2276 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO.

3, MARCH 2016

A Stochastic Geometry Model for Multi-Hop

Highway Vehicular Communication
Muhammad Junaid Farooq, Student Member, IEEE, Hesham ElSawy, Member, IEEE,
and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—Carrier sense multiple access (CSMA) protocol is traffic safety administration (NHTSA) reports that an average of
standardized for vehicular communication to ensure a distributed 90 deaths and more than 6000 injuries occur every day, across
and efficient communication between vehicles. However, several the US, due to car accidents [2]. Hence, there is a high demand
vehicular applications require efficient multi-hop information dis-
semination. This paper exploits stochastic geometry to develop to incorporate an automated and intelligent system to compen-
a tractable and accurate modeling framework to characterize sate for the human error, which is considered the major cause
the multi-hop transmissions for vehicular networks in a multi- for accidents. In this context, vehicles can communicate with
lane highway setup. In particular, we study the tradeoffs between each other to alert drivers, or even take automated actions, to
per-hop packet forward progress, per-hop transmission success avoid collisions and save lives. Vehicular communication can
probability, and spatial frequency reuse (SFR) efficiency imposed
by different packet forwarding schemes, namely, most forward also offer potential gains from an economical perspective. A
with fixed radius (MFR), the nearest with forward progress (NFP), recent report by The Economist estimated the total cost of traf-
and the random with forward progress (RFP). We also define a fic jams across four countries (US, UK, Germany, and France)
new performance metric, denoted as the aggregate packet progress to be 200 billion dollars [3]. In this case, vehicular networking
(APP), which is a dimensionless quantity that captures the afore- can be exploited to improve navigation systems and alleviate
mentioned tradeoffs. To this end, the developed model reveals the
interplay between the spectrum sensing threshold (ρ t h ) of the traffic jams, i.e., to develop real time and context-aware navi-
CSMA protocol and the packet forwarding scheme. Our results gation systems that collect real-time traffic information, via the
show that, contrary to ALOHA networks, which always favor vehicular network, to take appropriate navigation decisions [4].
NFP, MFR may achieve the highest APP in CSMA networks if ρ t h A typical vehicular communication system consists of
is properly chosen. three different links or modes of operation, namely, vehicle-
Index Terms—Interference characterization, multi-hop CSMA, to-vehicle (V2V), vehicle-to-roadside (V2R), and roadside-
stochastic geometry, vehicular communication. to-roadside. The V2V communication, also referred to as
inter-vehicle communication (IVC) [5], is an enabling tech-
I. I NTRODUCTION nology for a variety of applications such as adaptive traffic
control, coordinated braking, emergency messaging, peer-to-
T HE world wide interest to develop intelligent transporta-
tion systems (ITS), that improve the efficiency and safety
of transportation, increases the research focus on vehicular net-
peer networking for infotainment services, and automatic toll
collection. Further, IVC communication is preferred over V2R
working. While ITS cover all forms of transportation, including in high mobility environments as well as for applications with
road, rail, air, and water transportation, communication for road strict delay constraint such as the safety based dedicated short-
networks is of special interest due to its uncoordinated and less range communications (DSRC) [6]. According to the IEEE
controlled nature when compared to the other transportation standard, IVC is coordinated via carrier sense multiple access
systems. While planes, trains, and ships are usually driven by (CSMA) medium access control (MAC) protocol. CSMA is a
highly trained personnel, cars are driven by public varying in contention based spectrum access technique that is controlled
their skills, which raises serious safety issues for road trans- by means of random backoff counters. That is, each transmitter,
portation. For instance, according to the US census bureau, the that intends to transmit a packet, generates a random backoff
US has a round figure of 10 million road accident per year with timer which is decremented if the channel is sensed idle and
an average of 1.5% fatality rate [1]. The US national highway freezed if the channel is sensed busy. Only transmitters with
elapsed backoff timers can transmit. Hence, a transmission is
Manuscript received February 28, 2015; revised July 15, 2015 and October only initiated in an idle channel state to avoid packet collisions.
16, 2015; accepted November 10, 2015. Date of publication November 19,
2015; date of current version March 8, 2016. This work was supported by the The idle/busy state of the channel is determined by the carrier
King Abdullah University of Science and Technology (KAUST). The associate sensing threshold (ρth ) [7], which is a vital design parameter
editor coordinating the review of this paper and approving it for publication was for the CSMA protocol. The sensing threshold imposes a cru-
X. Zhou.
cial tradeoff between the transmission success probability or
The authors are with the Electrical Engineering Program, Computer,
Electrical, and Mathematical Sciences and Engineering (CEMSE) Division, simply the success probability, denoted by P, and the spatial
King Abdullah University of Science and Technology (KAUST), Thuwal frequency reuse (SFR) efficiency. A lower sensing thresh-
69000, Saudi Arabia (e-mail: muhammadjunaid.farooq@kaust.edu.sa; old enforces larger distances between concurrent transmitters,
hesham.elsawy@kaust.edu.sa; slim.alouini@kaust.edu.sa).
Color versions of one or more of the figures in this paper are available online
which decreases interference and increases the success prob-
at http://ieeexplore.ieee.org. ability, on the expense of degrading the SFR, and vice versa.
Digital Object Identifier 10.1109/TWC.2015.2501817 Hence, the sensing threshold (ρth ) should be carefully tuned to
1536-1276 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2277

expressions for the per-hop success probability as well as the

FP. Our developed model captures the tradeoffs among the suc-
cess probability, FP, and the SFR efficiency of each forwarding
scheme. In order to unify these three performance metrics, we
define the aggregate packet progress (APP).
Our results show that NFP achieves the highest APP in net-
works that require high link quality1 per transmission but with
relaxed delay constraint. On the contrary, MFR achieves the
Fig. 1. Message propagation in multi-lane highways using the MFR forwarding highest APP in networks with tight delay constraint and relaxed
scheme. A test transmitting vehicle A forwards the packet to a vehicle B that link quality. It is worth noting that ALOHA networks always
results in the most forward progress inside its transmission range. This process
continues until the packet reaches its destination.
favor NFP due to the high interference levels in the network
[8], [9]. In contrast, sufficient interference protection can be
achieved in CSMA networks by proper manipulation of the
balance the tradeoff between the success probability and SFR sensing threshold, which makes the MFR forwarding scheme
efficiency, thus achieving efficient local communication. preferable is some cases. The main contributions of this paper
Several vehicular applications require going beyond the are summarized as:
local communication and efficiently disseminating informa- 1) We develop a novel tractable framework to model inter-
tion across the roads via multi-hops, as shown in Fig 1. For vehicle communication in a multi-lane highway network.
instance, accidents on highways can be immediately reported 2) We compare between various abstraction models used for
to the nearest ambulance or emergency center using the multi- highway vehicular networks and show that the widely
hop vehicular network. Also in the case of highway congestion, used line abstraction model does not accurately model the
it is important to disseminate the information far enough so success probability for wide highways.
that other vehicles can take appropriate navigational decisions 3) We study different packet forwarding schemes for multi-
and change the selected trajectory to alleviate the congested hop communication in CSMA based IVC and compare
sector. Therefore, packets need to be efficiently propagated their performance under different traffic conditions.
from one hop to another until they arrive at the destination. 4) We define the aggregate packet progress (APP) metric
There are different packet forwarding schemes, that impose a which captures the tradeoff between success probability,
tradeoff between packet forward progress (FP) and per-hop suc- FP, and SFR efficiency.
cess probability. In particular, we focus on the most forward 5) We show that we can select a packet forwarding scheme
with fixed radius (MFR), the nearest with forward progress with an appropriate carrier sensing threshold to maximize
(NFP), and the random with forward progress (RFP) forward- the APP under specific QoS constraints.
ing schemes. As it is clear from the protocol names, the MFR
favors relaying the packets to a distant vehicle to maximize the
FP on the expense of lower per-hop success probability. On the II. R ELATED W ORK
contrary, the NFP favors communicating with the nearest vehi- Balancing the tradeoff between the success probability and
cle to attain a high per-hop success probability on the expense the spatial frequency reuse is a major challenge in wireless
of low FP. The RFP achieves a midpoint performance between ad hoc networks. In the literature, different metrics are used
the MFR and the NFP. It is worth mentioning that there is an to capture this fundamental tradeoff. For instance, transmission
interaction between the carrier sensing threshold and the packet capacity metric is used in [7], spatial throughput is used in [10],
forwarding protocols. For instance, to have an acceptable suc- and spatial reuse is used in [11] to quantify the number of suc-
cess probability, the MFR requires more conservative ρth due cessful transmissions per unit area. These metrics are mainly
to the large hop distance as compared to the NFP which can used to gain insights into the system behavior and to optimize
sustain an aggressive ρth due to the smaller hop distance. This the network performance. In addition to the success probabil-
directly has an impact on the end-to-end transmission delay ity and the spatial frequency reuse, the effect of the forward
or equivalently the throughput since the MFR requires lesser progress in multi-hop ad hoc networks is incorporated into the
number of transmissions to reach a particular destination as analysis in [9], [10]. However, [9], [10] consider conventional
compared to the NFP which requires more number of transmis- 2-D ad hoc networks. Note that vehicular ad hoc networks
sions. Note that, an efficient packet forwarding scheme should (VANETs) differ from general ad hoc networks by having their
achieve the best balance between the success probability, packet own characteristics (e.g., geographical limitation to roads) that
FP, and the SFR efficiency imposed by both the forwarding should be incorporated into the modeling framework.
scheme and the CSMA parameter ρth , while maintaining a In the context of VANETs, the authors in [12] and
particular end-to-end transmission delay. [13] develop an analytical model to maximize the network
In this paper, we focus on the IVC aspects in vehicular throughput2 . However, the presented models are based on
networks in a multi-lane highway scenario and study three dif- ALOHA spectrum access protocol. Although CSMA based
ferent packet forwarding schemes. In particular, we develop a 1 The required link quality is quantified by the signal-to-interference-plus-
novel analytical framework to model CSMA coordinated IVC noise-ratio required for correct signal reception.
in a highway setup using concepts from stochastic geome- 2 Note that the throughput is directly linked to the forward progress in
try and point process theory. We obtain simple yet accurate VANETs [14].
2278 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

