#  A Low-Complexity Location-Based Hybrid Multiple Access and Relay Selection
in V2X Multicast Communications

Chun-Yi Wei , Senior Member, IEEE, Yung Kai Wang  C.-Y. Wei is with the Dept.
of Communication Engineering, National Taipei University, 23741, Taiwan
(R.O.C.) (email: cywei@gm.ntpu.edu.tw).

###### Abstract

This study investigated relay-assisted mode 1 sidelink (SL) multicast
transmission, which encounters interference from mode 2 SL transmission, for
use in low-latency vehicle-to-everything communications. To accommodate mode
1–mode 2 SL traffic, we use the hybrid multiple access (MA) approach, which
combines orthogonal MA (OMA) and nonorthogonal MA (NOMA) schemes. We introduce
a low-complexity location-based hybrid MA algorithm and its associated relay
selection that can be used when SL channel state information is unavailable.

Index Terms—5G, multicast, multiple access, relay, V2X

##  I Introduction

Regarding 5th-generation (5G) vehicle-to-everything (V2X) communications, the
forthcoming 3GPP Rel-17 standards indicate that support of the sidelink (SL)
relay and of multicast transmission in the physical layer are essential [1,
2]. According to 3GPP specifications [2, 3], the gNB [i.e., the 5G base
station (BS)], may coordinate the SL communications of mode 1 user equipment
(UE), also known as the centralized mode, whereas the mode 2 UE is self-
originated and is known as the distributed mode. In this work, we consider a
case wherein mode 1 and mode 2 SL communication coexist [4]. In some
situations, such as public safety related use cases, the gNB requires the
delivery of traffic-related messages to a group of vehicle UEs (v-UEs) within
a designated proximity and in a specified time period. According to the 3GPP
long-term evolution (LTE) specifications [2], the gNB may designate a relay
v-UE to convey the messages using SL multicast transmission to specified group
members. Furthermore, a properly selected relay can not only offload the heavy
traffic of the Uu link but also achieve more efficient signal transmission in
V2X applications, especially when the traditional Uu-link and SL-link of V2X
can be allocated in different frequency bands. [5]. Unfortunately, the gNB is
usually far from the site of ongoing SL communications. It has minimal
knowledge regarding the SL channel state information (CSI) of all links among
the anticipated v-UEs (i.e., multicast members). The SL-CSI feedback in SL
multicast transmission is impractical, especially in high-mobility V2X
scenarios [2]. Furthermore, the complexity of relay selection in group
communications can exhibit nondeterministic polynomial (NP)-hard
characteristics [6].

Hence, this work is motivated by the premise that the gNB relies on a low-
complexity relay selection algorithm for low-latency V2X applications when no
SL CSI feedback is available. Furthermore, the relay selection of this work
aims to accommodate the mode 1 and mode 2 SL traffic, which to our knowledge,
has no precedent in the literature. That brings the novelty of this work. More
specifically, in a specified SL channel for the mode 1 SL multicast, the
ongoing mode 2 SL transmission may contain a source of interference. The
conventional approach to solving this problem is to employ the orthogonal
multiple access (OMA) to accommodates mode 1 and mode 2 SL transmission in
different time frames. On the other hand, the nonorthogonal multiple access
(NOMA) can be applied to cooperative network transmission [7, 8] for higher
spectral efficiency (SE) transmission. That says we may adopt NOMA to
accommodate the requirements of mode 1–mode 2 SL transmission for higher SE.
The significant contribution in this work is to present an efficient and
efficacious approach for determining the employ of the most suitable multiple
access (MA) scheme and the associated relay.

We propose a low-complexity, location-based hybrid MA and relay selection
(H-MARS) algorithm for use when no SL CSI information is available. The H-MARS
algorithm is executed in two steps. First, the suboptimal relay positions are
approximated for NOMA and OMA schemes subject to SL CSI being unavailable. The
v-UE closest to the approximated location is specified as the best relay
available for each MA scheme. Second, a proposed criterion based on the SE
outage of two MA schemes is employed in assisting the selection of MA schemes
and their associated relays. H-MARS is suitable for low-latency V2X
applications involving no SL CSI feedback. In this algorithm, the relay
location approximation and criterion developed for MA selection have low
complexity in the order of $\mathcal{O}(N)$, where $N$ is the number of mode 1
SL multicast members. By comparison, brute-force search (BFS)-based algorithms
have a complexity in the order of $\mathcal{O}(N^{2})$. With provided
simulation results, we verify that MA selection in the H-MARS algorithm is
consistent with those in BFS-based algorithms with significantly lower
complexity consumption.

The remainder of this paper is organized as follows. Section II lays out the
system model and problem formulation. Section III details our proposed
algorithm. Section IV presents the simulation results. Finally, Section V
concludes the paper.

##  II System Model and Problem Formulation

Figure 1: System layout.

