Received February 11, 2017, accepted March 11, 2017, date of publication March 16, 2017, date of current

version April 24, 2017.

Digital Object Identifier 10.1109/ACCESS.2017.2683718

Multicast Capacity for VANETs With Directional

Antenna and Delay Constraint Under
Random Walk Mobility Model
JIAJIE REN, GUANGLIN ZHANG, (Member, IEEE), AND DEMIN LI
College of Information Science and Technology, Donghua University, Shanghai, China, 201620
Engineering Research Center of Digitized Textile and Apparel Technology, Ministry of Education, Shanghai, China, 201620
Corresponding author: G. Zhang (glzhang@dhu.edu.cn)
This work was supported in part by NSF of China under Grant 61301118 and Grant 71171045, in part by the International
S&T Cooperation Program of Shanghai Science and Technology Commission under Grant 15220710600, in part by the Innovation
Program of Shanghai Municipal Education Commission under Grant 14YZ130, and in part by the Fundamental Research Funds for the
Central Universities.

1 ABSTRACT In this paper, we investigate the multicast capacity for vehicular ad hoc networks with
2 directional antennas and the end-to-end delay constraint. We consider a torus of unit area with n vehicles
3 (nodes), there are ns multicast sessions and each session contains one source vehicle which is associated with
4 p destinations. We study the 2D and 1D random walk mobility models with two different time scales, i.e., fast
5 and slow mobility. Given a delay constraint D and assuming that each vehicle is equipped with a directional
6 antenna, we obtain the multicast capacity of the two mobility models with two different time scales in the
7 order of magnitude, respectively. We then characterize the impact of the network parameters (i.e., the end-
8 to-end delay constraint D, the beamwidth of directional antenna θ, and the number of destinations p in each
9 session) on the multicast capacity. Moreover, we find that the unicast capacity can be considered as a special
10 case of our multicast results when the beamwidth of directional antenna θ tends to 2π and the number of
11 destinations p tends to 1 in the sense of probability.

12 INDEX TERMS Multicast capacity, VANETs, directional antenna, delay constraint, random walk.

I. INTRODUCTION models in VANETs. As large amounts of vehicular applica-

Emerging vehicular ad hoc networks (VANETs) have tions require vehicles to compete for data transmissions in a
attracted lots of researchers’ attention. A mass of work in limited network area, it is desirable to know the fundamental
the field of VANETs has made significant progress. With throughput capacity of VANETs, which is greatly important
applying wireless communication technologies, more and and provides the guidance in designing a real network.
more vehicles are equipped with on-board communication The asymptotic throughput capacity of static large-scale
facilities, enabling them to communicate with surrounding wireless networks was initiated by Gupta and Kumar in [5].
vehicles or road infrastructures efficiently. The advantages They prove that the throughput of each node under the
of using vehicular communication networks not only provide protocol model in random wireless ad hoc network is
information exchange service, but improve road safety by 2( √nWlog n )1 bits/sec, which means that while the number of
distributing incident warning signal. On account of these nodes trends to infinity, the per-node throughput decreases
merits of VANETs, literatures are abound in investigating the to zero. Later, Grossglauser and Tse show that constant
vehicular communications. Jiang et al. [1] study the efficient per-node capacity can be achieved by adopting a two-hop
multicast in vehicular networks based on vehicle trajectories. highly mobility model at the cost of tremendous delay [6].
Kong et al. [2] provide a frequency-divided approach to ana- Followed by their work, a plenty of studies have been done to
lyze vehicle density information in Dedicated Short Range
1 Given two non-negative functions f (n) and g(n): f (n) = O(g(n)) means
Communication (DSRC) vehicular networks. [3] investigates
message deliveries with privacy preservation in VANETs by there exists a constant c such that f (n) ≤ cg(n) for n large enough; f (n) =
(g(n)) if g(n) = O(f (n)); f (n) = 2(g(n)) means both f (n) = O(g(n))
proposing a novel routing scheme BusCast. Kerrache et al. [4] and f (n) = (g(n)); f (n) = o(g(n)) means limn→∞ f (n)/g(n) = 0; and
investigate an adversary-oriented overview on the main trust f (n) = ω(g(n)) means limn→∞ g(n)/f (n) = 0.

