Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc
Networks for BSM Safety Applications: Parallel Numerical
Solutions and Simulations
Jing Zhao
1,3
a
, Hao Zhou
1
, YanBin Wang
2
, HuaLin Lu
4
, Zhijuan Li
4 b
and XiaoMin Ma
5
1
School of Software Technology, Dalian University of Technology, Dalian 116024, China
2
Department of Industrial Engineering, Harbin Institute of Technology, Harbin 150001, China
3
Cyberspace Security Research Center, Peng Cheng Laboratory, Shenzhen 518052, China
4
Department of Computer Science and Technology, Harbin Engineering University, Harbin 150001, China
5
College of Science and Engineering, Oral Roberts University, Tulsa, OK 74171, U.S.A.
Keywords:
Vehicle-to-Vehicle Communication, MPI, Signal-to-Interference-to-Noise, Capacity, QoS.
Abstract:
Vehicular Ad-hoc Networks (VANETs) have been proposed and investigated for road safety applications.
Many safety applications are enabled by broadcasting basic safety message (BSM) periodically. Whether the
current IEE E802.11p communication system can meet the stringent quality of service (QoS) requirement for
safety applications is under discussion. Many analytical and si mulation models have been proposed to study
the reliability of DSRC (Dedicated Short Range Communication) I EEE802.11p broadcast services. However,
most analyses assume a deterministic communication range, which is unpractical. In this paper, we propose
an analytical model based on signal-to-interference-plus-noise ratio (SINR) to study of QoS and capacity of
VANET for BSM based safety applications. The analytical model considers the context of the more practical
vehicular communication environment: BSM broadcast, asynchronous timing between hidden terminals, Nak-
agami channel fading, and Non-Homogeneous Poisson Process vehicle distribution. For the proposed model,
the computation complexity of QoS and capacity metrics by numerical solutions is so high that the compu-
tation time is intolerable. Thus the efficient numerical way together with a parallel approach is needed to
evaluate these metrics. The Monte Carlo integration and MPI (Message Passing Interface) method are applied
for accelerating the computing process. The analysis of QoS metrics are validated by NS2 simulation.
1 INTRODU CTI ON
Intelligent Transportation System (ITS) (Andrisano
et al., 2000) is moving in the direction of safe
and comfortable driving. Vehicular ad hoc network
(VANET) plays an important role in ITS. Among the
many applications supported by VANET, safety appli-
cation is the most c ritical. Many safety applications
are accomplished by broadcasting basic safety mes-
sage (BSM), and these safety applications have strict
quality o f serv ic e (Qo S) require ments. Research on
whether the vehicle communication system based on
DSRC can meet the QoS requirements of safety ap-
plications is also under way. At present, many ana-
lytical m odels along with extensive simulations have
a
https://orcid.org/0000-0001-8529-3399
b
https://orcid.org/0000-0002-2162-5654
been proposed to stud y the performance and reliabil-
ity of DSRC IEEE 802.11p broa dcast traffic in one-
dimensional (Luong et al., 2017; Bazzi et al., 2018;
Ma et al., 2013b; Yin et al., 2013; Yao et al., 2013; Ma
et al., 20 21) and two-dime nsional inter sections (Ma
et al., 2013a; Steinmetz et al., 2015; Ma et al., 2016)
VANET. However, the current analysis of VANET
QoS and capacity mostly assumes that the c ommun i-
cation range is deterministic, and the communication
range and hidden terminals are important factors af-
fecting packet reception, which is impractical. In ad-
dition, for the purpo se of an alytical tractability, many
impractical assumptions are made, such as the expo-
nential vehicle distribution (Ma et al., 2011; Hafeez
et al., 2013; Yin et al., 2013; Ma et al., 2013b) and the
Raleigh fading channel mode l considering path loss
(Ye et a l., 2011 ), etc.
600
Zhao, J., Zhou, H., Wang, Y., Lu, H., Li, Z. and Ma, X.
Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc Networks for BSM Safety Applications: Parallel Numerical Solutions and Simulations.
DOI: 10.5220/0010472306000610
In Proceedings of the 7th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2021), pages 600-610
ISBN: 978-989-758-513-5
Copyright
c
 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