The scenario analyzed in this study is illustrated in Fig. 1. Specifically, a
BS aims to deliver messages to a designated multicast group ${\cal V}$
comprising $N$ v-UEs, which are denoted as $\\{v_{n}\in{\cal V}\\}_{n=1}^{N}$.
The v-UE $v_{n}\in{\cal V}$ is randomly located within a circle of radius $R$.
For convenience, we take the center of the circle to be the origin in two-
dimensional Cartesian coordinates. We denote $\theta_{n}=(x_{n},y_{n})$ as the
location of $v_{n}\in{\cal V}$ in the two-dimensional space. In the present
scenario, a v-UE $v_{r}\in{\cal V}$ is selected as a relay to relay messages
from the BS to the members of a multicast group ${\cal V}$. In practice, the
BS is not within the proximity of multicast group communications within ${\cal
V}$. Thus, per the 3GPP LTE specifications [2], the gNB may designate a relay
v-UE to execute mode 1 multicast transmission to the designated group members.
As mentioned, in the specified SL channel, mode 1 multicast transmission may
encounter interference from the ongoing mode 2 transmission. As illustrated in
Fig. 1, we assume that $v_{T}\notin{\cal V}$ is the mode 2–transmitting v-UE
closest to the multicast group; moreover, we designate $v_{D}$ as its
corresponding receiving v-UE. The signal transmission from $v_{T}$ to $v_{D}$
most likely causes the strongest interference to the ongoing transmission of
the multicast group ${\cal V}$.

###  II-A OMA Mode

First, we may schedule the mode 1 multicast SL traffic and mode 2 unicast SL
traffic in independent time frames—an approach known as time division MA, a
family of OMA schemes. Let $\eta^{\scaleto{OMA}{2.5pt}}_{1}$ denote the SE of
the link between a designated relay $v_{r}\in{\cal V}$ and a multicast member
$v_{n}\in{\cal V};$ it is given by

$\displaystyle\eta^{\scaleto{OMA}{2.5pt}}_{1,r,n}=\beta{\log_{2}}(1+g_{r,n}\rho_{n}d_{r,n}^{-\alpha}),$
(1)

where $g_{r_{n}}\sim\exp(1)$ is the exponentially distributed fading power
gain of the link between $v_{r}$ and $v_{n}$; $d_{r,n}$ is the distance
between $v_{r}$ and $v_{n}$; $\alpha$ is the path loss exponent;
$\rho_{n}=P/\sigma_{n}^{2}$, where $P$ is the transmission power of the v-UE
and $\sigma_{n}^{2}$ is the noise variance $v_{n}$; and $\beta=0.5$ represents
the $50\%$ occupancy time of the specified SL channel. Let
$\eta^{\scaleto{OMA}{2.5pt}}_{2}$ denote the SE of the link between $v_{T}$
and its receiving UE $v_{D}$, which is given by

$\displaystyle\eta^{\scaleto{OMA}{2.5pt}}_{2}=(1-\beta){\log_{2}}(1+g_{T,D}\rho_{\scaleto{D}{5pt}}d_{T,D}^{-\alpha}).$
(2)

In (2), we use $g_{T,D}\sim\exp(1)$ to denote the exponentially distributed
fading power gain of the link between $v_{T}$ and its receiving UE $v_{D}$
(spaced $d_{T,D}$ apart) with a rate of 1. We also define $\rho_{D}$ as
$P/\sigma_{D}^{2}$, where $P$ is the transmission power of the v-UE and
$\sigma_{D}^{2}$ is the noise variance $v_{D}$. To ensure the equal
distribution of the SL channel between multicast and unicast SL transmission,
the link between $v_{T}$ and $v_{D}$ should have a $50\%$ occupancy time of
the specified SL channel. Specifically, $(1-\beta)=0.5;\beta=0.5$.

To optimize reception under multicast SL communications, we select an optimal
relay $v_{\breve{r}}\in{\cal V}$ to maximize the total SE of the worst link
(i.e., that with the minimum SE) of multicast and unicast transmissions, as
indicated in (1) and (2), respectively. Thus, we obtain the desired SE
achieved with the relay $v_{\breve{r}}\in{\cal V},$ given as

$\displaystyle\eta^{\scaleto{OMA}{2.5pt}}=\max_{\breve{r}:v_{\breve{r}}\in{\cal
V}}(\min_{i:v_{i}\in{\cal
V}}\eta^{\scaleto{OMA}{2.5pt}}_{1,{\breve{r}},i}+\eta^{\scaleto{OMA}{2.5pt}}_{2}),$
(3)

where the minimum SE of $\eta^{\scaleto{OMA}{2.5pt}}_{1,{\breve{r}},i}$ is
achieved by $v_{i}$ and the optimal relay $v_{\breve{r}}$.