IEEE 802.11p protocol is the standardized access protocol for lane. Each vehicle transmits with a fixed power P, or equiv-
vehicular communication, it is shown in [15] that the perfor- alently has a fixed transmission range Rt . The signal power
mance of the IEEE 802.11p approaches the performance of decays with the distance according to the power law propa-
ALOHA protocol at sufficiently high node densities due to gation model with a path-loss exponent η > 1. Beside path
backoff timer ties3 The ALOHA based network models in [12], loss attenuation, the signal power experiences fading which
[13], [15] do not aptly capture the network performance when is assumed to be exponential with mean 1/μ. Therefore the
CSMA protocol operation is intact. Note that alleviating the power received from a transmitter located at vi to the receiver
backoff timer ties problem in the IEEE 802.11p to achieve an at v j is Ph i j vi − v j −η , where h i j represents the channel
intact CSMA spectrum access is viable by means of adaptive fading between vi and v j and . represents the Euclidean
contention window size techniques [16], [17]. norm. We assume that during each time slot, the network
VANETs with an intact CSMA spectrum access have been topology remains the same and is independent of other time
studied in [15], [18]–[20]. In [18], the maximum possible spa- slots. This is a reasonable assumption since the speed of
tial reuse in a VANET is investigated. The authors in [15], [19], packet transmission including the delays due to backoff is
[20] conduct performance and reliability analysis for vehicular much faster than the speed of vehicles. This is formally stated
networks. Most of the related works on VANETS using CSMA as:
protocol are based on the single line abstraction (SLA) model. Assumption 1: We assume that the network topology
The SLA model combines multiple highway lanes into a sin- remains constant for the duration of each time slot and changes
gle lane with the aggregated traffic, which may not be accurate independently from one time slot to the other.
for wide highways as well as for high traffic intensity [21]. The Remark 1: According to [22], the maximum average wait
models developed for CSMA based VANETs in the literature time before transmission is estimated to be 264μs and the total
are mostly limited to local vehicular communication. packet delivery time is in the order of milliseconds. Therefore
This paper focuses on VANETs with an intact CSMA spec- on an 80 km/h highway, a vehicle is displaced by much less
trum access protocol and develops a novel analytical paradigm than 1 m during a single transmission. This makes it reasonable
for multi-hop communication in a multi-lane highway. The to ignore the effect of mobility during contention and packet
developed model does not rely on the SLA model and cap- transmission time.
tures the effect of the number of traffic lanes and the inter-lane We consider a signal capture model, which assumes that a
separation. The paper also studies the tradeoff imposed by the packet can be correctly decoded by the receiver iff the signal-to-
CSMA sensing threshold and the packet forwarding scheme on interference-plus-noise-ratio (SINR) exceeds a target threshold
success probability, packet FP and SFR efficiency. To this end, T . All vehicles contend for channel access according to the
the developed analytical framework is used to select the for- slotted CSMA protocol. For the CSMA contention process, we
warding scheme and the carrier sensing threshold in order to have the following assumption:
optimize the performance tradeoffs. Assumption 2: We assume that no two vehicles in the same
contention domain have the same backoff counter value and
III. S YSTEM M ODEL hence the CSMA behaviour of IEEE 802.11p is intact.
Remark 2: The basic operation of the IEEE 802.11p is
In this section, we describe the network model, discuss similar to the conventional operation of the CSMA protocol
the considered packet forwarding schemes, and explain the except for the higher collision probability within the contention
methodology of analysis. domain due to discrete backoff counter values [23]. For this rea-
son, several efforts have been invested to adapt the contention
A. Network Model window (i.e., backoff timer size) to alleviate this problem [16],
We consider a multi-lane highway network constituted from [17]. Since the framework proposed in this paper focuses on
N parallel traffic lanes separated by a fixed distance d. It the spatial domain uncertainties and abstracts time dynamics,
is assumed that each traffic lane has an infinite length and we can assume that the contention domain collision problem is
is populated by vehicles according to an independent and solved by one of the techniques proposed in the literature (e.g.,
homogeneous 1-D Poisson point process (PPP) of intensity λ [16], [17]).
vehicles/km per lane, i.e., i = {vi j ; j = 0, 1, 2, . . .}, where The carrier sensing threshold ρth of the CSMA protocol is a
i = 1, . . . , N is the lane index and vi j represents the loca- system wide design parameter used by all vehicles. From a geo-
tion of the j th vehicle on the i th traffic lane. Note that for metric perspective, ρth can be directly translated to a sensing
ease of analysis, we assume a homogeneous traffic in which range, also denoted as the contention domain, represented by
each lane has the same intensity of traffic. A snapshot of the Rs = ( ρPth )1/η . Note that the transmission range or neighbour-
network model is shown in Fig. 1. For a test vehicle under anal- hood domain is similarly expressed as Rt = ( ρmin )
P 1/η
where
ysis, there are N R traffic lanes on its right side and N L traffic ρmin is the receiver sensitivity. The sensing range affects the
lanes on its left side where N = N R + N L + 1. The lane con- spatial frequency reuse because the CSMA protocol does not
taining the vehicle under analysis is referred to as the central allow the same channel to be used by two vehicles in the same
3 IEEE 802.11p is similar to the IEEE 802.11 protocol except for the dis- contention domain, where the contention domain of the j th
crete backoff timer values which reduces the contention time on the expense of vehicle on the i th lane (vi j ) is defined as Nvi j = {vmn : Pvi j −
having a flawed CSMA operation at high node density. vmn −η ≥ ρth , ∀m, n}. If vi j is in the contention domain of vmn ,
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2279

then using the assumption of reciprocity4 in wireless channels,

the converse is also true. Thus, for notational brevity, we drop
the subscripts of vi j , and hereafter refer to the contention
domain of a typical vehicle as N. The average number of vehi-
cles in the contention domain of a vehicle with N R traffic lanes
on its right side and N L traffic lanes on its left is given by:
⎛ ⎞
NR  NL 
E [|N|] = 2λ ⎝ Rs + Rs2 − (id)2 + Rs2 − (jd)2 ⎠ ,
i=1 j=1
(1)

where |.| represents the set cardinality. In a similar manner, we

can define the transmission neighbourhood5 of a typical node
as Nvi j = {vmn : Pvi j − vmn −η ≥ ρmin , ∀m, n}. The average
number of neighbours of a transmitting vehicle with transmis-
sion range Rt and with N R and N L traffic lanes on its right and
left side respectively is, therefore:
⎛ ⎞
NR  NL 

E [|N|] = 2λ ⎝ Rt + Rt2 − (id)2 + Rt2 − (jd)2 ⎠ .
i=1 j=1
(2)