A few analytic models have been used to evalu a te
the reliability metric s of PRP(Packet re ception prob-
ability)/PRR(Packet rece ption ratio) of 1-D broadcast
mobile ad-hoc networks (MANETs) (Ma et al. , 2011;
Ye e t al., 2011; Yin et al., 2013; Hassan et al., 2011;
Hafeez et al., 2013; Ma et al., 2013b; Tong et al.,
2016; Yao et al., 2013). These kin ds of analytical ap-
proach e s take the impact of th e hidden terminal prob-
lem, the fading channel, and the ch annel access pro-
tocol into consideration, investigate the performance
of VANET at different densities, different receiv-
ing distances, and different channel model (such as
Nakagami- m Fading, Weibull Fading, Rayleigh Fad-
ing and Rician Fading). However, few models could
provide the practical as well as a viable approach to
estimate the actual VANET capacity. Several studies
for VANET capacity using scaling-law based method
can only give per-node capacity scales in asymptoti-
cally large wireless networks (Wang et al., 2015; Lu
and Shen , 2014), which cannot be easily applied to
actual capacity e stima tion.
Most recently, a new interference-based capa c-
ity model was proposed for VANET safety message
broadc a st scenario (Ni et al., 2015; Ma et al., 2017).
The model approached the capacity analysis of one-
dimensional (1-D ) VANET safety message broad cast
under Na k agami fading channel through the deriva -
tion of SINR distribution afte r making reasonable ap-
proxim ations. This model enables the evaluation of
VANET ca pacity for safety applications in a more
practical way.
The SINR is the ratio of the strength of the re-
ceived useful signal to the strength of the received
interference signal (noise and interference). BER
(bit error rate) represents the probability that a bit is
misinterpreted at a receiver due to the propagation
process (Ya o et al., 2014 ; Molisch, 2012), which is
the function of the SINR. SINR threshold is defined
as the value whose mapping BER is small eno ugh(
usually 10
−5
) for the tolerable tran sm ission error.
The actual physical communication system such as
a real radio hardware USRP (Gotsis et al., 2 017), or
the simulation components for DSRC such as popu-
lar tools N S2 (Chen et al., 200 7), NS3 (Eckermann
et al., 201 9; Shaban et al., 2020) , and Matlab (Got-
sis et al., 2017; Bazzi et al., 2019) employ the SINR
threshold co mmunication mechanism to receive the
data packet. Accordingly, estimating entire network
capacity or evaluating the performance of VANET
Based on SINR distribution could be obtained. There-
fore, the SINR based modeling approach to analy ze
the QoS VANET not only re present the actual com-
munication system features, but also establish the
quantization standard such as Capacity and QoS.
Although the SINR based m odeling approach for
VANET has some advantages compar e d with the de-
terministic distance based modeling appr oach, the
computation complexity of the SINR based far ex-
ceeds the deterministic distance approach. Message
Passing In terface (MPI) (Gropp et al., 1996) is a
message-passing standard that is widely used to so lve
scientific computing problems on parallel computers.
It provides a r ic h collection of interfaces for com-
munication between processes. MPI supports point-
to-poin t communication and collective communica-
tion. Thanks to the parallel model MPI, the com -
plex high dimensional integrations could be trans-
formed into solvable problem. VANET QoS met-
rics have no efficent numerical solutions based on
SINR model(Ma et al., 2017), since it needs efforts
to find an efficient numerical way to evaluate those
metrics. To acce le rate th e computing process, ther e
are two directions for optimiz a tion, reducing integral
sampling points and computing integrand in parallel.
For reducingg integral sampling points, several deli-
cate approaches can be adopted, such as Monte Carlo
integration(Morokoff and Caflisch, 1995), Sparse
grids(Gerstne r and G riebel, 1998), Bayesian quadra-
ture(Gunter et al., 2014), etc. Some of the m such as
Bayesian quadrature is hard to parallelism since each
iteration of algorithm is related to the last iteration be-
fore. For computing integrands in p arallel, many re-
search try to c ompute integrands in par a llel by GPU
(Arumu gam et al. , 2013; Zo ng et al., 2019) or FPGA
(Razak et al., 2017), which is significantly faster than
compute by CPU. The hardware featur e of GPU and
FPGA make them can only do the simple evalua tion,
while the integrands of model by SINR is too com-
plex to implement on them. Therefore, in this paper,
we choose Monte Carlo integration as well as MPI
method to acceler ate computing process.
In this paper, to build a firm and complete frame-
work of the SINR based approach to the capacity
and QoS of VANET for safety message broadcast, we
come up with a new systematic approach to derivation
of the transmission probability and the SINR distri-
bution in the context of BSM safety applications with
more general vehicle distribution. The new approach
considers the impact of IEEE 802.11p MAC channel
access and asynchronous timing between hidden ter-
minals. Then the SINR based analysis is further ex-
tended to the a nalysis of QoS metrics for the safety
applications.
Main Contributions of this paper are summar iz ed
as follows:
• An analytical model based on SINR is built
with No n-Homo geneou s Poisson Process (NHPP)
node distribution in 1-D, unsatura ted message
Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc Networks for BSM Safety Applications: Parallel Numerical Solutions
and Simulations
601

generation, Nakagami channel fading with path
loss, and impac t of hidden terminal.
• The new model derives th e SINR distribution
through MPI Mo nte Carlo program ming model,
which transform the complex numerical compu-
tation to an actual solvable problem.
• Simulations and experiments are proposed for the
analysis of validity, computational efforts, accu-
racy of the model b y SINR.
2 PROBLEM FORMULATION
AND ASSUMPTIONS
2.1 Problem Formulation
Given a h ighway vehicular e nvironment on which
all vehicles are equipped with IEEE 802.11p DSRC
wireless communication capability, each vehicle
broadc a sts BSM containing measured mo bility infor-
mation to all surrounding vehicles in its tran smission
range periodically with message generation rate λ,
and receives the BSMs from the surroundin g vehicles.
In this way, awareness range of drivers can be ex-
tended and more accidents can be avoided (Schmidt-
Eisenlohr, 2010). The saf e ty-related message broad-
cast requires high reliab ility and performance. How-
ever, the quality of service (QoS) is degraded by mes-
sage collisions and fading communication channel.
We are concerned a bout if th e current DSRC system,
under c ertain vehicular environment, is able to pr o-
vide the broad cast service with guaranteed QoSs for
all selected safety applications. In order to evalua te
the system in this regard, several QoS metrics and
capacity nee d to be eva luated: 1) Messag e (packet)
delivery probability defined a s the probability that a
receiver successfully decodes the message (packet)
from a source n ode with a distance. 2) Packet (mes-
sage) reception ratio defined as the percentage of re-
ceivers in a range tha t are free fr om transmission er-
rors once a broadcast m essage is sent out. 3) Link
capacity defined as the maximum message (packet)
transmission rate betwe en two nodes in the comm u-
nication chann el.
2.2 Assumptions
We assume that IEEE 802.1 1p beacon m essage broad-
cast works under the following scenarios. (1) A 2-D
strip-like network area can be approximated to a 1-
D single lane. (3) All nodes are treated as homog e-
neous with identical vehicle length L
V
and transmis-
sion power P
t
. (4) Mobile nodes are placed on the
lines accord ing to NHPP with the density of vehicles
at a distance x from the tagged node: β(x) (in nodes
per meter). Then, the probability of finding i vehicles
in a space interval (x,x + l) is given by
P[i,(x,x + l)] =