###  II-B NOMA Mode

As noted in [7, 8], we can exploit the NOMA to accommodate the
multicast–unicast SL transmission in the specified SL channel. In our
scenario, because the multicast members $v_{n}\in{\cal V}$ are close to each
other, $v_{n}$ usually receives a stronger signal from the relay
$v_{r}\in{\cal V}$ than from $v_{T}$. Thus, according to the NOMA principle
[9], a v-UE $v_{n}\in{\cal V}$ can directly decode the signal from $v_{r}$. As
for $v_{D}$, its signal from $v_{T}$ can be decoded using the successive
interference cancellation technique.111According to [4], SL CSI feedback is
only supported in unicast transmission. That said, $v_{D}$ has the CSI of
unicast transmission for performing the signal decoding in the NOMA procedure.
However, multicast CSI feedback is not supported.

Under the NOMA scheme [9], $v_{D}$ successively decodes and cancels signals
transmitted from $v_{r}\in{\cal V}$ prior to decoding its own signals from
$v_{T}$. Thus, the SE of the link between the relays $v_{r}\in{\cal V}$ and
$v_{n}\in{\cal V}$ can be written as

$\displaystyle\eta^{\scaleto{NOMA}{2.5pt}}_{1,r,n}={\log_{2}}(1+\frac{g_{r,n}\rho_{n}d_{r,n}^{-\alpha}}{\varrho
g_{T,n}\rho_{n}d_{T,n}^{-\alpha}+1}),$ (4)

where $\varrho$ is the power factor of NOMA [9] and
$\rho_{n}=P/\sigma_{n}^{2}$ is defined in (1).

Let $\eta^{\scaleto{NOMA}{2.5pt}}_{1,r,D}$ denote the SE of the link between
the relay $v_{r}\in{\cal V}$ and $v_{D}\notin{\cal V}$, which is given by

$\displaystyle\eta^{\scaleto{NOMA}{2.5pt}}_{1,r,D}={\log_{2}}(1+\frac{g_{r,D}\rho_{D}d_{r,D}^{-\alpha}}{\varrho
g_{T,D}\rho_{D}d_{T,D}^{-\alpha}+1}),$ (5)

where $\rho_{D}=P/\sigma_{D}^{2}$ is defined in (2) and we define
$\varrho\triangleq\frac{(1+g_{T,D}\rho_{D}d_{T,D}^{-\alpha})^{0.5}-1}{g_{T,D}\rho_{D}d_{T,D}^{-\alpha}}$.222We
set
$\varrho\triangleq\frac{(1+g_{T,D}\rho_{D}d_{T,D}^{-\alpha})^{0.5}-1}{g_{T,D}\rho_{D}d_{T,D}^{-\alpha}}$
assuming $v_{D}$ can have the same SE in OMA and NOMA, that is,
$0.5{\log_{2}}(1+g_{T,D}\rho_{D}d_{T,D}^{-\alpha})={\log_{2}}(1+\varrho
g_{T,D}\rho_{D}d_{T,D}^{-\alpha})$. Thus, the SE of the link between the
relays $v_{T}$ and $v_{D}$ can be written as

$\displaystyle\eta^{\scaleto{NOMA}{2.5pt}}_{2}={\log_{2}}(1+\varrho
g_{T,D}Pd_{T,D}^{-\alpha}).$ (6)

Similar to the OMA scheme, the optimal relay $v_{\hat{r}}\in{\cal V}$ should
maximize the total SE of the worst-performing link in multicast and unicast
transmission, as described in (4) and (5), respectively, to optimize reception
in multicast SL communications. Using such an approach, we can obtain the
desired SE achieved with the relay $v_{\hat{r}}\in{\cal V},$ which is given as

$\displaystyle\eta^{\scaleto{NOMA}{2.5pt}}=\max_{{\hat{r}}:v_{\hat{r}}\in{\cal
V}}(\min_{i:v_{i}\in{\cal
V}}(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i},\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D})+\eta^{\scaleto{NOMA}{2.5pt}}_{2}),$
(7)

where the minimum SE of $\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\breve{r}},i}$ is
achieved by $v_{i}$ and the optimal relay $v_{\breve{r}}$.

On the basis of the results of (3) and (7), we can activate the NOMA scheme
with the selected relay $v_{\hat{r}}\in{\cal V}$ if the SE of NOMA is greater
than that of OMA; that is, if
$\eta^{\scaleto{NOMA}{2.5pt}}\geq\eta^{\scaleto{OMA}{2.5pt}}$.

However, the optimal determination of the objective functions of (3) and (7)
requires SL CSI. Moreover, the calculation complexity of conducting a BFS-
based approach is at least in the order of $\mathcal{O}(N^{2})$. Compounded
with the issue of SL CSI feedback, this is impractical for SL multicast
communications and low-latency V2X applications. Thus, we introduce a low-
complexity location-based approach for situations when SL CSI is unavailable.

