}+I_{X_{R}}+I_{Y_{R}}}\geq\Theta_{1}\Bigg{\\}}\Bigg{]}$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SR}|^{2}r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon(a_{1}-\Theta_{1}a_{2})\geq\Theta_{1}\big{[}I_{X_{R}}+I_{Y_{R}}\big{]}\Bigg{\\}}\Bigg{]}.$

We can notice from (26) that, when $\Theta_{1}\geq a_{1}/a_{2}$, the success
probability $\mathbb{P}(\textit{O}^{C}_{R_{1}})$ is always zero, that is,
$\mathbb{P}(\textit{O}_{R_{1}})=1$. Then, when $\Theta_{1}<a_{1}/a_{2}$, and
after setting $\Psi_{1}=\Theta_{1}/(a_{1}-\Theta_{1}a_{2})$, then

$\mathbb{P}(\textit{O}^{C}_{R_{1}})=\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SR}|^{2}\geq\frac{\Psi_{1}}{r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{R}}+I_{Y_{R}}\big{]}\Bigg{\\}}\Bigg{]}.$

Since $|h_{SR}|^{2}$ follows a gamma distribution, its complementary
cumulative distribution function (CCDF) is given by

$\bar{F}_{|h_{SR}|^{2}}(X)=\mathbb{P}(|h_{SR}|^{2}>X)=\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})},$
(27)

hence

$\displaystyle\mathbb{P}(\textit{O}^{C}_{R_{1}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{LOS}}_{X_{R}}+I^{\textrm{LOS}}_{Y_{R}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}$
(28) $\displaystyle\times$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{NLOS}}_{X_{R}}+I^{\textrm{NLOS}}_{Y_{R}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\prod_{{\textrm{K}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\frac{\Gamma\Big{(}m_{\textrm{Z}},\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{{\textrm{Z}}}}\Upsilon}(I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}})\Big{)}}{\Gamma(m_{\textrm{Z}})}\Bigg{]}$

The exponential sum function when $m_{\textrm{Z}}$ is an integer is defined as

$e_{(m_{\textrm{Z}})}=\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{(\frac{m_{\textrm{Z}}}{\mu}X)^{k}}{k!}=e^{X}\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})},$
(29)

then

$\frac{\Gamma(m_{\textrm{Z}},\frac{m_{\textrm{Z}}}{\mu}X)}{\Gamma(m_{\textrm{Z}})}=e^{-\frac{m_{\textrm{Z}}}{\mu}X}\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}{\big{(}\frac{m_{\textrm{Z}}\>X}{\mu}\big{)}}^{k}.$
(30)

We denote the expectation in equation (28) by $\mathcal{E}(I_{X},I_{Y})$, then
$\mathcal{E}(I_{X},I_{Y})$ equals

$\displaystyle\mathcal{E}(I_{X},I_{Y})$ $\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}(I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}})\Big{)}\times\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}(I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}})\Big{)}^{k}\Bigg{]}$
(31) $\displaystyle=$
$\displaystyle\sum_{k=0}^{m-1}\frac{1}{k!}\>\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}^{k}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{(}I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}}\big{)}\Big{)}\big{(}I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}}\big{)}^{k}\Bigg{]}.$

Applying the binomial theorem in (31), we get

$\displaystyle\mathcal{E}(I_{X},I_{Y})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\Big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\Big{)}^{k}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}}\big{]}\Big{)}\sum_{n=0}^{k}\binom{k}{n}{(I^{\textrm{K}}_{X_{R}})}^{k-n}\>{(I^{\textrm{K}}_{Y_{R}})}^{n}\Bigg{]}$
(32) $\displaystyle=$
$\displaystyle\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\Omega^{k}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\exp\Big{(}-\Omega\big{[}I^{\textrm{K}}_{X_{R}}+I^{\textrm{K}}_{Y_{R}}\big{]}\Big{)}\sum_{n=0}^{k}\binom{k}{n}{(I^{\textrm{K}}_{X_{R}})}^{k-n}\>({I^{\textrm{K}}_{Y_{R}}})^{n}\Bigg{]},$

where
$\Omega=\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}$.
To calculate the expectation in (32) we process as follows