R
x+l
x
β(y)dy

i
e
−
R
x+l
x
β(y)dy
i!
.
(5) Same Nakagami fading mo del is assumed for ve-
hicular commun ic a tion channel. (6) The distance be-
tween an interf ere and the tagged transmitter should
be no longer than 2r
E
(Ni et al., 2015), where r
E
is the
average sensin g range r
E
= d
0
α
p
P
t
η/P
th
, and P
th
is
the clear channel assessment(CCA) sensitivity. Then,
PDF of the power P
r
received from a receiver with
distance d away from a sou rce node is rewritten as
f
P
r
|d
(x) =
1
Γ(m)

m
¯
P
r
(d)

m
x
m−1
exp

−
mx
¯
P
r
(d)

,
where Γ() is the Gamma function, and m is the fad-
ing parameter.
¯
P
r
(d) = P
t
η

d
0
d

α
(η is a transceiver-
determined constant, d
0
is the reference distance for
the far-zone field, α is the pathloss exponent) is the
mean value determined by the pathloss.
3 ANALYSIS OF SINR
DISTRIBUTION
As shown in Figure 1 , given that there are l nodes
in the shaded interferenc e region, and th e inte rfere
I is the (l − i)-th node within the right shaded re-
gion (i = 1,. .. ,l − 1), Denote d
l
be the distance be-
tween the tagged node T and the l −th node (the far-
thest interfering node) within the right shaded region
[r
E
,d
max
] where dmax is the maximum range of all in-
tended interfering nodes. Given NHPP distribution of
distance between nodes, according to (Ma and Chen,
2008) and (Haenggi, 2005), the complimentary cumu -
lative probability distribution P (d
l
> r) is given by the
probability that there is at least one node in the range
of d
max
− r divided by the probability that is at least a
node in the range of d
max
:
P(d
l
> r) =
1 − e
−
R
d
max
r
β(y)dy
1 − e
−
R
d
max
0
β(y)dy
,
Then, the cu mulative distribution function (CDF) of
distance d
l
(r
E
< d
l
< 2 r
E
) can be calculated as
F
d
l
(τ) = P(d
l
< τ) =
e
−
R
2r
E
τ
β(y)dy
− e
−
R
2r
E
0
β(y)dy
1 − e
−
R
2r
E
0
β(y)dy
,
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
602

r
E
r
E
r
E
r
T
r
T
D
I
D
s
r
E
T R I
I
1
st
2
nd
l
th
D
1
D
2
D
B
...
k
th
i
th
...
(i+1)
th
(l-1)
th
...
Figure 1: General interfering scenario for VANET safety
message broadcast.
Since d
l−i
= d
l−i+1
− Y
l−i+1
(i = 1,. .. ,l − 1),
where Y
l−i+1
is distance between l − i + 1th node and
l − ith node with distribution
F
Y
l−i+1
|d
l−i+1
(y|d) = 1 − e
−
R
d
d−y
β(z)dz
,
f
Y
l−i+1
|d
l−i+1
(y|d) =
dF
Y
l−i+1
|d
l−i+1
(y|d)
dy
= β(d − y)e
−
R
d
d−y
β(z)dz
.
Consequently, the PDFs of distances of individ-
ual nodes l − i in the shaded area can be d erived as
(Trivedi, 2002)
f
l−i
(τ) =
Z
2r
E
τ
f
l−i+1
(x) f
Y
l−i+1
|d
l−i+1
((x − τ)|x)dx,
i = 1,...,l − 1 .
Given D
S
is the distan ce between T and R, the dis-
tance between (l − i)-th inte rfering node and node R
is d enoted as D
I
(i = 0,...,l − 1), where r
E
− D
s
≤
D
I
≤ 2 r
E
− D
s
.
F
D
I
|(d
s
,l−i)
(x) = P(D
I
< x|r
E
− d
s
≤ D
I
≤ 2 r
E
− d
s
)
=
F
l−i
(d
s
+ x)
R
2r
E
r
E
f
l−i
(τ)dτ
,i = 0,. .. ,l − 1.
The PDFs of the distances of the in dividual interfe ring
nodes to the receiver R are obtained as
f
D
I
|(d
s
,l−i)
(x) =
f
l−i
(d
s
+ x)
R
2r
E
r
E
f
l−i
(τ)dτ
,i = 0, .. .,l − 1.
(1)
From (1), the pro bability that there are l nodes in the
shaded area is
P[l, (r
E
,2r
E
)] =