##  III Location-Based H-MARS Algorithm

Figure 2: Layout of proposed location-based H-MARS algorithm.

The steps of our proposed scheme are detailed as follows. First, according to
the locations of all anticipated v-UEs, the suboptimal relay location
approximations subject to SL CSI being unavailable are proposed for OMA and
NOMA and detailed in Prop. 1 and 2, respectively. Second, Prop. 3 addresses
the proposed criterion for determining the most suitable MA scheme, derived on
the basis of the SE outage of OMA and NOMA, and calculated with the
approximated relay locations acquired from Prop. 1 and 2. Propositions 1–3,
the three cornerstones of the proposed H-MARS, are presented as follows.

###### Proposition 1

Given the location of $v_{D}\notin{\cal V}$, which is spaced $d_{0,D}$ away
from the center shown in Fig. 2, the suboptimal relay location of OMA subject
to SL CSI being unavailable is approximated as

$\displaystyle d_{\breve{r},i}=0.5d_{i,j},$ (8)

where $v_{i}\in{\cal V}:i=\arg\max_{1\leq n\leq N}d_{n,D}$ is the v-UE with
the greatest distance to $v_{D}$, $v_{j}\in{\cal V}$ is the v-UE with the
greatest distance to $v_{i}$, and $d_{i,j}$ is the distance between $v_{i}$
and $v_{j}$.

###### Proof:

Because $\eta^{\scaleto{OMA}{2.5pt}}_{2}$ in (3) is irrelevant to the
selection of relay $v_{\breve{r}}$, (3) is dominated by

$\displaystyle\max_{\breve{r}:v_{\breve{r}}\in{\cal V}}(\min_{i:v_{i}\in{\cal
V}}\bar{\eta}^{\scaleto{OMA}{2.5pt}}_{1,{\breve{r}},i})\Rightarrow\min_{{\breve{r}}:v_{\breve{r}}\in{\cal
V}}(\max_{i:v_{i}\in{\cal V}}d_{{\breve{r}},i}),$ (9)

where
$\bar{\eta}^{\scaleto{OMA}{2.5pt}}_{1,{\breve{r}},i}=\beta{\log_{2}}(1+\rho_{i}d_{{\breve{r}},i}^{-\alpha})$
is a modification of (1) given that $g_{{\breve{r}},i}=1$ because of the lack
of SL CSI. The calculation complexity of (9) is in the order of
$\mathcal{O}(N^{2})$. To find an approximated solution to (9) for a random
topology, we add the following assumptions,

###### Assumption 1

$v_{i}\in{\cal V}:i=\arg\max_{1\leq n\leq N}d_{n,D}$ is the v-UE with the
greatest distance to $v_{D}$ (Fig. 2).

###### Assumption 2

$v_{j}\in{\cal V}:j=\arg\max_{1\leq n\leq N}d_{n,i}$ is the v-UE with the
greatest distance to $v_{i}$ (Fig. 2).

The first assumption gives us an anchor $v_{i}$. The second assumption allows
us to find $v_{\breve{r}}$ along the line between $v_{i}$ and $v_{j}$. Thus,
the $v_{\breve{r}}$ that satisfies (9) should be in the middle of $v_{i}$ and
$v_{j}$; that is, $d_{{\breve{r}},i}=d_{{\breve{r}},j}=0.5d_{i,j}$. ∎

###### Proposition 2

Given the location of $v_{D}\notin{\cal V}$, the suboptimal relay location of
NOMA is approximated as

$\displaystyle
d_{\hat{r},i}=\frac{c_{1}^{-1/\alpha}}{c_{1}^{-1/\alpha}+c_{2}^{-1/\alpha}}\cdot
d_{i,D},$ (10)

where $c_{1}={\varrho d_{T,i}^{-\alpha}+\rho_{i}^{-1}}$ and $c_{2}={\varrho
d_{T,D}^{-\alpha}+\rho_{D}^{-1}}$.

###### Proof:

In (7), $\eta^{\scaleto{NOMA}{2.5pt}}_{2}$ is irrelevant to the selection of
relay $v_{\hat{r}}$; moreover, $\eta^{\scaleto{NOMA}{2.5pt}}$ is dominated by
the first term of (7), which is

$\displaystyle\max_{\hat{r}:v_{\hat{r}}\in{\cal V}}(\min_{i:v_{i}\in{\cal
V}}(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i},\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}).$
(11)

If the inequality
$\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i}\geq\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}$
holds, (11) becomes $\max_{\hat{r}:v_{\hat{r}}\in{\cal
V}}\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}$. This entails that
$v_{\hat{r}}$ should be as close to $v_{D}$ as possible, whereas $v_{\hat{r}}$
should be as far away from $v_{i}$ as possible, to maximize the minimum of
$\eta^{\scaleto{NOMA}{2.5pt}}_{1,\hat{r},i}$. Thus, $v_{i}$ should also be
away from $v_{D}$, making Assumption 1 reasonable. Furthermore,
$\eta^{\scaleto{NOMA}{2.5pt}}_{1,\hat{r},D}$ is upper bounded by the minimum
of $\eta^{\scaleto{NOMA}{2.5pt}}_{1,\hat{r},i}$. To maximize
$\eta^{\scaleto{NOMA}{2.5pt}}_{1,\hat{r},D}$, we have the associated relay
$v_{\hat{r}}$ to satisfy