B. Packet Forwarding Schemes

From a transmitter’s perspective, there are several strategies
to select the relaying vehicle from its neighbors. The most sim-
ple strategies are to transmit to either the nearest node, the
farthest node, or to a random node inside the transmission Fig. 2. Modeling approaches for multi-lane highways. The simulations in (d) &
range [9]. We assume that the packet’s destination is located (e) show that the single-lane model is not an accurate model for multi-lane
far enough such that the progress can be defined as the horizon- highways. The deviation of the single-lane model from the multi-lane and 2D-
tal distance moved along the highway towards the receiver. We PPP model increases as the highways become wider.
only allow FP in which the message direction is known (e.g.,
the location of the nearest emergency center in case of acci- 3) Random With Forward Progress (RFP): In this forward-
dents). The different packet forwarding schemes are described ing scheme, every potential receiver, that results in forward
as follows: packet progress, has an equal probability of being selected
1) Most Forward With Fixed Radius (MFR): In this for- by the protocol. This forwarding scheme achieves a midpoint
warding scheme, a typical transmitting node forwards the performance between the MFR and the NFP.
packet to the neighbour that results in the most FP within its
transmission range. MFR maximizes the FP on the expense of
C. Modeling Approaches
low success probability due to the imposed large transmitter-
receiver spacing. Note that MFR requires a conservative sensing There are three main approaches to model vehicles on a
threshold (i.e., ζ  Rs /Rt ≥ 1) such that the sensing range can multi-lane highway as illustrated in Fig. 2. The first approach,
provide sufficient interference protection for the receiver. Note which is widely used in literature, is known as the single lane
also that a conservative sensing threshold implies low spatial abstraction (SLA) model or simply the line abstraction model
frequency reuse. shown in Fig. 2(a) in which all the traffic lanes are merged into
2) Nearest With Forward Progress (NFP): In this forward- a single lane with the aggregated traffic intensity. This greatly
ing scheme, a typical transmitting node forwards the packet simplifies the analysis. However, it has been shown in [21] that
to its closest neighbour. This would increase the per-hop suc- the line abstraction model cannot accurately characterize the
cess probability on the expense of minimizing the FP. However, performance of highway vehicular networks when the highway
the NFP forwarding scheme can sustain an aggressive sensing is significantly wide and under high traffic intensity. The second
threshold (i.e., ζ  Rs /Rt  1) due to the small link distance approach is to consider that the traffic is restricted into individ-
(i.e., short transmitter-receiver separation) which improves the ual lanes separated by a fixed inter-lane distance, as illustrated
desired signal received power. in Fig. 2(b). The third approach models the highway traffic
4 The wireless channel between vehicles is assumed to be symmetric w.r.t.
using a 2-D PPP restricted to the area covered by the highway,
the roles of transmitter and receiver.
as illustrated in Fig. 2(c). Compared to the second approach, the
5 Vehicles inside the transmission range of each other are considered to be third approach gives one more degree of freedom on the loca-
neighbours. tion of the vehicles. However it is not tractable. A simulation
2280 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

study comparing the results of the three approaches under the scheme, the set  ˜ i \v0 ⊆ i denotes the set of active transmit-
RFP forwarding scheme (see Fig. 2(d) and 2(e)) shows that the ters, excluding the intended transmitter v0 , which is obtained
multi-lane approach and the 2-D PPP approach give approxi- after dependent thinning of the original PPP due to the CSMA
mately the same results while the SLA model overestimates the contention. vi j ∈ R2 is the location of the j th interfering vehi-
probability of successful transmission. The same conclusion is cle on the i th traffic lane. Finally, h o and h i j are i.i.d. random
drawn for the MFR and NFP forwarding scheme. Therefore, we variables representing the channel fading between the test trans-
use the second approach to conduct the analysis in this paper. mitting vehicle and the selected receiver, and between the
receiver and i j th interfering vehicle respectively, and κ is the
noise variance. From (3), it can be directly observed that SINR
D. Methodology of Analysis
is a random variable that includes several uncertainties, i.e.,
Our objective is to study the tradeoff between the suc- the vehicle locations, the channel gains, the CSMA contention
cess probability, forward packet progress, and the spatial fre- based access, as well as the transmitter-receiver separation. The
quency reuse efficiency imposed by the forwarding schemes complementary cumulative distribution function (ccdf) of the
and the CSMA parameter ρth . Hence, our main performance SINR defines the per-hop success probability, which can be
metrics are per-hop success probability, the FP, and the aggre- calculated as follows (see equation (5) in [24]):
gate packet progress. Before conducting the main analysis, 
three auxiliary parameters need to be calculated, namely, the Pχ = P S I N Rχ > T ,
    
transmitter-receiver distance distribution, the intensity of con- T μκr η T μr η
current transmitters, and the Laplace transform (LT) of the = exp − L Iagg frχ (r )dr, (4)
r P P
probability density function (pdf) of the aggregate interference.
While the transmitter-receiver distance distribution is necessary where L Iagg (.) is the LT of Iagg 6 . Equation (4) shows the
to characterize the desired signal power at the test receiver, auxiliary metrics that are required to evaluate the per-hop suc-
the intensity of concurrent transmitters is required to charac- cess probability, namely, the transmitter-receiver separation pdf
terize the interference power at the test receiver. Note that the frχ (.) and the LT of the aggregate interference Iagg . Note that
interference power is characterized via its LT. the LT of the aggregate interference is a function of the intensity
of concurrent transmitters (or interferers).
IV. SINR C HARACTERIZATION
Due to the shared nature of the wireless medium, SINR is A. Distance Distribution
a major performance metric. The per-hop transmission is con-
Since the receiver selection is based on the underlying packet
sidered successful (i.e., correctly decoded by the receiver) if
forwarding protocol, the transmitter-receiver distance distribu-
the SINR is above a predefined threshold. The SINR can be
tion is different for MFR, NFP, and RFP. For the MFR for-
written as:
warding scheme the transmitter-receiver distance distribution is
−η obtained via the following lemma:
Ph o rχ
SINRχ = , (3) Lemma 1: The distance distribution between the transmitter

N   −η
κ+ Ph i j vi j  and receiver for the MFR forwarding scheme, fr M , is given in
i=1 vi j ∈
˜ i \v0 (5), where E [|N|] is given in (2), A(.) is given in (30), which

  is shown at the bottom of page 13, and A (.) is the derivative of
Iagg A(.), which is always negative.
where χ ∈ {M, N , R}, referring to the MFR, NFP, and RFP Proof: See Appendix A 
forwarding schemes respectively. The distance rχ denotes 6 With a slight abuse of terminology, we denote the LT of the pdf of I
agg as
the transmitter-receiver separation when using χ forwarding the LT of Iagg .

⎧
⎪
⎪ λe−λA(r )
⎪
⎪ − A (r ), 0 < r ≤ d,
⎪
⎪ N (1 − eE[|N|]/2 )  √ 
⎪
⎪
⎪
⎪ −λA(r ) 2λ exp −λA r 2 − d2
√
⎪
⎪− λe  (r ) −  ( r 2 − d 2 ),
⎪
⎪ A A d < r ≤ 2d,
⎪
⎪ N (1 − eE[|N|]/2 ) N (1 − eE[|N|]/2 )
⎪
⎨ ..
fr M (r ) = .    (5)
⎪
⎪
⎪
⎪ λe −λA(r ) N R λ exp −λl A r 2 − (id)2

⎪
⎪
⎪
⎪ − A (r ) − A ( r 2 − (id)2 )
⎪ N (1 − e
⎪ E[|N|/2] ) i=1  N (1 − e
E[|N|]/2 )
⎪
⎪  
⎪
⎪ N L λ exp −λl A r − (id)
2 2
⎪
⎪  
⎪
⎩− A ( r 2 − (id)2 ), max(N R , N L )d < r ≤ Rt .
N (1 − e E[|N|]/2 )
i=1
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2281

For the NFP forwarding scheme, the distribution of the parent PPP, in which points are given a uniformly distributed
transmitter-receiver distance is stated via the following lemma: mark [0,1] and only the points which have the least mark in their
Lemma 2: The distribution of the transmitter-receiver dis- neighbourhood are retained. The uniform mark corresponds to
tance for the NFP forwarding scheme, fr N , is given in (6), the backoff counter value chosen by each node in the CSMA
where E [|N|] is given in (2), B(.) is given in (37), which is protocol. Note that the interference is coming from the set
shown at the bottom of page 14, and B (.) is the derivative ˜ i which constitutes a MHCPP. According to [27]–[29] the
of B(.). intensity of 
˜ i is given by:
Proof: See Appendix B  1 − e−αE[|N|]/2
The transmitter-receiver distance distribution for the RFP is =λ , (8)
αE [|N|] /2
stated via the following lemma:
Lemma 3: The distribution of the transmitter-receiver dis- where E [|N|] is given in (1) and 1 ≤ α ≤ 2 is a correction
tance for the RFP forwarding scheme, fr R , is given in (7). factor that is used to mitigate the underestimation problem
Proof: See Appendix C  of the MHCPP-II. That is, the MHCPP-II uses α = 2 in (8),
which implies a contention domain length of 2Rs along with
retaining one transmitter per contention domain. However, the
MHCPP-II suffers from the intensity underestimation prob-
B. The Intensity of Concurrent Transmitters & LT of the lem due to the role of unselected points (see [29] for details).
Aggregate Interference Exploiting the natural order of points in 1-D lines, we argue that
For the sake of simple presentation, we first present the reducing the contention domain to Rs by α = 1, as opposed
analysis of the intensity of concurrent transmitters and LT of to 2Rs imposed by α = 2 for the MHCPP-II, mitigates the
aggregate interference for the single lane model. Then, we intensity underestimation problem.
generalize the analysis for the multi-lane scenario. Remark 3: The intuition behind the choice of α can be
1) Single Lane Model: The simplest highway consists of a explained as follows; i) starting from a retained transmitter and
single traffic lane and we deal with this case first. A test trans- moving in each direction along the line, the first transmitter
mitter transmits to a receiver, located on the same lane, based after a void distance of Rs contends only with the nodes in the
on the forwarding scheme selected. The remaining vehicles, next Rs distance; ii) the MHCPP-II saturates at the intensity of
1
which have packets to send, contend for spectrum access via 2Rs which is a loose packing density. This is because if Rs is
the CSMA protocol, and transmitters who won the contention the void region on each side of the transmitters, the intensity of
(i.e., have elapsed timers) transmit simultaneously. The receiver concurrent transmitters should saturate at R1s . Using this order
experiences interference from the set of concurrent transmitters of selected points, we can deduce that in general, the contention
(i.e., the transmitters selected by the CSMA protocol) excluding domain is reduced by half. Thus for the multi-lane case, the
the intended transmitter. The Matérn hard core point process of average void distance between two retained points in any lane
type II (MHCPP-II) is widely used in literature to model the set is on average greater than or equal to the sum of the forward
of concurrent transmitters in CSMA networks [24], [25]. The line segments inside the circle of radius Rs . For example, in a
MHCPP-II captures the repulsive behavior of the point process 2-lane highway, theaverage void distance per lane in the satu-
due to the effect of the protection created around the transmit- rated case is Rs + Rs2 − d 2 . Therefore, throughout the paper,
ting nodes [26]. It is obtained via dependent thinning of the we use α = 1.