R
2r
E
r
E
β(y)dy

l
e
−
R
2r
E
r
E
β(y)dy
l!
.
Then, the total D
I
’s conditional PDF can be expressed
as
f
D
I
|d
s
(x) =
∞
∑
l=1
P[l, (r
E
,2r
E
)]
l−1
∑
j=0
f
D
I
|(d
s
,l− j)
(x)p
j
,
(2)
where p
j
is the probability that the inte rfere I is
the j − th node within the right shaded area, which is
evaluate d as
p
j
= 1 /l, j = 0,1,. .. ,l − 1.
Similar to the derivation of D
I
’s PDF, D
S
’s PDF
can be solved as follows. Assign D
i
i = 1,2, ..., as
distances of ith vehicle to the tagged vehicle T , then,
F
D
1
(x) = 1 − e
−
R
x
0
β(z)dz
; f
D
1
(x) = βxe
−
R
x
0
β(z)dz
.
Since D
i
= D
i−1
− Y
d
i−1
(i = 2 , .. .,l), where Y
d
i−1
is
distance between i − 1th node and ith (i = 2,...,l)
node with distribution
F
Y
d
i−1
|d
i−1
(y|d) = 1 − e
−
R
d+y
d
β(z)dz
,
f
Y
d
i−1
|d
i−1
(y|d) = β(d + y)e
−
R
d+y
d
β(z)dz
,
f
D
i
(τ) =
Z
τ
0
f
D
i−1
(x) f
Y
d
i−1
|d
l−1
((τ− x)|x)dx.
Then, the total D
S
’s PDF can be expressed as
f
D
S
(x) =
∞
∑
l=1
P[l, (0,r
E
)]
l
∑
j=1
f
D
j
(x)
1
l
.
The CDF of I’s interference power P
I
received at
R could b e presented as
F
P
I
|d
s
(x) = Pr(P
I
< x|D
S
= d
s
)
=
Z
x
t
′
=0
Z
2r
E
−d
s
r
E
−d
s
f
P
r
|D
I
(t
′
) f
D
I
|d
s
(t)dtdt
′
,
f
P
I
|d
s
(x) =
Z
2r
E
−d
s
r
E
−d
s
f
P
r
|D
I
(x) f
D
I
|d
s
(t)dt,
(3)
Next, we evaluate effect of transmissions from
node I
′
at left hand side of T on R’s reception. In sim-
ilar way, CDF and PDF of I
′
’s interference power at R
can also be derived. Given D
S
is the distance between
T and R, the d istance between (l
′
− i) −th interfering
node and node R is denoted as D
I
′
(i = 0, .. ., l
′
− 1),
where r
E
+ D
s
≤ D
I
′
≤ 2 r
E
+ D
s
.
F
D
I
′
|(d
s
,l
′
−i)
(x) = P(D
I
′
< x|r
E
+ d
s
≤ D
I
′
≤ 2 r
E
+ d
s
)
=
F
l
′
−i
(x − d
s
)
R
2r
E
r
E
f
l
′
−i
(τ)dτ
.
The PDFs of the distances of the individual inter-
fering nod es to the receiver R a re obtained as
f
D
I
′
|(d
s
,l
′
−i)
(x) =
dF
D
I
(d
s
,l
′
−i)
(x)
dx
=
f
l
′
−i
(x − d
s
)
R
2r
E
r
E
f
l
′
−i
(τ)dτ
,
f
D
I
′
|d
s
(x) =
∞
∑
l=1
P[l, (r
E
,2r
E
)]
l−1
∑
j=0
f
D
I
′
|(d
s
,l− j)
(x)p
j
,
(4)
F
P
I
′
|d
s
(x) = Pr(P
I
′
< x|D
S
= d
s
)
=
Z
x
t
′
=0
Z
2r
E
+d
s
r
E
+d
s
f
P
r
|D
I
′
(t
′
) f
D
I
′
|d
s
(t)dtdt
′
,
Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc Networks for BSM Safety Applications: Parallel Numerical Solutions
and Simulations
603