$\displaystyle\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i}=\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}$
(12) $\displaystyle\Rightarrow$
$\displaystyle\frac{g_{\hat{r},i}\rho_{i}d_{\hat{r},i}^{-\alpha}}{\varrho
g_{T,i}\rho_{i}d_{T,i}^{-\alpha}+1}=\frac{g_{\hat{r},D}\rho_{D}d_{\hat{r},D}^{-\alpha}}{\varrho
g_{T,D}\rho_{D}d_{T,D}^{-\alpha}+1}.$ (13)

Subject to SL CSI being unavailable, it is reasonable to specify
$g_{\hat{r},i},g_{\hat{r},D},g_{T,i}$ and $g_{T,D}$ all equal 1, and we may
rewrite (13) as

$\displaystyle\frac{d_{{\hat{r}},i}^{-\alpha}}{\varrho
d_{T,i}^{-\alpha}+\sigma_{i}^{2}}=\frac{d_{{\hat{r}},D}^{-\alpha}}{\varrho
d_{T,D}^{-\alpha}+\sigma_{D}^{2}},$ (14) $\displaystyle\Rightarrow$
$\displaystyle
c_{2}d_{{\hat{r}},i}^{-\alpha}=c_{1}(d_{i,D}-d_{{\hat{r}},i})^{-\alpha}$ (15)
$\displaystyle\Rightarrow$ $\displaystyle
d_{{\hat{r}},i}=\frac{c_{1}^{-1/\alpha}d_{i,D}}{c_{1}^{-1/\alpha}+c_{2}^{-1/\alpha}},$
(16)

where $c_{1}={\varrho d_{T,i}^{-\alpha}+\rho_{i}^{-1}}$ and $c_{2}={\varrho
d_{T,D}^{-\alpha}+\rho_{D}^{-1}}$.

Given $v_{\hat{r}}$, if the inequality of
$\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i}<\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}$
holds, (12) can be solved, leading to (16). For brevity, this process is not
shown. ∎

Figure 3: Recorded SEs and total calculation complexity of the H-MARS
algorithm with $d_{0,T}/R=3$.

Figure 4: Duty cycle of NOMA in the H-MARS algorithm with different values of
$(r_{e},d_{T}/R)$.

Figure 5: Recorded SEs of the H-MARS algorithm with different values of
$(r_{e},d_{T}/R)$.

###### Proposition 3

Given the $v_{\breve{r}}$ and $v_{\hat{r}}$ selected from Prop. 1 and 2 for
the OMA and NOMA schemes, respectively, with a specified $\gamma$, we can
activate the NOMA mode if

$\displaystyle\min(\frac{d_{{\hat{r}},i}^{-\alpha}e^{-\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma}}{d_{{\hat{r}},i}^{-\alpha}+\varrho
d_{T,i}^{-\alpha}\gamma},\frac{d_{{\hat{r}},D}^{-\alpha}e^{-\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}}{d_{{\hat{r}},D}^{-\alpha}+\varrho
d_{T,D}^{-\alpha}\gamma})\geq(e^{-\rho_{D}d_{\breve{r},i}^{-\alpha}\gamma})$
(17)

is true. Otherwise, the OMA mode will be activated.

Notably, (17) is dominated by the locations of all anticipated v-UEs and
results in the calculation complexity in the order of $\mathcal{O}(N)$.

###### Proof:

If the outage probability of NOMA is smaller than that of OMA, the SE of NOMA
$\eta^{\scaleto{NOMA}{2.5pt}}$ is likely higher than the SE of OMA
$\eta^{\scaleto{OMA}{2.5pt}}$ for a given $\gamma$. In such circumstances, the
NOMA mode should be activated; otherwise, the OMA mode should be activated. We
have the outage probabilities of NOMA and OMA approximated in Lemma 1 of
Appendix A and Lemma 2 of Appendix B, respectively. Thus, the NOMA mode is
activated when

$\displaystyle\max(1-\frac{\rho_{i}d_{{\hat{r}},i}^{-\alpha}e^{-\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma}}{\rho_{i}d_{{\hat{r}},i}^{-\alpha}+\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma},1-\frac{\rho_{D}d_{{\hat{r}},D}^{-\alpha}e^{-\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}}{\rho_{D}d_{{\hat{r}},D}^{-\alpha}+\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma})$
$\displaystyle\leq(1-e^{-\rho_{D}d_{\breve{r},i}^{-\alpha}\gamma}),$ (18)

which may be rewritten as (17). ∎

On the basis of Lemma 3 in Appendix C, the association of $\gamma_{i}$ with
$v_{i}$ is specified to be $\gamma_{i}=(2^{(A-B)}-1)$, and the association of
$\gamma_{D}$ with $v_{D}$ can be similarly set. Now, we can specify
$\gamma=\min(\gamma_{i},\gamma_{D})$ in Prop. 3. The steps of the H-MARS
algorithm are summarized in Algorithm 1.