⎧
⎪
⎪ λe−λB(r )
⎪
⎪
⎪ B (r ) 0 ≤ r ≤ d,
⎪
⎪ N (1 − eE[|N|]/2 )  √ 
⎪
⎪
⎪
⎪ λe−λB(r ) 2λl exp −λB r 2 − d2 √
⎪
⎨  (r ) +
E[|N|]/2
B E[|N|]/2
B ( r 2 − d 2 ), d < r ≤ 2d,
fr N (r ) = N (1 − e ) N (1 − e )
⎪
⎪ ..
⎪
⎪ .
⎪
⎪   
⎪
⎪
⎪
⎪ λe −λB(r ) (N −1)/2
 2λr exp −λB r 2 − (id)2

⎪
⎪  (r ) +  ( r 2 − (id)2 ), max(N R , N L )d < r ≤ Rt ,
⎩ B B
N (1 − eE[|N|]/2 ) i=1 N (1 − eE[|N|]/2 )
(6)
⎧ 1
⎪
⎪ , 0 ≤ r ≤ d,
⎪
⎪
⎪
⎪
N Rt
⎪
⎪ 1 2r
⎪
⎨ + √ , d < r ≤ 2d,
N Rt N R r 2 − d2
fr R (r ) = ..
t (7)
⎪
⎪
⎪
⎪ .
⎪
⎪
⎪ 1
⎪ (N −1)/2
 2r
⎪
⎩ +  , max(N R , N L )d < r ≤ Rt .
N Rt i=1 N Rt r 2 − (id)2
2282 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

Fig. 4. Intensity Scaling during projection. A homogeneous set of points expe-

riences a compression if mapped to the central lane with equal distance from
the origin. The compression  is a function of the distance γ from the origin.
Fig. 3. Multi-lane highway with test transmitter and receiver located at the
central lane with N R lanes on their right side and N L lanes on the left. The
interference comes from outside the sensing range of the transmitter, marked is given as follows:
with bold lines.
1
c(γ , i) =   , (10)
Finding the LT of interference associated with a MHCPP
γ 2 + 2 γ 2 − (id)2 + 1 − γ
is an open research problem because there is no known
expressions of the probability generating functional for the
where i corresponds to the i th traffic lane away from the
MHCPP. Therefore, only approximate expressions for the LT
central lane. We can apply this principle while mapping the
are obtained by approximating the MHCPP with an equi-dense
homogeneous poisson distributed vehicles to the central lane.
PPP existing outside the interference protection region created
Since the compression is non-linear in γ , so we obtain a non-
by the test transmitter [30]. This is formally stated as:
homogeneous PPP after the mapping operation. Therefore the
Assumption 3: We assume that the interfering set of vehicles
intensity of the PPP used to model concurrent transmitting
˜ i constitutes a PPP with intensity existing outside the aver-
vehicles is given by the following lemma.
age interference protection imposed by the CSMA protocol as
Lemma 5: The intensity of the PPP of concurrent transmit-
shown in Fig. 3. The PPP assumption has proven to be accu-
ting vehicles on the i th traffic lane, mapped to the central lane
rate if the intensity and interference boundaries are carefully
on a multi-lane highway is given by:
calculated [15], [27], [29].
The Laplace Transform of Iagg for the single lane model is
(γ
˜ , i) = c(γ , i) . (11)
given by the following lemma:
Lemma 4: The Laplace transform of the aggregate interfer-
Note that c(γ , 0) = 1 and hence (γ
˜ , 0) = , i.e., the inten-
ence at a test receiver placed at the origin in the case of a single
sity of the PPP on the central lane remains unchanged under
lane highway is expressed as:
the mapping. It is worth pointing out that the intensity calcu-
  1  
Ps η 1 lation in (11) accounts for the contention between vehicles on
L Iagg (s) = exp − η
dt , different lanes. This is because the expression for given in (8)
μ R1 1 + t
  √   √ 
contains the contention domain size E[N] that includes vehicles
(η=2) − Ps −1 Ps/μ
− Ps −1 Ps/μ
in all lanes’ sections covered by the sensing range as shown
μ tan μ tan
= e Rs +r Rs −r
. (9)
in Fig. 3. Using this methodology, we can extend the frame-
Proof: See Appendix D  work used in the single lane model to the multi-lane case. The
2) Multi-Lane Model: The multi-lane model illustrated in LT of the aggregate interference for the multi-lane highway is,
Fig. 3 is complicated and tricky to handle. We propose a three therefore given by the following lemma:
step process to deal with the interference in the multi-lane Lemma 6: The Laplace transform of the aggregate interfer-
model. The first step is to identify concurrent transmitters via ence observed from a test vehicle with N R traffic lanes on its
dependent thinning of the set of parallel PPPs to form a set right and N L lanes on its left can be expressed as:
of MHCPPs. We then approximate the MHCPP on each traffic 
NR 
NL
lane with an equi-dense PPP. Finally we project the homoge- L Iagg (s) = L(0) (s) × L(n) (s) × L(m) (s), (12)
neous PPPs on all the traffic lanes to nonhomogeneous PPPs on n=1 m=1
a single lane via a transformation of intensity. We use an anal-
ogy to calculate the intensity after projection and the boundaries where
of the PPP as shown in Fig. 3 and Fig. 4. Consider a set of ⎛ ⎛ ⎞ ⎞

uniformly spaced points on a line parallel to the line passing ⎜ ⎜ (γi , i) ⎟ ⎟
through the origin, also referred to as the central lane. From L(i) (s) = exp ⎝− ⎝  η ⎠ dγi ⎠ , (13)
γi
the perspective of the origin, the uniformly spaced points can
ψi ∈R 1 1 + r T 1/η
be mapped to the central lane such that the distance from the
origin to the points remains the same. We can observe clearly and ψi is the interference region on the i th traffic lane as illus-
from Fig. 4 that the points experience a compression which is trated in Fig. 3. Thus the limits of integration are from γ̂i to ∞
a function of the distance γ from the corresponding point and and from −∞ to γ̃i . The parameters γ̂i and γ̃i can be expressed
the origin. The compression factor for the multi-lane scenario as follows:
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2283

!  neglects the non-uniform compression effect on the intensity of

γ̂i = r 2 + Rs2 + 2r Rs2 − (id)2 , interferers due to projection. Referring to Fig. 4, the aggres-
!  sive approximation means that the projected points are very
N −1
γ̃i = r 2 + Rs2 − 2r Rs2 − (id)2 , i = 0, 1, . . . , . close to each other experiencing maximum compression and
2 the distance between them does not increase. On the other hand,
(14)
the conservative approximation means that all projected points
Proof: See Appendix D  have the same inter-point distance as the unprojected points.
Exploiting these approximations, a single integral expression
V. P ERFORMANCE A NALYSIS for the success probability is obtained in the following two
corollaries.
In this section we evaluate the system performance for each Corollary 1: The success probability for an N −lane high-
of the packet forwarding schemes based on the following way is aggressively approximated by overestimating the inter-
metrics: ference by its maximum value and hence the expression in (15)
1) Transmission success probability (P) reduces to:
2) Normalized average forward progress (NAFP) Rt  
3) Aggregate packet progress (APP) 2 (1/η) tan−1 r T (1/η) +tan−1 r T (1/η)
− μTPr κ − r T
Pχ ≈ e e Rs −r Rs +r
In the following subsections, we define these metrics along
with the analysis for the different packet forwarding schemes. 0
 

NR
˜ γ̂i ,i)r T (1/η) tan−1 r T (1/η) +tan−1 r T (1/η)
− (
× e γ̂
i γ̃
i
A. Transmission Success Probability
i=1
 