$\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{X_{R}}}\>e^{-\Omega\>I^{\textrm{K}}_{Y_{R}}}\sum_{n=0}^{k}\binom{k}{n}{(I^{\textrm{K}}_{X_{R}})}^{k-n}\>{(I^{\textrm{K}}_{Y_{R}})}^{n}\Bigg{]}=\sum_{n=0}^{k}\binom{k}{n}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{X_{R}}}\>e^{-\Omega\>I^{\textrm{K}}_{Y_{R}}}{(I^{\textrm{K}}_{X_{R}})}^{k-n}\>{(I^{\textrm{K}}_{Y_{R}})}^{n}\Bigg{]}$
(33) $\displaystyle=$
$\displaystyle\sum_{n=0}^{k}\binom{k}{n}\mathbb{E}_{I_{X}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{X_{R}}}{(I^{\textrm{K}}_{X_{R}})}^{k-n}\Bigg{]}\mathbb{E}_{I^{\textrm{K}}_{Y_{R}}}\Bigg{[}e^{-\Omega\>I^{\textrm{K}}_{Y_{R}}}\>{(I^{\textrm{K}}_{Y_{R}})}^{n}\Bigg{]}$
(34) $\displaystyle\overset{(a)}{=}$
$\displaystyle\sum_{n=0}^{k}\binom{k}{n}(-1)^{k-n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{R}}}(\Omega)}{\textrm{d}^{k-n}\Omega}(-1)^{n}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{R}}}(\Omega)}{\textrm{d}^{n}\Omega}$
$\displaystyle=$
$\displaystyle(-1)^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{R}}}(\Omega)}{\textrm{d}^{k-n}\Omega}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{R}}}(\Omega)}{\textrm{d}^{n}\Omega}.$

where (a) stems form the following property

$\mathbb{E}_{I}\Big{[}e^{-\Omega
I}{I}^{N}\Big{]}=(-1)^{N}\frac{\textrm{d}^{N}\mathbb{E}_{I}\Big{[}e^{-\Omega\>I}{I}^{N}\Big{]}}{\textrm{d}^{N}\Omega}=(-1)^{N}\frac{\textrm{d}^{N}\mathcal{L}_{I}(\Omega)}{\textrm{d}^{N}\Omega},$
(35)

Finally, the expectation becomes

$\sum_{k=0}^{m_{\textrm{Z}}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}.$
(36)

Then plugging (36) in (28) yields

$\mathbb{P}(\textit{O}_{R_{1}}^{C})=\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\times\\\
\prod_{\textrm{K}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\sum_{k=0}^{m_{L}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{1}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\\\
$ (37)

The expression of
$\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X}}(s)/\textrm{d}^{k-n}(s)$ and
$\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y}}(s)/\textrm{d}^{n}(s)$ are
given by (54) and (55). The probability $\mathbb{P}(\textit{O}^{C}_{D_{1}})$
can be calculated following the same steps above.

In the same way we express $\mathbb{P}(\textit{O}_{(2)})$ as a function of a
success probability $\mathbb{P}(\textit{O}_{(2)}^{C})$, where
$\mathbb{P}(\textit{O}_{(2)}^{C})$ is given by

$\mathbb{P}(\textit{O}_{(2)})=1-\mathbb{P}(\textit{O}^{C}_{(2)}).$ (38)

The probability $\mathbb{P}(\textit{O}^{C}_{(2)})$ is expressed as

$\mathbb{P}(\textit{O}^{C}_{(2)})=1-\mathbb{P}(\textit{O}^{C}_{R_{2}}\cap\textit{O}^{C}_{D_{2}})\\\
=\mathbb{P}(\textit{O}^{C}_{R_{2}})\mathbb{P}(\textit{O}^{C}_{D_{2}}),$ (39)

where

$\textit{O}^{C}_{R_{2}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\bigcap_{i=1}^{2}\big{\\{}\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{i}}\geq\Theta_{i})\big{\\}}$
(40)
$\textit{O}^{C}_{D_{2}}\triangleq\bigcup_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\bigcap_{i=1}^{2}\big{\\{}\textrm{Z}_{RD_{2}}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{D_{2-i}}<\Theta_{i})\big{\\}}.$
(41)

To calculate $\mathbb{P}(\textit{O}^{C}_{R_{2}})$ we proceed as follows

$\displaystyle\mathbb{P}(\textit{O}^{C}_{R_{2}})$ $\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\bigcap_{i=1}^{2}\big{\\{}\textrm{Z}_{SR}\cap(\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{i}}\geq\Theta_{i})\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\bigcap_{i=1}^{2}\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{i}}\geq\Theta_{i}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\sum_{{\textrm{Z}}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\>\>\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\
\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{1}}\geq\Theta_{1}\cap\textrm{SIR}^{(\alpha_{\textrm{Z}})}_{R_{2}}\geq\Theta_{2}\Bigg{\\}}\Bigg{]}.$

Following the same steps as for $\mathbb{P}(\textit{O}_{R_{1}}^{C})$, we get

$\mathbb{P}(\textit{O}_{R_{2}}^{C})=\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}\frac{|h_{SR}|^{2}r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{1}}{|h_{SR}|^{2}r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{2}+I_{X_{R}}+I_{Y_{R}}}\geq\Theta_{1},\frac{|h_{SR}|^{2}r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon
a_{2}}{I_{X_{R}}+I_{Y_{R}}}\geq\Theta_{2}\Bigg{\\}}\Bigg{]}.$