Input: Locations of $v_{i}\in{\cal V},\forall i$, $v_{T}$ and $v_{D}$.

Output: The determined MA scheme and its associated relay, either
$v_{\breve{r}}\in{\cal V}$ for NOMA or $v_{\hat{r}}\in{\cal V}$ for OMA.

Step 1(a): Find the relay of OMA $v_{\breve{r}}\in{\cal V}$, which is the v-UE
nearest to the relay location approximation from Prop. 1.

Step 1(b): Find the relay of NOMA $v_{\hat{r}}\in{\cal V}$, which is the v-UE
nearest to the relay location approximation from Prop. 2.

Step 2: Use Prop. 3 to determine whether to activate the NOMA mode with its
associated relay $v_{\hat{r}}$ or the OMA mode with its associated relay
$v_{\breve{r}}$.

Algorithm 1 Location-Based H-MARS Algorithm

##  IV Simulation Results

The system parameters used in the simulations are summarized as follows. On
the basis of the spatial Poisson process [3], given a density of 0.25$\%$,
$20$ v-UEs on average were randomly distributed in a circle with a radius
$R=50(m)$. In addition, we experimented with various locations of $v_{T}$ and
$v_{D}$. Each recorded SE was the average result of $10^{5}$ experiments. The
transmission power $P$ of each v-UE was 21 dBm [4], and the noise variance was
$-89$ dBm. The path loss exponent $\alpha=4$ is used in Prop. 1–3, along with
the path loss model presented in [4].333The path loss model used for
simulations is
$PL=40\log_{10}(d)+7.65-17.3\log_{10}(h_{t}-1)-17.3\log_{10}(h_{r}-1)+2.7\log_{10}(f_{c})$,
where $h_{t}=h_{r}=1.5$ m and $f_{c}=5.9$ GHz.

Fig. 5 displays the recorded SE (a) and the total calculation complexity (b)
of the H-MARS and counterpart algorithms in terms of the location of $v_{D}$,
which was determined to exert a strong influence on the results. To our
knowledge, there is no other similar work, which also targets a joint design
of the MA selection and the associated relay location approximation in V2X
networks. Hence, the SEs of the optimal BFS-based OMA and NOMA provide the
most suitable performance references to our work. As shown in Fig. 5(a), BFS-
based NOMA having higher SE when $(d_{0,D}/R)\leq 1.4$ (i.e., $d_{0,D}\leq
70m$) is a logical choice over BFS-based OMA. Subject to SL CSI being
unavailable, the proposed H-MARS, having the MA switch point at
$(d_{0,D}/R)=1.2$, still closely adapts to the changes of BFS-based MA scheme.
In addition, the total calculation complexity recorded in Fig. 5(b) shows that
the H-MARS has significantly lower calculation complexity compared to BFS-
based MA scheme in terms of the number of multiplications and additions. For
example, with the setting of $N=20$, the recorded calculation complexity of
H-MARS is at least $20$ times lower than that of the BFS-based MA scheme.
These results indicate that the H-MARS algorithm, with low complexity, is
extremely efficient and efficacious in guiding the MA scheme selection
provided with the effective associated relay location approximation.

The H-MARS provides an efficacious MA switch. Hence, Fig. 5 presents the duty
cycle of NOMA in the H-MARS algorithm in terms of $(r_{e},d_{0,T}/R)$, where
$r_{e}$ indicates the maximum value of the uniformly randomized vehicle’s
location estimation error (in meters), which may be resulted by variant
localization algorithms, mobility, etc. [10] As shown in Fig. 5, the larger
value of $d_{0,T}/R$ will result in more duty cycles of NOMA. It entails that
OMA is preferred when the mode-2 $v_{T}$ is closer to the mode-1 V2X group,
and NOMA will be employed when $v_{T}$ is away from the mode-1 V2X group. More
importantly, Fig. 5 shows that the MA selection of H-MARS only depends on the
relative locations of mode-2 $v_{T}$ and $v_{D}$. In addition, Fig. 5 recorded
that the location estimation error, especially for $r_{e}=5$m, results in a
negligible impact to the MA’s duty-cycle of H-MARS. Hence, the MA schemes do
not frequently switch due to rapid CSI variations and have moderate
sensitivity to the vehicle’s location estimation error. Those facts make the
proposed H-MARS practical in use.