The transmission success probability, or success probability 
NL ˜ γ̂ j , j)r T (1/η) tan−1 r T (1/η) +tan−1 r T (1/η)
− ( γ̂ γ̃
is defined as the probability that the SINR at the receiver is × e j j
frχ (r )dr,
above a particular threshold. The general framework for evalu- j=1
ating the success probability has been explained in Section IV. (16)
For an N −lane highway with an inter-lane separation d, the
success probability is given by the following theorem. for χ = {M, N , R}. 1l{.} is the indicator function that is equal
Theorem 1: The transmission success probability evaluated to 1, if the condition in the braces is true, and 0 otherwise. The
at a receiving vehicle on an N -lane highway with N R traffic parameters γ̂i and γ̃i are given in Lemma 6 and the function
lanes on its right and N L traffic lanes on its left and an inter-lane (γ
˜ , i) is given by (11).
separation d is expressed as: Corollary 2: The success probability for an N -lane highway
with an inter-lane separation of d can be conservatively approx-
Rt  (1/η) −1 r T (1/η)

imated by assuming the same intensity of interferers on each
μT r 2 κ − r T (1/η) tan−1 r T
Rs −r +tan
Pχ = e− P e Rs +r
traffic lane, i.e., and thus can be expressed as:
0
⎛ ⎛ ⎞ ⎞ Rt √ √
μT r 2 κ −1 Tr

NR  Pχ ≈ e− P e−N T r tan Rs −r
⎜ ⎜ (γi , i) ⎟
˜ ⎟
× exp ⎝− ⎝  η ⎠ dγi ⎠ 0
γ
i=1 ψi ∈R 1
1 + r T 1/η
i √ √
−1 Tr
⎛ ⎛ ⎞ ⎞ e−N T r tan Rs +r frχ (r )dr, (17)
N 
L
⎜ ⎜ (γ˜ j , j) ⎟ ⎟ for χ = {M, N , R}.
× exp ⎝− ⎝  η ⎠ dγ j ⎠ frχ (r )dr,
γ
j=1 ψ j ∈R1 1 + r T 1/ηj

B. Normalized Average Forward Progress

(15)
The forward progress is defined as the average distance trav-
for χ = {M, N , R}. The interference region on the i th traffic eled by the packet towards its final destination. Note that we
lane, ψi is given in Lemma 6. assume the destination is located infinitely far away and there-
Proof: The proof follows by substituting the laplace trans- fore the horizontal distance covered can be reasonably used
form of the aggregate interference from (12) in (4).  as a measure of progress towards the destination. The packet
However the exact analytical evaluation of the expression progress is an important measure because it affects the num-
in (15) for the multi-lane case is not possible. Note that the ber of transmissions required for transmitting a packet from the
intractability comes from the fact that the projection of inter- source to the destination over a multi-hop network and hence
ferers leads to a non-homogeneous PPP with a complicated directly relates to the transmission time. The average per hop
distance dependent factor c(γ , i). Therefore, for tractability, FP is denoted by Z̄χ = E[Zχ ], and is evaluated in (18), (19)
we suggest two approximations to the success probability. The and (20), which are shown at the bottom of the next page, for
first is an aggressive approximation in which we assume that the MFR, NFP and the RFP forwarding schemes respectively.
(γ
˜ , i) = sup(c(γ , i)) . The second is a conservative approx- Similar to [31], to make the average FP dimensionless,
imation in which we assume that (γ ˜ , i) = . That is, the we normalize it by the average distance between two nearest
aggressive approximation neglects the intensity decay with the neighbours. In this way, we transform the notion of distance (in
distance from the receiver and the conservative approximation meters) to the notion of hops (or number of nodes crossed over
2284 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

under a single transmission). The average distance between two The APP is a dimensionless quantity that tells the average
nearest nodes is given by (40). Thus we can conveniently use forward distance traveled by successfully transmitted packets
Z̄χ λ as a measure of the normalized average forward progress per unit length in the road. Note that the success probability
(NAFP) made in a single transmission in terms of the number depends on both χ and ρth , the intensity of concurrent trans-
of nodes crossed over. mitters depends only on ρth and the average FP depends only
on χ . The challenge then is to jointly select the optimal values
of χ and ρth that achieve the best balance between the success
C. Aggregate Packet Progress probability, spatial frequency reuse efficiency and average FP
given specific QoS requirements. The problem is formulated as
In order to have a unifying performance measure for multi- follows:
hop transmission we define a combined metric called the
aggregate packet progress (APP). The APP captures the trade- max A P Pχ (ρth ),
χ ,ρth  
off between the success probability, the average FP, and the λ τδ
(22)
s.t. (ρth ) Pχ (ρth )Z̄χ
≤ ,
spatial frequency reuse efficiency. It is defined as follows:
where δ is the distance from the source to destination, and τ is
A P Pχ (ρth ) = Pχ (ρth ) × (ρth ) × Z̄χ . (21)
the duration of each time slot. The QoS constraint comes from

N
E[Z M ] ≈ E[N]
e 2 −1
⎡ ⎛ ⎞
    i
  λ(N − (1+1l { −N |} ))ω(i)
⎢
i
⎜ i j>|N R L ⎟
⎢ exp λ (1 + 1l{ j>|N R −N L |} )ω( j) ⎝ 1 − λ(N − (1 + 1l{ j>|N R −N L |} ))ω(i) e j=1
⎠
⎢
⎢ β j=0 j=1
×⎢
⎢
⎢ i=0 
i
⎢ λ(N − (1 + 1l{ j>|N R −N L |} ))2
⎣ j=1

  ⎤

i

i λ(N − (1+1l{ j>|N R −N L |} ))ω(i+1)
λ(N − (1 + 1l{ j>|N R −N L |} ))ω(i + 1) − 1 e j=1 ⎥
⎥
j=1 ⎥
+ ⎥, (18)

i ⎥
⎥
λ(N − (1 + 1l{ j>|N R −N L |} ))2 ⎦
j=1

N
E[Z N ] ≈ E[|N|]
1 − e− 2
⎡    
i
i 
i −λ(N − (1+1l{ j>|N R −N L |} ))ω(i)
⎢
β exp −λ (1 + 1l{ j>|N R −N L |} )ω( j) 1 + λ(N − (1 + 1l{ j>|N R −N L |} ))ω(i) e j=1
⎢
⎢ j=0 j=1
×⎢
⎢
⎢ i=0 
i
⎣ λ(N − (1 + 1l{ j>|N R −N L |} ))2
j=1

  ⎤

i

i −λ(N − (1+1l{ j>|N R −N L |} ))ω(i+1)
1 + λ(N − (1 + 1l{ j>|N R −N L |} ))ω(i + 1) e j=1 ⎥
⎥
j=1 ⎥
− ⎥, (19)

i ⎥
⎥
λ(N − (1 + 1l{ j>|N R −N L |} ))2 ⎦
j=1

β   
1 1
E[Z R ] ≈ + (i) ω(i + 1)2 − ω(i)2 (20)
2 N Rt
i=1

Proof: See Appendix E ⎧ 

( i =β
0, i =0 ⎨ 0,
Note: ω(i) =  2 (i) = β−i−1
 (1+1l{i>|N R −N L |} )
Rt − ((β − i − 1)d) , other wise.
2 ⎩ √ 2 , other wise.
i=1
2 N Rt −(id)
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2285

the transmission delay or the end-to-end throughput require- This follows from using the approximation for A(z) given
ments. The constraint reflects the average end-to-end transmis- by (35) under the assumption that Rt max(N R , N L )d 7
sion delay from the source to the destination and enforces it Similarly, using the approximation for B(z) given by (38) leads
to be less than a certain threshold . Note that 1/Pχ signi- to the approximate average FP for the NFP forwarding scheme
fies the average number of re-transmissions required to achieve as:
successful per-hop packet transmission, δ/Z̄χ is the average
1 − exp(−N λRt )(N λRt + 1)
number of hops that are required for the packet to reach a des- E [Z N ] ≈ . (26)
tination located at a distance of δ, and τ is the duration of each Nλ
time slot. It is important to highlight that the additional delay Finally, under the same assumption, the expected forward
imposed by the CSMA contention to access the spectrum can progress for the RFP forwarding scheme can be obtained using
be easily incorporated to the constraint in (22) by multiplying (42) as:
λ
by the factor , which captures the average number of time
slots spent in CSMA contention to access the spectrum for a Rt
E [Z R ] ≈ . (27)
single transmission. 2