f
P
I
′
|d
s
(x) =
Z
2r
E
+d
s
r
E
+d
s
f
P
r
|D
I
′
(x) f
D
I
′
|d
s
(t)dt,
(5)
Therefore, the total interference power accumu-
lated at th e receiver R if the interferences fro m two
sides occu r at same time (Ni et al., 2015):
f
P
I+I
′
|d
s
(x) =
Z
∞
0
f
P
I
|d
s
(x − t) f
P
I
′
|d
s
(t)dt,
Considering node T and node R are out of mutual
carrier sensing range, T ’s transmission could occur at
any time of T ’s transmission. According to (Yin et al.,
2013), the probability that a node in the shaded area
transmits during the vulnerable period of the trans-
mission from the tagged node T is evaluated as
p
t
= π
XMT
2(T − DIFS)
T
,
where π
XMT
is the steady-state pro bability that a ve-
hicle is in transmission state, which is derived in (Yin
et al., 2013). T is the time duration for one packet
transmission, DIFS is a time period of distributed
inter-frame space of IEEE 802.11p MAC.
Hence, considering three possible interference oc-
currenc e cases from two sides of the receiver with dif-
ferent probabilities (sin gle interferer from one side,
and two interferers from both sides), the total interfer-
ence powe r accumulated at the rec e iver R is the sum
of two independen t rando m variables from two sides
of R.
At the sam e time, we should consider the distri-
bution of interference power when there is no inter-
ference on both sides. In this case, the interference
power is the power of the basic noise, which is as-
sumed to be constant and expressed by P
I
n
. The CDF
and PDF of the interference power is:
F
P
I
n
|d
s
(x) =
(
1, i f x ≥ P
I
n
0, i f x < P
I
n
,
f
P
I
n
|d
s
(x) = dF
P
I
n
|d
s
(x)/dx. (6)
Thus, PDF of interference power on R is expressed
as
f
P
Σ
|d
s
(x) =
h
1 −e
−∆
R
ih
1 −e
−∆
L
i
f
P
I+I
′
|d
s
(x)
+ e
−∆
L
h
1 −e
−∆
R
i
f
P
I
|d
s
(x)
+ e
−∆
R
h
1 −e
−∆
L
i
f
P
I
′
|d
s
(x)
+ e
−∆
L
e
−∆
R
f
P
I
n
|d
s
(x)
where ∆
R
= p
t
R
2r
E
r
E
β(y)dy; ∆
L
= p
t
R
−r
E
−2r
E
β(y)dy.
Given D
s
= d
s
, the SINR at R is the ratio of two
random variables, an d its conditional PDF and CDF
could be presented as
f
SINR|d
s
(x) =
Z
∞
0
t · f
P
r
|d
s
(t · x) f
P
Σ
|d
s
(t)dt,
F
SINR|d
s
(x) =
Z
x
0
f
SINR|d
s
(t)dt,x > 0,
Then, the SINR’s PDF can be derived as
f
SINR
(x) =
Z
r
E
0
f
SINR|d
s
(x) f
D
s
(t)dt,
F
SINR
(x) =
Z
x
0
f
SINR
(t)dt,x > 0.
4 QoS AND CAPACITY
DERIVATION
Having derived SINR distribution, the following Qo S
metrics and capacity can be defined and evaluated.
4.1 QoS Derivation
First, the probability that receiver with distance ds
to the tagged node accepts the message successfully
if the measured conditional SINR is bigger than the
given threshold and the received signal is stronger
than the reception threshold P
th
, which is expressed
as
PRP(d
s
,θ) = Pr(SINR ≥ θ|d
s
) · Pr(P
r
≥ P
th
|d
s
)
= (1 − F
SINR|d
s
(θ))(1 −
Z
P
th
0
f
P
r
|d
s
(x)dx ), d
s
≤ d
ROI
.
(7)
Define region o f inter e st (ROI) of a safety application
as size of the geograph ical region covered by those
entities participating in the application, which is de-
noted as d
ROI
. Different kinds of safety applications
have different ROI sizes (Bai et al., 2006). Second,
packet reception ratio (P RR) (the percentage of re-
ceivers that are free fr om transmission errors) within
ROI can be evaluated as
PRR(d,θ) =
R
d
0
β(x)PRP(x,θ)dx
R
d
0
β(x)dx
,d ≤ d
ROI
.
4.2 Capacity Eva luation
The CDF of link capa c ity can be o btained from the
Shannon’s Theorem(Ni et al., 2015):
F
C
(x) = Pr(W log
2
(1+SINR) < x) = F
SINR
(2
x/W
−1),
where W is the bandwidth allocated to the observed
communication pa ir. The PDF of link capacity is as
follow:
f
C
(x) =
ln 2
W
· (2
x/W
) f
SINR
(2
x/W
− 1).
Then Expected link capacity is calcu la ted ac cording
to the f ollowing formula:
E(C) =
Z
∞
0
x f
C
(x)dx .
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
604