The SEs of H-MARS recorded in Fig. 5 also show the consistency of the above
discoveries. The SE of H-MARS presents negligible degradation for $r_{e}=5$m
and shows a moderate loss for $r_{e}=10$m for all settings of $d_{0,T}/R$.
More importantly, Fig. 5 shows that with the same value of $d_{D}$, the mode-2
$v_{T}$ closer to the mode-1 V2X group, in general, will result in higher SE.
Along with the discovery acquired from Fig. 5, NOMA enhances the SE with
larger $d_{0,T}/R$ and OMA plays its role when $d_{0,T}/R$ is small. In
summary, NOMA is preferred when $v_{T}$ is far away from the multicast group,
whereas OMA performs better when $v_{T}$ is sufficiently close to that group.
The location of $v_{D}$ in between is key.

##  V Conclusions

This paper introduced a low-complexity hybrid MA scheme and its associated
relay selection algorithm when SL-CSI is not available. The proposed scheme
has low calculation complexity in the order of $\mathcal{O}(N)$ compared with
the BFS-based algorithm, which is in the order of $\mathcal{O}(N^{2})$. As a
result, the proposed H-MARS algorithm closely adapts to the BFS-based
benchmark with low SE loss. Furthermore, it has moderate sensitivity to the
vehicle’s location estimation error, which makes the proposed H-MARS practical
in use.

## Appendix A

###### Lemma 1

Given the selected $v_{\hat{r}}$ from Prop. 2, we can obtain

$\displaystyle\Pr(\eta^{\scaleto{NOMA}{2.5pt}}<\Gamma)$
$\displaystyle\propto\max(1-\frac{\rho_{i}d_{{\hat{r}},i}^{-\alpha}e^{-\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma}}{\rho_{i}d_{{\hat{r}},i}^{-\alpha}+\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma},1-\frac{\rho_{D}d_{{\hat{r}},D}^{-\alpha}e^{-\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}}{\rho_{D}d_{{\hat{r}},D}^{-\alpha}+\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}),$
(19)

where $\gamma=(2^{\Gamma}-1)$.

###### Proof:

According to (7), the selection of $v_{\hat{r}}$ is irrelevant to
$\eta^{\scaleto{NOMA}{2.5pt}}_{2}$ in (6). Thus, we have

$\displaystyle\Pr(\eta^{\scaleto{NOMA}{2.5pt}}<\Gamma)$
$\displaystyle\propto\max(\Pr(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i}<\Gamma),\Pr(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}<\Gamma)).$
(20)

There exists a $\gamma=(2^{\Gamma}-1)$ such that

$\displaystyle\Pr(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},i}<\Gamma)\propto\Pr(\frac{g_{{\hat{r}},i}\rho_{i}d_{{\hat{r}},i}^{-\alpha}}{\varrho
g_{T,i}\rho_{i}d_{T,i}^{-\alpha}+1}<\gamma),$ (21)
$\displaystyle\Pr(\eta^{\scaleto{NOMA}{2.5pt}}_{1,{\hat{r}},D}<\Gamma)\propto\Pr(\frac{g_{{\hat{r}},D}\rho_{D}d_{r,D}^{-\alpha}}{\varrho
g_{T,D}\rho_{D}d_{T,D}^{-\alpha}+1}<\gamma).$ (22)

We let $X=g_{{\hat{r}},i}\rho_{i}d_{{\hat{r}},i}^{-\alpha}$ and $Y=\varrho
g_{T,i}\rho_{i}d_{T,i}^{-\alpha}$ with
$\lambda_{y}=\varrho\rho_{i}d_{T,i}^{-\alpha}$ and
$\lambda_{x}=\rho_{i}d_{{\hat{r}},i}^{-\alpha}$ such that

$\displaystyle\Pr(\frac{g_{{\hat{r}},i}Pd_{{\hat{r}},i}^{-\alpha}}{\varrho
g_{T,i}\rho_{i}d_{T,i}^{-\alpha}+1}<\gamma)=1-\frac{\rho_{i}d_{{\hat{r}},i}^{-\alpha}e^{-\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma}}{\rho_{i}d_{{\hat{r}},i}^{-\alpha}+\varrho\rho_{i}d_{T,i}^{-\alpha}\gamma}.$
(23)

Similarly, we have

$\displaystyle\Pr(\frac{g_{{\hat{r}},D}\rho_{D}d_{r,D}^{-\alpha}}{\varrho
g_{T,D}\rho_{D}d_{T,D}^{-\alpha}+1})<\gamma)=1-\frac{\rho_{D}d_{{\hat{r}},D}^{-\alpha}e^{-\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}}{\rho_{D}d_{{\hat{r}},D}^{-\alpha}+\varrho\rho_{D}d_{T,D}^{-\alpha}\gamma}.$
(24)

In (A), we substitute (21) with (23) and (22) with (24), respectively, to
obtain (1). ∎

## Appendix B

###### Lemma 2

Given the selected $v_{\breve{r}}$ from Prop. 1, we can obtain

$\displaystyle\Pr(\eta^{\scaleto{OMA}{2.5pt}}<\Gamma)\propto
1-e^{-\rho_{i}d_{{\breve{r}},i}^{-\alpha}\gamma},$ (25)

where $\gamma=(2^{\Gamma}-1)$.