VI. S IMPLIFYING A PPROXIMATIONS VII. N UMERICAL R ESULTS AND D ISCUSSION

While the PPP approximation in Assumption 3 is mandatory In this section, we first validate our analysis and then pro-
for our analysis, there are some approximations which can sim- vide numerical results to compare the performance of the three
plify the complicated expressions in (5), (6) and (18)–(20) to forwarding schemes on the metrics defined in Section V. Note
gain more insight [(5) and (6) are shown at the bottom of pages that the analytical expressions validated in this section are those
5 and 6, respectively]. We begin with the expressions of the which are based on the approximations given in Section VI. To
distance distribution between the transmitter and receiver. The this end, we show that there is an optimum sensing threshold
distance distribution for the MFR forwarding protocol given that maximizes the success probability, SFR as well as the FP
by (5) can be approximated as in (23), which is shown at the under given transmission delay constraints.
bottom of the page.
This approximation is based on the fact that the receiver, in
the case of the MFR forwarding scheme, is located near the A. System Parameters and Model Validation
edge of the transmission range with high probability and thus Unless otherwise stated, the simulation parameters are
one part of the pdf dominates, resulting in this approximation selected as; transmission power (P = 1 W), noise floor (κ =
which is tight for higher intensities. Similarly for the NFP for- −104 dBm), road length in simulation (L = 10 km), number of
warding scheme, we can approximate the distance distribution traffic lanes (N = 11, with N R = N L = 5), inter-lane distance
given by (6) as follows: (d = 5 m), transmission range (Rt = 200 m), sensing range to
transmitter range ratio (ζ = 1), path loss exponent (η = 2), dis-
N λe−λNr tance to destination (δ = 10 km), time slot duration (τ = 1 ms).
fr N (r ) ≈ , 0 < r ≤ Rt . (24)
1 − eE[|N|/2] The vehicles are distributed on the roads as linear Poisson point
processes with an intensity ranging from 0 to 30 vehicles/km in
This approximation is based on the nature of the NFP forward- each traffic lane.
ing scheme which dictates a receiver located very close to the The results in Fig. 5(a), 5(b) and 5(c) effectively validate our
transmitter, thus there is a higher probability of small transmit- analysis. It can be observed that the aggressive approximation is
ter receiver distances which leads to the above approximation. accurate for small values of ζ . However for higher values of ζ ,
The approximate simplified expression for frχ (.) not only sim- both the conservative approximation and the aggressive approx-
plifies the calculation for the success probability in (17), but imation are very close. This is because increasing ζ implies a
also simplifies the expressions for the expected FP given in (18), higher sensing range which increases the interference protec-
(19) and (20). The expected FP for the MFR forwarding can be tion around the receiver. In other words, the distance to the
approximated as: nearest interferer increases and since the compression factor
1 + exp(N λRt )(N λRt − 1) 7 This is a reasonable assumption since a vehicle on one edge of the highway
E [Z M ] ≈ . (25)
N λ(exp(E[N]/2)) should be able to communicate to the roadside base stations located near the
opposite edge of the highway.

⎧
⎪
⎪ 0, 0 < r ≤ max(N R , N L )d,
⎪
⎨ 
E[|N|]
 
E[|N|] √ 2


fr M (r ) ≈ −λ −Nr −λ −N r −(id)2 (23)

⎪
⎪ λe 2λ (N −1)/2
 2λr e 2λ
⎪
⎩ +  , max(N R , N L )d < r ≤ Rt .
1 − eE[|N|/2] i=1 (1 − eE[|N|/2] ) r 2 − (id)2
2286 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

Fig. 5. Model validation for the three packet forwarding schemes. It can be observed that for small values of ζ , the aggressive approximation is very close.
However with increasing ζ , the aggressive approximation approaches the conservative approximation. Moreover, increasing ζ increases the success probability in
general.

MFR at aggressive values of the sensing threshold, while the

converse is true at conservative values of the sensing threshold.
Finally, the figure shows the effect of the traffic intensity on the
FP. As the traffic intensity increases, the FP increases because
the probability to find a vehicle that achieves high FP increases.
In Fig. 6(b), we plot the normalized average FP against dif-
ferent values of λ and different values of ζ to gain further
insights into the system behavior. The figure shows that for
smaller values of ζ , the RFP forwarding scheme outperforms
the MFR and the NFP for all traffic intensities due to the insuf-
ficient interference protection. However, increasing the value of
Fig. 6. Normalized average forward progress against carrier sensing threshold ζ increases the interference protection and leads to the outper-
and traffic intensity. formance of the MFR forwarding scheme. Exact critical values
of ζ at which the MFR outperforms the RFP can be calculated
given in (10) decays with the distance, therefore the effects from the intersecting points in Fig. 6(a). From Fig. 6(a) and
of compression diminish for larger ζ . Hence the aggressive Fig. 6(b), we observe that the NFP performs poorly in terms
approximation becomes similar to the conservative approxima- of the normalized average FP for all traffic intensities at larger
tion, which neglects compression effects. On the other hand, for values of ζ .
smaller values of ζ , the interference protection of the receiver is
low so the model becomes sensitive to the intensity and due to
significant compression effects at lower ζ , both approximations B. Maximizing Aggregate Packet Progress
are different. Additionally, increasing the value of ζ improves Looking into Figs. 6(a) and Fig. 6(b), it is hard to judge
the success probability, which is expected, due to the increased the best forwarding scheme and the associated carrier sens-
interference protection. ing threshold that optimizes the tradeoff between the three
We now exploit our developed model to compare perfor- performance objectives. This is because for a given success
mance of the three packet forwarding schemes (i.e., MFR, NFP, probability, the MFR requires a conservative sensing threshold
and RFP) in terms of the normalized average FP. Note that in all that may degrade the spatial frequency reuse across the road. On
the results, we use the aggressive approximation of the success the other hand, the NFP can allow an aggressive sensing thresh-
probability since it better approximates the success probability. old that increases the spatial frequency reuse but results in small
Figure. 6(a) reveals several insights to the design of forwarding FP. Note that the key objective is to increase the number of suc-
schemes in CSMA coordinated vehicular networks. The figure cessful parallel transmissions per unit length of road segment
manifests the effect of the CSMA sensing threshold on the per- along with ensuring an acceptable end-to-end delay. Therefore,
formance of forwarding schemes. The impact of the sensing we define the APP with the delay constraint in (22), which is
threshold on the MFR forwarding scheme is significant due to plotted in Fig. 7 for 10 vehicles/km, to capture these tradeoffs.
the large transmitter-receiver separation. Hence, MFR scheme The APP is plotted for two different values of the SINR
requires a conservative CSMA operation. On the other hand, threshold, i.e., T = 0 dB and T = 10 dB, and number of
the effect of the sensing threshold on the NFP is not prominent traffic lanes, i.e., N = 8 (N R = 3, N L = 4) and N = 11
due to the short transmitter-receiver separation. Hence, the NFP (N R = 5, N L = 5) under high and low delay tolerances. In
can sustain an aggressive CSMA operation. It is also interesting general, it is observed that the APP is higher for N = 8, as
to note that even for aggressive sensing threshold, the RFP out- compared to N = 11, due to higher success probability as
performs the NFP. The figure also shows that we cannot judge a result of lower interference. Note that the solid line parts
the performance of the forwarding scheme without looking to of the curves denote the feasible range of ρth (i.e., when
the sensing threshold. For instance, the RFP outperforms the the constraint in (22) is satisfied), while the dotted parts of
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2287