5 ACCELERATION OF
NUMERICAL COMPUTATION
5.1 Problem Description
Formula (7) o f PRP(d
s
,θ) can not be simplified, so it
needs to be solved by numerical calculation.
The F
SINR|d
s
(x) and F
P
r
|d
s
(x) need to be calculated
for computing the PRP , in which the computational
overhead o f F
P
r
|d
s
(x) c a n be neglected. Thus the for-
mula mainly took in F
SINR|d
s
(x) calculation.
5.2 Influence of f
D
I
|d
s
(x) and f
D
I
′
|d
s
(x)
on C omputational Efficiency
This section lists a ll the formulas involved in ca lc u-
lating F
SINR|d
s
(θ):
F
SINR|d
s
(θ) =
Z
θ
0
f
SINR|d
s
(k)dk,
f
SINR|d
s
(k) =
Z
∞
p
th
k
t · f
P
r
|d
s
(t · k) · ϕ
∑
(t, d
s
)dt,
ϕ
∑
(t, d
s
) = f
P
∑
|d
s
(t)
= [1 − e
−∆
R
][1 − e
−∆
L
]
Z
∞
0
f
P
I
|d
s
(t − m) f
P
I
′
|d
s
(m)dm
+ e
−∆
L
[1 − e
−∆
R
] f
P
I
|d
s
(t)
+ e
−∆
R
[1 − e
−∆
L
] f
P
I
′
|d
s
(t)
+ e
−∆
L
e
−∆
R
f
P
I
n
|d
s
(t).
Formula f
P
I
|d
s
(x) and f
P
I
′
|d
s
(x) are shown in (3)
and (5), re spectively. In order to calculate formulas
f
P
I
|d
s
(x) and f
P
I
′
|d
s
(x), f
D
I
|d
s
(x) and f
D
I
′
|d
s
(x) n eed to
be calculated first, their expressions are shown in for-
mulas (2) and (4).
Formulas ( 2) and (4) show that their co mputa-
tional time complexity is O(n
2
). If they are not sim-
plified, it will c ost a lot in the subsequent calcula-
tion process. Fortunately, by changing the summation
order, the formulas can be reduced to the following
forms, and their computationa l time complexity is re-
duced to O(n):
f
D
I
|d
s
(x) =
n
∑
l=0
f
D
I
|(d
s
,l−l)
(x)
n+1
∑
j=l+1
P[ j − 1,(r
E
,2r
E
)]p
j
,
p
j
= 1/ j
(8)
f
D
I
′
|d
s
(x) =
n
∑
l=0
f
D
I
′
|(d
s
,l−l)
(x)
n+1
∑
j=l+1
P[ j − 1,(r
E
,2r
E
)]p
j
,
p
j
= 1/ j
(9)
Variable n is the upper limit of the number of vehicles
in the communic a tion range.
5.3 Implementation Scheme of MPI and
Monte Carlo Method
5.3.1 Representation of Objective Funct ion
For the convenience of description, we represent the
following variables with c
1
, c
2
, c
3
:
c
1
=[1 − e
−∆
L
][1 − e
−∆
R
],
c
2
=e
−∆
L
[1 − e
−∆
R
],
c
3
=e
−∆
R
[1 − e
−∆
L
].
Let:
f
1
= f
1
(t, k) =t · f
P
r
|d
s
(t · k),
f
2
= f
2
(t, j) = f
P
r
|D
I
(t) f
D
I
|d
s
( j),
f
3
= f
3
(t, j) = f
P
r
|D
I
′
(t) f
D
I
′
|d
s
( j).
Expanding the formu la of F
SINR|d
s
(θ), the formula
(10) is ob ta ined:
F
SINR|d
s
(θ) =
Z
θ
0
f
SINR|d
s
(k)dk
= c
1
Z
θ
0
Z
∞
p
th
k
f
1
∞
Z
0
(
2r
E
−d
s
Z
r
E
−d
s
f
2
(t − m, j)d j
2r
E
+d
s
Z
r
E
+d
s
f
3
(m,l)dl)
dmdtdk + c
2
Z
θ
0
Z
∞
p
th
k
f
1
2r
E
−d
s
Z
r
E
−d
s
f
2
d jdtdk
+ c
3
Z
θ
0
Z
∞
p
th
k
f
1
2r
E
+d
s
Z
r
E
+d
s
f
3
d jdtdk
+ e
−∆
L
e
−∆
R
Z
θ
0
Z
∞
p
th
k
t · f
P
r
|d
s
(t · k) · f
P
I
n
|d
s
(t)dtdk.
(10)
Let, F
SINR|(d
s
,P
I
n
)
(θ) =
R
θ
0
R
∞
p
th
k
t · f
P
r
|d
s
(t · k) ·
f
P
I
n
|d
s
(t)dtdk. Noise is the only sou rce of interference
at this time. Because its power P
I
n
is assumed to be
constant, so the value of the formula can b e obtained
by using the definition of SINR. Its calculation time
is constant, so it is not considered in subsequent nu-
merical calculatio n.
And use BI, RI, LI, NI to represent
c
1
θ
Z
0
∞
Z
p
th
k
f
1
∞
Z
0
(
2r
E
−d
s
Z
r
E
−d
s
f
2
(t − m, j)d j
2r
E
+d
s
Z
r
E
+d
s
f
3
(m,l)dl)dmdtdk,
c
2
Z
θ
0
Z
∞
p
th
k
f
1
Z
2r
E
−d
s
r
E
−d
s
f
2
d jdtdk,
Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc Networks for BSM Safety Applications: Parallel Numerical Solutions
and Simulations
605