###### Proof:

According to (3), the selection of $v_{r}$ is irrelevant to
$\eta^{\scaleto{OMA}{2.5pt}}_{2}$. Thus, we have

$\displaystyle\Pr(\eta^{\scaleto{OMA}{2.5pt}}<\Gamma)\propto\Pr(\eta^{\scaleto{OMA}{2.5pt}}_{1,\breve{r},i}<\Gamma).$
(26)

Furthermore, there exists a $\gamma=(2^{\Gamma}-1)$ such that

$\displaystyle\Pr(\eta^{\scaleto{OMA}{2.5pt}}_{1,\breve{r},i}<\Gamma)\propto\Pr(g_{\breve{r},i}\rho_{i}d_{\breve{r},i}^{-\alpha}<\gamma).$
(27)

If $X$ is an exponentially distributed random variable with a rate
$\lambda_{x}$, we have $\Pr(X<x)=1-e^{-\lambda_{x}x}$. Thus, we let
$X=g_{\breve{r},i}\rho_{i}d_{\breve{r},i}^{-\alpha}$ with
$\lambda_{x}=\rho_{i}d_{\breve{r},i}^{-\alpha}$ such that

$\displaystyle\Pr(g_{\breve{r},i}\rho_{i}d_{\breve{r},i}^{-\alpha}<\gamma)=1-e^{-\rho_{i}d_{\breve{r},i}^{-\alpha}\gamma}.$
(28)

Then, in (26), we substitute (27) with (28) to obtain (25). ∎

## Appendix C

###### Lemma 3

Given that $X=g_{{\hat{r}},i}\rho_{i}d_{{\hat{r}},i}^{-\alpha}$ and $Y=\varrho
g_{T,i}\rho_{i}d_{T,i}^{-\alpha}$ are independent and exponentially
distributed with rates $\lambda_{y}=\varrho\rho_{i}d_{T,i}^{-\alpha}$ and
$\lambda_{x}=\rho_{i}d_{{\hat{r}},i}^{-\alpha}$, respectively, we have

$\displaystyle{\mathbb{E}}\\{\log_{2}(1+\frac{X}{Y+1})\\}=A-B,$ (29)

where

$\displaystyle
A=\frac{e^{1/\lambda_{x}}E_{1}(1/\lambda_{x})}{\log_{e}(2)(1-\frac{\lambda_{x}}{\lambda_{y}})}+\frac{e^{1/\lambda_{y}}E_{1}(1/\lambda_{y})}{\log_{e}(2)(1-\frac{\lambda_{y}}{\lambda_{x}})}\text{
and}$ (30) $\displaystyle
B=\frac{e^{1/\lambda_{y}}E_{1}(1/\lambda_{y})}{\log_{e}(2)}$ (31)

###### Proof:

First, (29) can be rewritten as

$\displaystyle{\mathbb{E}}\\{\log_{2}(1+\frac{X}{Y+1})\\}={\mathbb{E}}\\{\log_{2}(1+(X+Y)\\}-{\mathbb{E}}\\{\log_{2}(1+Y)\\}.$
(32)

According to [11, (C.15)], we have

$\displaystyle{\mathbb{E}}\\{\log_{2}(1+\sum_{k=1}^{K}g_{k})\\}=\sum_{k=1}^{K}\frac{e^{1/\mu_{k}}E_{1}(1/\mu_{k})}{\log_{e}(2)\prod_{\ell\neq
k}(1-\frac{\mu_{\ell}}{\mu_{k}})},$ (33)

where $g_{k}$ is exponentially distributed with a rate $\mu_{k}$ and
$E_{1}(x)=\int_{1}^{\infty}e^{-xt}/tdt$ is the exponential integral.

Next, two terms of (32) are given by

$\displaystyle
A={\mathbb{E}}\\{\log_{2}(1+(X+Y)\\}=\frac{e^{1/\lambda_{x}}E_{1}(1/\lambda_{x})}{\log_{e}(2)(1-\frac{\lambda_{x}}{\lambda_{y}})}+\frac{e^{1/\lambda_{y}}E_{1}(1/\lambda_{y})}{\log_{e}(2)(1-\frac{\lambda_{y}}{\lambda_{x}})},$
(34) $\displaystyle
B={\mathbb{E}}\\{\log_{2}(1+Y)\\}=\frac{e^{1/\lambda_{y}}E_{1}(1/\lambda_{y})}{\log_{e}(2)}.$
(35)

∎

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