is presented. The developed model is based on stochastic

geometry and point process theory. Approximate yet accurate
expressions for the intensity of concurrent transmitters and
packet success probability have been obtained. We introduce
the aggregate packet progress metric for multi-hop CSMA net-
works, which is a dimensionless quantity that shows the average
forward progress traveled by successfully transmitted pack-
ets per unit length in the road. The APP is used to optimize
the CSMA threshold and compare the performance of differ-
ent packet forwarding schemes. By virtue of the interference
coordination of the CSMA protocol, VANETs favor the MFR
forwarding scheme under high throughput requirements due to
the associated longer forward progress. The NFP, however can
achieve the best performance under the high SINR regime and
at the expense of high transmission delay, which is not feasi-
ble. This is contradictory to the results given in [8], [9] which
show that ALOHA networks always favor NFP due to the
uncoordinated spectrum access. In a nutshell, the interference
Fig. 7. Aggregate packet progress for the MFR, NFP and RFP forwarding protection provided by the CSMA protocol enables a longer
schemes against carrier sensing threshold under different SINR threshold T ,
number of traffic lanes N , and delay tolerance  for λ = 10 vehicles/km.
per-hop forward progress via the MFR while the ALOHA pro-
Increasing T allows the NFP scheme to achieve a higher APP performance tocol sacrifices the forward progress via the NFP to maintain
as compared to the MFR, but only at the expense of a higher , which results in acceptable per-hop transmission success. For future work, this
lower throughput. framework can be extended for modeling urban traffic scenarios
where the traffic is restricted to crossing lanes.
the curves denote the infeasible range of ρth . Note that the
constraint is not satisfied for very low and very high values
of the sensing threshold. For very low sensing threshold, the A PPENDIX A
number of contending neighbours in the CSMA protocol are P ROOF OF L EMMA 1
very large leading to high contention delay while for very
Let X + be the event that there exists a receiver inside the
high sensing threshold, the number of retransmissions required
forward transmission range, illustrated by the shaded half circle
for transmission success becomes large resulting in higher
in Fig. 8, of the test transmitter. The probability of the event can
transmission delay. Regardless of the number of traffic lanes N ,
be calculated as follows:
it can be observed from Figs. 7(a) and 7(b) that the MFR, with
) *
the proper choice of ρth is preferable for lower link quality P [X +] = P At least one vehicle in {Â(z) + B̂(z)} ,
requirement (e.g., 0 dB) due to the associated long forward ) *
= 1 − P No vehicle in {Â(z) + B̂(z)} ,
progress. However, Figs. 7(c) and 7(d) show that the NFP is the
preferred forwarding scheme for high link quality requirement = 1 − e−E[|N|]/2 , (28)
(e.g., 10 dB), but on the expense of longer delay. This is because
the MFR will require much more retransmissions to achieve where E [|N|] /2 is the average number of forward neighbours
the target SINR due to the distance between the transmitter of the transmitter. Now let Zχ be a random variable denoting
and the receiver as compared with the NFP. Therefore the best the FP made by the packet in one transmission. The subscript
forwarding scheme depends on the required link quality per χ ∈ {M,N,R} referring to the MFR, NFP and RFP forwarding
transmission. We believe that vehicular applications require schemes respectively. Using the methodology in [8], the condi-
lower delay, e.g., safety messaging [6], so the requirements for tional CDF of the FP in the MFR scheme Z M can be expressed
link quality are not expected to be very aggressive and hence as follows:
the MFR forwarding scheme is well suited for such networks. FZ M (z|X +)
It is pertinent to mention here that increasing the carrier
sensing threshold reduces the average protection region around P{No vehicle in Â(z) and X +}
= P{Z M ≤ z|X +} = ,
a node and therefore, at sufficiently higher values of ρth , the P{X +}
CSMA protocol starts behaving as an ALOHA. We can observe P{No vehicle in Â(z) and at least one vehicle in B̂(z)
from Fig. 7 that at very high values of ρth , the NFP always = ,
P{X +}
performs better than the MFR, thus verifying the result in [8]
P{No vehicle in Â(z)}P{At least one vehicle in B̂(z)
[9] that the NFP is the best forwarding scheme in ALOHA = ,
networks. P{X +}
e−λA(z) (1 − e−λB(z) )
= ,
VIII. C ONCLUSION AND F UTURE W ORK 1 − e−E[|N|]/2
A novel analytical framework for modeling CSMA coor- e−λA(z) − e−E[|N|]/2
= , 0 ≤ z ≤ Rt , (29)
dinated inter-vehicle communication in multi-lane highways 1 − e−E[|N|]/2
2288 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

where Â(z) is the shaded region illustrated in Fig. 8, and the

length of road segment inside the shaded region, denoted by
A(z), is given in (30), where β = max(N R , N L ). The condi-
tional probability distribution of FP Z M can then be evaluated
by taking the derivative of (29):

−λe−λA(z)
f ZM (z|X +) = × A (z), 0 ≤ z ≤ Rt ,
1 − e−E[|N|]/2
(31)
Fig. 8. Transmitter-Receiver distance for MFR forwarding scheme

where A (z) = dA(z) E[|N|]

dz .
For a test transmitter at the origin, a
A(z) ≈ − N z, 0 ≤ z ≤ Rt . (35)
potential receiver can be located either on the same traffic lane 2λ
or on an adjacent lane. If the receiver is on the same lane, the This approximate result can be very useful in evaluating
conditional pdf can be written as: approximate expressions for the complicated distance distribu-
tion of the MFR forwarding scheme given in (5). It is pertinent
−λe−λA(r ) to mention here that the approximation is also valid for Rt >
fr M (r |l0 ) = f Z M (r ) = A (r ), 0 ≤ r ≤ Rt ,
1 − e−E[|N|]/2 βd. However it becomes more accurate as Rt βd.
(32)

A PPENDIX B
If the receiver is located on the i th adjacent traffic lane, the
P ROOF OF L EMMA 2
conditional pdf fr M (z|li ) for i = 1, . . . , max(N R , N L ), can be
obtained using the following procedure: Similar to the proof in Appendix A, we can derive the pdf for
the NFP case as follows:
r M = z2 + (id)2 , FZN (z|X +)

Fr M (r |li ) = P(r M ≤ r ) = P( z2 + (id)2
≤ r ), P{No vehicle in B̂(z)}P{At least one vehicle in Â(z)
  = ,
= P(z ≤ r − (id) ) = Fz ( r 2 − (id)2 ),
2 2 P{X +}
   e−λB(z) − e−E[|N|]/2
exp −λA( r 2 − (id)2 ) − exp (−E [|N|] /2) = , 0 ≤ z ≤ Rt , (36)
= . 1 − e−E[|N|]/2
1 − e−E[|N|]/2
(33) where B(z) is the length of road segment inside the shaded
region B̂(z) illustrated in Fig. 8, and is expressed in (37). Using
the same procedure as in Appendix A, the probability distribu-
From (33), we can obtain the conditional pdf as:
tion for the transmitter-receiver distance r N can be obtained in
   the form given in (6). For the case when Rt βd, B(z) can be
−λr exp −λA( r 2 − (id)2 )  approximately expressed as:
fr M (r |li ) =  A ( r 2 − (id)2 ),
(1 − e−E[|N|]/2 ) r 2 − (id)2 B(z) ≈ N z. (38)
(id) ≤ r ≤ Rt .
(34) Again, the approximation is also valid when Rt > βd. The
approximate pdf of the FP in the NFP case is therefore given
as:
Finally the complete pdf of the transmitter-receiver distance r M
can be obtained in the form given in (5). For the case when N λe−λN z
f Z N (z) ≈ , 0 ≤ r ≤ Rt . (39)
Rt βd, A(z) can be approximately expressed as: 1 − eE[|N|]/2

⎧ 
⎪
⎪ E[|N|]
⎪
⎪ − N z, 0 < z ≤ Rt2 − (βd)2 ,
⎪
⎪ 2λ
⎪
⎪   
⎪
⎨ E[|N|] − (1 + 1l{N =N } ) R 2 − (βd)2 − (N − (1 + 1l{N =N } ))z,
⎪
R 2 − (βd)2 < z ≤ Rt2 − ((β − 1)d)2 ,
R L t R L t
A(z) = 2λ
⎪
⎪ ..
⎪
⎪ .
⎪
⎪
⎪
⎪ E[|N|] NR 
 NL 
 
⎪
⎪ − Rt2 − (id)2 − Rt2 − (id)2 − z. Rt2 − d 2 < z ≤ Rt
⎩
2λ i=1 i=1
(30)
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2289

The average distance between any two closest nodes, using the to γ˜i as shown in Fig. 3. For the case where the intensity of the
approximation in (24) is given as: interfering PPP is constant, the expression in (43) reduces to:
 ∞   1  
E[r N ] = r fr N (r )dr ≈
1
. (40) N
Ps η 1
Nλ L Iagg (s) = exp − η dti . (44)
0 μ ti ∈R1 1 + ti
i=1
A PPENDIX C
P ROOF OF L EMMA 3 Substituting (44) in (4) and putting appropriate limits of inte-
gration for the interference region proves the lemma. Note that
For the RFP forwarding forwarding scheme, the relay node N = 1 in the case of Lemma 4. In Lemma 6, the product has
can be any of the vehicles in the forward neighbourhood been broken down in terms of N R and N L for model generality.
of the transmitter. Similar to the proofs in Appendix A and
Appendix B, it can be shown that the transmitter-receiver dis-
tance for the RFP forwarding scheme can be given as in (7), A PPENDIX E
which is shown at the bottom of page 6. The probability dis- P ROOF OF AVERAGE F ORWARD P ROGRESS
tribution of the FP in the case of the RFP forwarding scheme, The proof of the average FP Zχ is done by simply taking
denoted by Z R , is given in (41). If Rt βd, f Z R (z) can be the expectation of the FP for each forwarding scheme. The dis-
approximated as: tribution of the FP is given in (32), (39) and (41), which is
1 shown at the bottom of the page, respectively. For notational
f Z R (z) ≈ , 0 ≤ z ≤ Rt . (42) convenience, we define the function ω which is defined as:
Rt

A PPENDIX D ω(i) = Rt2 − ((β − i − 1)d)2 . (45)
P ROOF OF L EMMA 4
The average FP for the MFR forwarding scheme is evaluated as
Since the set of interfering vehicles is approximated by a PPP follows:
with intensity , the LT of the aggregate interference can be
Rt ω(1)
 E[|N|]
expressed as:
z N λe−λ( 2λ −N z)
E[Z M ] ≈ z f Z M (z)dz = dz
L Iagg (s) = E˜ l {e−s Iagg } 1 − e−E[|N|]/2
⎡ ⎛ ⎞⎤ 0 0

N  ω(2)
  