When $\Theta_{1}>a_{1}/a_{2}$, then $\mathbb{P}(\textit{O}_{R_{2}})=1$,
otherwise we continue the derivation We set $\Psi_{2}=\Theta_{2}/a_{2}$, then

$\displaystyle\mathbb{P}(\textit{O}_{R_{2}}^{C})$ $\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SR}|^{2}\geq\frac{\Psi_{1}}{r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{R}}+I_{Y_{R}}\big{]},|h_{SR}|^{2}\geq\frac{\Psi_{2}}{r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{R}}+I_{Y_{R}}\big{]}\Bigg{\\}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{I_{X},I_{Y}}\Bigg{[}\mathbb{P}\Bigg{\\{}|h_{SR}|^{2}\geq\frac{\mathrm{max}(\Psi_{1},\Psi_{2})}{r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{[}I_{X_{R}}+I_{Y_{R}}\big{]}\Bigg{\\}}\Bigg{]}.$

Following the same steps above, $\mathbb{P}(\textit{O}_{R_{2}}^{C})$ equals

$\mathbb{P}(\textit{O}_{R_{2}}^{C})=\sum_{\textrm{Z}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\mathbb{P}(\textrm{Z}_{SR})\times\\\
\prod_{\textrm{K}\in\\{\textrm{LOS},\textrm{NLOS}\\}}\sum_{k=0}^{m_{L}-1}\frac{1}{k!}\>\big{(}-\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}^{k}\sum_{n=0}^{k}\binom{k}{n}\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{k-n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{R}}}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}{\textrm{d}^{n}\big{(}\dfrac{m_{\textrm{Z}}\>\Psi_{\mathrm{max}}}{\mu\>r_{SR}^{-\alpha_{\textrm{Z}}}\Upsilon}\big{)}}\\\
$ (43)

where $\Psi_{\mathrm{max}}=\mathrm{max}(\Psi_{1},\Psi_{2})$. The probability
$\mathbb{P}(\textit{O}_{D_{2}}^{C})$ can be calculated following the same
steps above.

## Appendix B

The Laplace transform of the interference originating from the X road at $M$
is expressed as

$\displaystyle\mathcal{L}_{{I^{\textrm{K}}_{X_{M}}}}(s)$ $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}{\exp\big{(}-sI^{\textrm{K}}_{X_{M}}\big{)}}\Bigg{]}$
(44) $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}{\exp\Bigg{(}-\sum_{x\in\Phi^{\textrm{K}}_{X_{M}}}s|h_{{M}x}|^{2}r_{{M}x}^{-\alpha_{\textrm{K}}}\Bigg{)}}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{M}}}\exp\Bigg{(}-s|h_{{M}x}|^{2}r_{{M}x}^{-\alpha_{\textrm{K}}}\Bigg{)}\Bigg{]}$
$\displaystyle\overset{(a)}{=}$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{M}}}\mathbb{E}_{|h_{{M}x}|^{2},p}\Bigg{\\{}\exp\Bigg{(}-s|h_{{M}x}|^{2}r_{Mx}^{-\alpha_{\textrm{K}}}\Bigg{)}\Bigg{\\}}\Bigg{]}$
$\displaystyle\overset{(b)}{=}$
$\displaystyle\mathbb{E}\Bigg{[}\prod_{x\in\Phi^{\textrm{K}}_{X_{M}}}\dfrac{p}{1+sr_{{M}x}^{-\alpha_{\textrm{K}}}}+1-p\Bigg{]}$
$\displaystyle\overset{(c)}{=}$
$\displaystyle\exp\Bigg{(}-\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\Bigg{[}1-\bigg{(}\dfrac{p}{1+sr_{{M}x}^{-\alpha_{\textrm{K}}}}+1-p\bigg{)}\Bigg{]}\textrm{d}x\Bigg{)}$
$\displaystyle=$
$\displaystyle\exp\Bigg{(}-p\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\dfrac{1}{1+1/sr_{{M}x}^{-\alpha_{\textrm{K}}}}\textrm{d}x\Bigg{)}$
$\displaystyle=$
$\displaystyle\exp\Bigg{(}-p\lambda^{\textrm{K}}_{X}\displaystyle\int_{\mathbb{R}}\dfrac{1}{1+r_{{M}x}^{\alpha_{\textrm{K}}}/s}\textrm{d}x\Bigg{)},$
(45)

where (a) follows from the independence of the fading coefficients; (b)
follows from performing the expectation over $|h_{{M}x}|^{2}$ which follows an
exponential distribution with unit mean, and performing the expectation over
the set of interferes; (c) follows from the probability generating functional
(PGFL) of a PPP. The expression of $\mathcal{L}_{{I^{\textrm{K}}_{Y_{M}}}}(s)$
can be acquired by following the same steps. The Laplace transform of the
interference originating from the X road at the received node denoted $M$, is
expressed as