c
3
Z
θ
0
Z
∞
p
th
k
f
1
Z
2r
E
+d
s
r
E
+d
s
f
3
d jdtdk,
e
−∆
L
e
−∆
R
Z
θ
0
Z
∞
p
th
k
t · f
P
r
|d
s
(t · k) · f
P
I
n
|d
s
(t)dtdk.
After the tran sformation , as shown in Equation (11).
F
SINR|d
s
(θ) = BI + RI + LI + NI. (11)
Formula (10) shows, tha t solving F
SINR|d
s
(θ) n eed
to calculate multidimensional integrals. The real-
izability and efficiency of numerical methods is a
very important problem. The Monte Carlo integration
method can calculate multidimensional integrals, and
the integration sp eed is only related to the number of
sampling poin ts, and is independent of the dimension.
So we use Monte Carlo integral to solve the m ultidi-
mensional integral problem in this paper.
In order to speed up the solution, we use MPI to
solve PRP(d
s
,θ).
Figure 2 shows the process of dividing/calculat-
ing, synchronizing and r e ducing for the MPI. We use
the process numbered 0 as the main process and use
N processes to calculate PRP(d
s
,θ). Figure 2 gives
the flow of the entire program.
First, the main process calculates the parts that are
indepen dent of the integral functio n of the integral ac-
cording to the input parameters, such as c
1
, c
2
, c
3
, and
the integral area volume v
1
, v
2
, v
3
;
Secondly, according to the total Monte Car lo num-
ber of samp le s N, the mean values and the errors of
the LI, BI and RI are calculated, respectively;
Finally, the main process calculates the value
PRP(d
s
,θ) based on the value of F
SINR|d
s
(θ).
The detailed implementation is presented by
pseudo code in Algorithm 1.
Algorithm 2 shows the process of MPI parallel
coupled with Monte Carlo numerical method to esti-
mate the mean E( f ; N) and the e rror σ
2
(E; N), by in-
creasing sampling points to ensure the erro r σ
2
(E; N).
The detailed implementation is presented by
pseudo code in Algorithm 2.
5.4 Experiments
We d evelop the experiments based on MPI cluster
which include 20 cores CPU for numerical integra-
tion. The hardware of nodes in MPI cluster is Intel
E5-26 60 2.60GHz CPU and 32GB memor y, and each
node in cluster is organize d by IntelMPI 5. 1.2. Our
developed numerical programs create one MPI pro-
cess which is allocated 64MB local memory to com-
pute for each core. The programs apply the G N U nu-
merical computing library to generate random num-
ber and calculate integral, the seed of random num-
bers in each process should be different f or various
Algorithm 1: Scalable algorithm for VANET QoS analysis.
Require: QoS analysis problem, the total number of
samples N of Monte Carlo , the number of MPI
calculated processes k;
Ensure: Numeric al solution, Monte Carlo error eps;
1: Detach QoS analysis problem into three part LI,
RI, BI ;
2: Calculate the part that is independent of the inte-
grands of the integral;
3: The numbe r of samples to be calculated for each
process is divided equally into N/k;
4: for each subproblem ∈ LI, RI,BI do
5: Use the k process call algorithm 2 to get the
summation value sum;
6: Accumulate the sum of these k proc esses and
get the final summation value sum
f inal;
7: Sum
f inal divided by N, get the mean of the
integrands;
8: The main process calculates the error eps and
outputs it;
9: end for
Algorithm 2: Parallel algorithm of Monte Carlo method.
Require: Th e integrands function f , the total number
of samples N of Monte Carlo , the number of MPI
calculated processes k;
Ensure: The estimate of the integral E( f ; N), the er-
ror on this e stima te σ
2
(E; N);
1: for i = 1 to k do
2: Generate ⌈N/k⌉ sampling points P
i
by each
process i;
3: Compute f (p
i, j
) for each sampling point p
i, j
∈
P
i
sequentially in e ach parallel proce ss i;
4: Keep a ll result of f (p
i, j
) in shared memory;
5: end for
6: Calculate the average
ˆ
f of f (p
i, j
), 1 ≤ i ≤ k,1 ≤
j ≤ ⌈N/k⌉;
7: Calculate the estimated integral E( f ; N) =
V
ˆ
f and estimated absolute error σ
2
(E; N) =
N
2
V
2
∑
k
i=1
∑
⌈N/k⌉
j=1
( f (p
i, j
) −
ˆ
f )
2
;
cycles. Parameters are set as follows:SINR value θ
is set as 4, and signal propagation distance d
s
is set
as 50. The average time of single samplin g for LI,RI
and BI is 105.07 , 105.18 and 736462 ms, respec tively.
Due to the convolution, the sampling of BI spe nds
thousands of times than LI, RI.
The experiments need large enough sampling
points to ensure accuracy for Monte Carlo integration.
Table 1 is the statistics of integration errors of LI,
RI and BI parts under different number of sampling
points to obtain QoS metric PRP. The number of sam-
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
606