= E˜ i ⎣exp ⎝−s Ph i j vi j −η ⎠⎦ , zNλ E[|N|]
+ exp −λ − (1 + 1l{1>|N R −N L |} )
i=1 vi j ∈
˜ i \v0 1 − e−E[|N|]/2 2λ
⎡ ⎤ ω(1)

N  
μ
= E˜ i ⎣ ⎦, ω(1)) −(N − (1 + 1l{1>|N R −N L |} ))z dz + . . . +
μ + s Pvi j −η
i=1 j∈
˜i
⎛ ⎛
     Rt β

N ˜ i , i)Psγ −η
(γ zNλ E[|N|] 
= exp − i
dγi , exp⎝−λ⎝ − (1 + 1l{i>|N R −N L |} )
ψi ∈R 1 μ+ Psγi
−η 1 − e−E[|N|]/2 2λ
i=1 ω(β) i=1
(43) ⎞⎞

ω(i) − z ⎠⎠ dz. (46)

where ψi represents the interference region on the i th traffic
lane. Thus the limits of integration are from γˆi to ∞ and −∞

⎧ 
⎪
⎪ N z, 0 < z ≤ Rt2 − (βd)2 ,
⎪
⎪   
⎪
⎪
⎪
⎨ (1 + 1
l {N R =N L } ) Rt2 − (βd)2 + (N − (1 + 1l{N R =N L } ))z, Rt2 − (βd)2 < z ≤ Rt2 − ((β − 1)d)2 ,
B(z) = .. (37)
⎪
⎪ .
⎪
⎪ 
⎪
⎪ 
N NL 
 
⎪
⎩
R
Rt2 − (id)2 + Rt2 − (id)2 + z, Rt2 − d 2 < z ≤ Rt ,
i=1 i=1
⎧ (N −1)/2 
⎪ 
⎪
⎪ 1
+ √ 2
, 0 ≤ z ≤ Rt2 − (βd)2 ,
⎪ N Rt
⎪ N Rt2 −(id)2
⎪
⎪ i=1
⎪
⎪  
⎨ 1 (N −3)/2

f Z R (z) = N Rt + √ 2
2 −(id)2
, R 2 − (βd)2 < z ≤
t Rt2 − ((β − 1)d)2 , (41)
⎪
⎪ i=1 N R t
⎪
⎪ ..
⎪
⎪ .
⎪
⎪ 
⎪
⎩ 1
N Rt , Rt2 − d 2 < z ≤ Rt .
2290 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016

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throughput in multi-antenna cognitive underlay networks with multi-hop B.S. degree in electrical engineering from the
relaying,” IEEE J. Sel. Areas Commun., vol. 31, no. 8, pp. 1543–1558, School of Electrical Engineering and Computer
Aug. 2013. Science (SEECS), National University of Sciences
[11] Y. Kim, F. Baccelli, and G. de Veciana, “Spatial reuse and fairness of and Technology (NUST), Islamabad, Pakistan, the
ad hoc networks with channel-aware CSMA protocols,” IEEE Trans. Inf. M.S. degree in electrical engineering from the King
Theory, vol. 60, no. 7, pp. 4139–4157, Jul. 2014. Abdullah University of Science and Technology
[12] B. Blaszczyszyn, P. Muhlethaler, and Y. Toor, “Maximizing throughput (KAUST), Thuwal, Saudi Arabia, in 2013 and 2015,
of linear vehicular ad-hoc networks (VANETs): A stochastic approach,” respectively. Currently, he is a Research Assistant
in Proc. Eur. Wireless Conf., May 2009, pp. 32–36. with the Qatar Mobility Innovations Center (QMIC),
[13] B. Błaszczyszyn, P. Mühlethaler, and Y. Toor, “Stochastic analysis of Qatar Science and Technology Park (QSTP), Doha,
ALOHA in vehicular ad hoc networks,” Ann. Telecommun., vol. 68, Qatar. His research interests include modeling, analysis and optimization of
no. 1–2, pp. 95–106, 2013. wireless communication systems, stochastic geometry, and green communi-
[14] I.-H. Ho, K. Leung, and J. W. Polak, “Optimal transmission probabilities cations. He was the recipient of the President’s Gold Medal for the best aca-
in VANETs with inhomogeneous node distribution,” in Proc. IEEE 20th demic performance from the National University of Sciences and Technology
Int. Symp. Pers. Indoor Mobile Radio Commun. (PIMRC), Tokyo, Japan, (NUST).
Sep. 2009, pp. 3025–3029.
[15] T. Nguyen, F. Baccelli, K. Zhu, S. Subramanian, and X. Wu, “A perfor-
mance analysis of CSMA based broadcast protocol in VANETs,” in Proc.
IEEE Int. Conf. Comput. Commun. (INFOCOM), Turin, Italy, Apr. 2013, Hesham ElSawy (S’10–M’14) received the B.Sc.
pp. 2805–2813. degree in electrical engineering from Assiut
[16] A. Souza, A. L. Barros, A. S. Vieira, F. Roberto, and J. C. Júnior, “An University, Assiut, Egypt, the M.Sc. degree in
adaptive mechanism for access control in VANETs,” in Proc. 10th Int. electrical engineering from the Arab Academy for
Conf. Netw. (ICN), St. Maarten, Sweden, Jan. 2011, pp. 183–188. Science and Technology, Cairo, Egypt, and the
[17] R. Stanica, E. Chaput, and A.-L. Beylot, “Enhancements of IEEE 802.11p Ph.D. degree in electrical engineering from the
protocol for access control on a VANET control channel,” in Proc. IEEE University of Manitoba, Winnipeg, MB, Canada, in
Int. Conf. Commun. (ICC), Kyoto, Japan, Jun. 2011, pp. 1–5. 2006, 2009, and 2014, respectively. Currently, he is
[18] A. Giang and A. Busson, “Modeling CSMA/CA in VANET,” in a Postdoctoral Fellow with the Computer, Electrical,
Analytical and Stochastic Modeling Techniques and Applications, K. Al- and Mathematical Sciences and Engineering
Begain, D. Fiems, and J.-M. Vincent, Eds. New York, NY, USA: Springer, Division, King Abdullah University of Science
2012, pp. 91–105. and Technology (KAUST), Thuwal, Saudi Arabia, and an Adjunct Faculty
[19] V. D. Khairnar and K. Kotecha, “Performance of vehicle-to-vehicle with the School of Computer Science and Engineering, York University,
communication using IEEE 802.11p in vehicular ad-hoc network envi- North York, ON, Canada. From 2006 to 2010, he worked with the National
ronment,” Int. J. Network Security & Its Appl. (IJNSA), vol. 5, no. 2, Mar. Telecommunication Institute, Cairo, Egypt, where he conducted professional
2013, DOI: 10.5121/ijnsa.2013.5212. training both at the national and international levels, as well as research on
[20] Y. Yao, L. Rao, and X. Liu, “Performance and reliability analysis of IEEE network planning. From 2010 to 2014, he worked as a Student Researcher
802.11p safety communication in a highway environment,” IEEE Trans. with TRTech, Winnipeg, MB, Canada. His research interests include statistical
Veh. Technol., vol. 62, no. 9, pp. 4198–4212, Nov. 2013. modeling of wireless networks, stochastic geometry, and queueing analysis
[21] M. J. Farooq, H. ElSawy, and M. -S. Alouini, “Modeling inter- for wireless communication networks. He is recognized as an Exemplary
vehicle communication in multi-lane highways: A stochastic geometry Reviewer by the IEEE T RANSACTIONS ON COMMUNICATION. For his
approach,” in Proc. IEEE Veh. Technol. Conf. (VTC), Boston, MA, USA, academic excellence, he was the recipient of the several academic awards,
Sep. 2015, to be published. including the NSERC Industrial Postgraduate Scholarship during the period of
[22] S. Eichler, “Performance evaluation of the IEEE 802.11p WAVE commu- 2010–2013, and the TRTech Graduate Students Fellowship during the period
nication standard,” in Proc. IEEE 66th Veh. Technol. Conf. (VTC), Sep. of 2010–2014. He was also the recipient of the Best Paper Award in the ICC
2007, pp. 2199–2203. 2015 workshop on small cells and 5G networks.
FAROOQ et al.: A STOCHASTIC GEOMETRY MODEL FOR MULTI-HOP HIGHWAY VEHICULAR COMMUNICATION 2291

Mohamed-Slim Alouini (S’94–M’98–SM’03–F’09)

was born in Tunis, Tunisia. He received the Ph.D.
degree in electrical engineering from the California
Institute of Technology (Caltech), Pasadena, CA,
USA, in 1998. He served as a Faculty Member
with the University of Minnesota, Minneapolis, MN,
USA, then at the Texas A&M University at Qatar,
Education City, Doha, Qatar, before joining King
Abdullah University of Science and Technology
(KAUST), Thuwal, Makkah Province, Saudi Arabia
as a Professor of Electrical Engineering in 2009. His
research interests include the modeling, design, and performance analysis of
wireless communication systems.