$\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}(s)=\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{X}\int_{\mathbb{R}}\dfrac{1}{1+\|\textit{x}-{M}\|^{\alpha_{\textrm{K}}}/s}\textrm{d}x\Bigg{)},$
(46)

where

$\|\textit{x}-{M}\|=\sqrt{\big{[}m\sin(\theta_{{M}})\big{]}^{2}+\big{[}x-m\cos(\theta_{M})\big{]}^{2}}.$
(47)

The Laplace transform of the interference originating from the Y road at $M$
is given by

$\mathcal{L}_{I^{\textrm{K}}_{Y_{M}}}(s)=\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{Y}\int_{\mathbb{R}}\dfrac{1}{1+\|\textit{y}-{M}\|^{\alpha_{\textrm{K}}}/s}\textrm{d}y\Bigg{)},$
(48)

where

$\|\textit{y}-{M}\|=\sqrt{\big{[}m\cos(\theta_{{M}})\big{]}^{2}+\big{[}y-m\sin(\theta_{M})\big{]}^{2}},$
(49)

where $\theta_{M}$ is the angle between the node ${M}$ and the X road.

In order to calculate the Laplace transform of interference originated from
the X road at the node $M$, we have to calculate the integral in (46). We
calculate the integral in (46) for $\alpha_{\textrm{K}}=2$. Let us take
$m_{x}=m\cos(\theta_{M})$, and $m_{y}=m\sin(\theta_{M}$), then (46) becomes

$\displaystyle\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}(s)$ $\displaystyle=$
$\displaystyle\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{X}\int_{\mathbb{R}}\dfrac{1}{1+m_{y}^{2}+(x-m_{x})^{2}/s}\textrm{d}x\Bigg{)},$
(50) $\displaystyle=$
$\displaystyle\exp\Bigg{(}-\emph{p}\lambda^{\textrm{K}}_{X}s\int_{\mathbb{R}}\dfrac{1}{s+m_{y}^{2}+(x-m_{x})^{2}}\textrm{d}x\Bigg{)},$

and the integral inside the exponential in (50) equals

$\int_{\mathbb{R}}\dfrac{1}{s+m_{y}^{2}+(x-m_{x})^{2}}\textrm{d}x=\dfrac{\pi}{\sqrt{m_{y}^{2}+s}}.$
(51)

Then, plugging (51) into (50), and substituting $m_{y}$ by
$m\sin(\theta_{{M}})$ we obtain

$\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}(s)=\exp\Bigg{(}-\dfrac{\emph{p}\lambda^{\textrm{K}}_{X}s\,\pi}{\sqrt{m^{2}\sin(\theta_{{M}})^{2}+s}}\Bigg{)}.$
(52)

Following the same steps above, and without details for the derivation with
respect to $s$, we obtain

$\mathcal{L}_{I^{\textrm{K}}_{Y_{M}}}(s)=\exp\Bigg{(}-\dfrac{\emph{p}\lambda^{\textrm{K}}_{Y}s\,\pi}{\sqrt{m^{2}\cos(\theta_{{M}})^{2}+s}}\Bigg{)}.$
(53)

Then, when compute the derivative of (52) and (53), we obtain

$\frac{\textrm{d}^{k-n}\mathcal{L}_{I^{\textrm{K}}_{X_{M}}}\big{(}s\big{)}}{\textrm{d}^{k-n}s}=\Bigg{[}-\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi}{\sqrt{t^{2}\sin(\theta_{{M}})^{2}+s}}+\frac{1}{2}\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi
s}{(m^{2}\sin(\theta_{{M}})^{2}+s)^{3/2}}\Bigg{]}^{k-n}\\\
\times\exp\Bigg{(}-\frac{\emph{p}\lambda^{\textrm{K}}_{X}\pi
s}{\sqrt{m^{2}\sin(\theta_{{M}})^{2}+s}}\Bigg{)}.$ (54)
$\frac{\textrm{d}^{n}\mathcal{L}_{I^{\textrm{K}}_{Y_{M}}}\big{(}s\big{)}}{\textrm{d}^{n}s}=\Bigg{[}-\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi}{\sqrt{m^{2}\cos(\theta_{{M}})^{2}+s}}+\frac{1}{2}\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi
s}{(m^{2}\cos(\theta_{{M}})^{2}+s)^{3/2}}\Bigg{]}^{n}\\\
\times\exp\Bigg{(}-\frac{\emph{p}\lambda^{\textrm{K}}_{Y}\pi
s}{\sqrt{m^{2}\cos(\theta_{{M}})^{2}+s}}\Bigg{)}.$ (55)

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