Themainprocessreceivesinput
parameters:ds,ɽ,N
Overallprocess
Themainprocesscalculatesc1,
c2,c3andtheintegralarea
volumeofLI,RI,BI˖
v1,v2,v3
CalculatethemeanvalueofLI
underthetotalnumberof
samplesN
CalculatethemeanvalueofRI
underthetotalnumberof
samplesN
CalculatethemeanvalueofBI
underthetotalnumberof
samplesN
ThemainprocessgetsXXXX
ThemainprocessgetsthePRP
Averagenumberofsamplesaccordingto
thenumberofprocessesk
LI,RIandBIspecificprocesses
Calculate:
f1(orf2orf3)
underN/k
Calculate:
f1(orf2orf3)
underN/k
Synchronize
Reducethelocal_sumtogettheglobal_sum
Calculatethemeanoftheintegrandvalue
undertheN
Calculateerror
0
...
NͲ1
S
INR| ds F
Figure 2: MPI program description.
pling points applied in our exper iment is 1000, 3000
and 5000 times, respectively. The results in Table 1 is
the average error of 7 times of eva luation. As shown
in table 1, the estima te d absolute error decr eases a s
the number of sampling points N. However, more
sampling means mo re computing resources.
Table 1: Integral error under different number of samples.
LI RI BI
The error of 1000 sampling 1.49e-05 6.55e-0 5 1.38e-04
The error of 3000 sampling 8.02e-06 1.91e-05 3.41e-05
The error of 5000 sampling 8.47e-06 1.12e-05 2.13e-05
The total running time of the pr ogram under 1000,
3000 and 5000 samples is 20.08, 41.82 and 61.36
hours, respectively. The ru nning time means the real
time of program running in parallel by 20 cores. The
process time increases significantly with the number
of sampling, since it’s important to trade off com-
puting resources with evaluation accu rracy when a p-
plied the model based on SINR. Thus, we employ the
Monte Carlo integration with 3000 sampling points to
compute various SINR settings.
6 COMPARISON OF
THEORETICAL AND
SIMULATION RESULTS
To validate the new proposed theoretical analysis, the
Matlab and C++ are deployed for theoretical com-
putations w ith MPI Monte Carlo method, and NS2
is deployed for network simulations. We con sid er a
specific DSRC VANET in highway for safety mes-
sage dissemination. Each vehic le in the network is
equippe d with DSRC capability. The communication
network parameters as set as follows. W = 10MHz
(DSRC channel b andwidth), P
t
= 0.28183815Watts
(transmission power of each node), P
th
= 2 .28289e −
11Watts (carrier sensing power strength or clear chan-
nel assessment sensitivity), d
0
= 100meters (the ref -
erence distance for the far-zone), η = 7.29e − 10,
P
I
n
= −99dBm, r
E
= 300m (average sensing range),
σ = 16s (Slot time), DIFS = 64s, CW = 15, T
H1
=
40s (PHY preamble), T
H2
= 272bits (MAC header) ,
T
H3
= 4s, (PLCP he a der), T
c
= 0.1s (Packet gen-
eration interval), R
d
= 24Mbps (Data r ate), PL =
200bytes (Packet length), α = 2 (path loss expone nt),
Fading Parameter m for r <50m, 50m<r<150m and
r ≥ 150m, th e value is 3, 1.5 and 1, respectively. The
communication nodes a re Poisson distributed with
piecewise constant densities on highway with length
of 1000m on each of the crossing roads.
The density distribution, as function of distances
(X) to a tagged veh icle , in the case of x ≤ 50m,
50m < x ≤ 10 0m and x > 100m, is 3β
av
/2, β
av
and
β
av
/2, respectively. where β
av
is a constant average
road de nsity durin g a certain time period . Figure 3
and 4 shows the CDF and the PDF of SINR at the
receivers with different values of the signal propa-
gation distance d
s
and SINR thresholds, respectively.
β
av
=0.1 vehicles/me ter. I t can be seen from Fig. 3,
the farther propagation distance, the smaller received
signal power, and accord ingly the sm a ller SINR at re-
ceiver obtaine d. Thus, w e can see the CDF’s increas-
ing trends with the propagation distance equaling with
50m, 250m, 350m and 450m, i.e., the SINR at the re-
ceiver with propagatio n distance 50m could be largest
compare d with the other pr opagation distance while
the SINR a t the receiver with 450m could be small-
est. As with the Fig.3, it is also observed the PDFs
varying trends with propagation distance with 50m,
250m, 350m and 450m in Fig. 4. It first increase
with the propagation distance when SINR is relatively
small and then decrease with SINR. The PDFs vary-
ing trends indicate that the closer the transmission dis-
tance is, the greater the SINR value at the receiving
node has.
Figure 5 and 6 sh ows the PRP and PRR at the re-
ceivers with different values of the signal propagation
distance d
s
and SINR thresholds. It is shown from
Figure 5 and 6 that analytical results practically coin-
cide with the simulation results, which verify correct-
ness of th e proposed model. Both PRP and PRR with
short propagation distance have better performance
than that with long distance, i.e ., the performance of
PRP a s well as PRR with propagation distance 50m is
better than that of 150m, and the performance of PRP
as well a s PRR with 150m is better than that of 250m,
and hence with 350m or 450m. On the other hand, it is
observed that fixed propagation distance 50m, 150m,
250m and 350m, the performa nce of PRP as well as
Accelerating Interference-based QoS Analysis of Vehicular Ad Hoc Networks for BSM Safety Applications: Parallel Numerical Solutions
and Simulations
607

PRR is decreasing with SINR thresholds, respectively.
because the modulation and coding mechanism of the
receiver could be better applied at the small SINR
threshold, and thus the packet loss is smaller. Fig-
ure 7 shows link c a pacity of the local VANET . From
figure 7 we can see that the ra nge of link capacity is
among [78, 105] Mbps with high probability.
0 200 400 600 800 1000 1200 1400
SINR
0.0
0.2
0.4
0.6
0.8
1.0
FSINR
Figure 3: Conditional CDF of SINR at receiver.
0 200 400 600 800 1000 1200 1400
SINR
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
fSINR
x10
2
Figure 4: Conditional PDF of SINR at receiver.
7 CONCLUSION
In this pape r, a start-of-the- art framework based on
SINR for safety message broadca st are proposed,
which is mor e practical and has the ability to analyze
link capacity and reliability metrics of PRR and PRP.
Assumptions such as NHPP vehicle distributions and
Nakagami fadin g channel model with path loss make
proposed model more practical and general. The de-
tailed description a bout the assumptions and deriva -
tion of the SINR is in section 2 to 4. However,
the model based on SINR introduces complex equa-
tion which spends sign ifica nt computation resources
to evaluate, Monte Carlo and MPI methods are pro-
posed for accelerate computa tion process. At the end
Figure 5: PRP of VANET with communication range.
Figure 6: PRR of VANET with communication range.
0 20 40 60 80 100 120 140
Link capacity (Mbps)
0.00
0.50
1.00
1.50
2.00
pdf
x10
2
Figure 7: Link capacity of VANET broadcast.
of paper, we analyze the computation efforts and eva l-
uation error of model by several exper iments and val-
idate its correctness by simulation. The ana lytical re-
sults give the numerical CDF and the PDF of SINR
at the receivers with signal pro pagation distance d
s
and SINR thresholds, respectively. The results could
further be utilized by the engineer to m e asure the
VANET communication system, and then optimize
the system parameters which should be the future re-
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search direction. The analytical results of PRP a nd
PRR practically coinc ide with the simulation results,
which verify corre ctness of the proposed m odel.
ACKNOWLEDGEMENT
This work is supp orted by the Natio nal Nature Sci-
ence Foundation of China under Grant 61572150, the
Fundamental Research Funds for the Central Univer-
sities of DU T, No.DU T17RC(3)097.
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