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J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

investigate the capacity of mobile ad hoc networks (MANETs) In the analysis of the impact of mobility model on the
under a variety of mobility models [7]–[10]. Other works multcast capacity of VANETs, we first study the two-
focus on the delay-throughput tradeoffs for MANETs with dimensional random walk mobility model with fast mobility.
variety scenarios [11]–[15]. All the above works focus on Then, we derive the multicast capacity of VANETs under the
the investigation of unicast traffic flow, which appear low two-dimensional random walk mobility model with slow
effective and little beneficial in both theory and applica- mobility. Next, we analyze the one-dimensional random walk
tion. For instance, multicast information flow can be uti- mobility model under conditions of fast and slow mobile
lized for delivering live multimedia contents in a variety of vehicles. We find that the throughput capacity is improved
applications, e.g., weather prediction, wireless video con- in the one-dimensional mobility model, which is intuitively
ference etc. In [16], they study the multicast capacity of the same because the probability that the source chooses the
static wireless networks based on a comb routing scheme. location of its destinations becomes larger compared with the
Li et al. [17] prove the matching upper bound and lower two-dimensional model. Finally, we obtain the upper bounds
bound for multicast capacity via a Euclidean minimum span- on the multicast capacity of VANETs, which show how the
ning tree method. Zhou and Ying [25] study the multicast asymptotic results rely on the number of destinations p in a
capacity of large-scale MANETs under delay constraints by multicast session, the delay constraint D and the beamwidth
adopting raptor code technique. of directional antenna θ . The main contributions of our paper
The investigation of capacity and delay for VANETs is of are as follows:
great importance in both theory and its potential applications. • We present an asymptotic study of the multicast capacity
Pishro-Nik et al. [18], Nekoui and Pishro-Nik [19] initiate for VANETs with directional antenna and delay con-
the study of scaling laws for VANETs. They show that the straint under random walk mobility model. We give the
road geometry has great effect on the throughput capacity asymptomatic upper bounds on the muticast capacity.
of VANETs. They propose a new concept of sparseness for • We adopt random walk mobility model with two time
analyzing the influence of road geometry on the capacity. scales to characterize the vehicles mobility patterns.
Lu et al. [20] show the asymptotic capacity of social- We analyze the impact of mobility models on the multi-
proximity VANETs. They consider that vehicles move around cast capacity for two-dimensional and one-dimensional
a specific social spot in a restricted region. Wang et al. [21] scenarios, which isn’t considered in the state-of-art
analyze the throughput capacity by proposing a novel packet research.
forwarding strategy. In the previous work [22], we investigate • We investigate the impact of system parameters on the
the multicast capacity of VANETs with directional antennas multicast capacity, i.e., the number of vehicles, the num-
and end-to-end delay constraints under the i.i.d. mobility ber of destinations of each session, the beamwidth of
model. directional antenna, and the delay constraint. We also
In this paper, we study the multicast capacity of VANETs find that some of the previous work can be consid-
under the random walk mobility model. Specifically, we ered as a special case of our results. Compared with
assume that every vehicle is equipped with a directional existing work, our results perform better in the capacity
antenna. Data transmission is under the delay constraint, i.e., while other system parameters are fixed. We validate our
the source node transmits to its destinations directly within results by providing extensive simulations.
the transmission radius. Otherwise, data is transmitted by The rest of the paper is organized as follows. In Section II,
multihop fashion under a delay constraint D. The packets we introduce the definitions and notations. In Section III,
can be transmitted successfully if and only if each pair of we present the system models. In Section IV, we analyze
transmitter and receiver is within their directional antenna the multicast capacity and give the main results for different
beam coverage ranges. scenarios. Finally, we conclude the paper in Section V.
Different from previous work, we adopt the random walk
mobility model, which is a more general model with con-
II. DEFINITIONS AND NOTATIONS
siderable applications. The random walk mobility model can
In this section, we give the definitions and related notations
be used to characterize the social characteristic of vehicle’s
for problem formulation and analysis.
mobility in a localized city region. For example, a vehi-
cle often moves within a bounded area that is in the prox-
A. FEASIBLE MULTICAST THROUGHPUT
imity of the driver’s company or neighbourhood. In this
mobility model, vehicles can move to adjacent regions or For a VANET with n vehicles, we say that the multicast
stay at the current region in the next time slot. Further- throughput, denoted by λ(n), is feasible if there exists a spatial
more, in real life, a vehicle usually returns to its starting and temporal scheduling scheme that yields a throughput of
point after a long time travel. In [23], Polya have demon- λ(n) bits/second.
strated that the node following random walk model on a
d-dimensional (d ≤ 2) surface returns to its origin with a B. AGGREGATE MULTICAST THROUGHPUT CAPACITY
probability of 1, which is similar with the mobile trajectory of We say the aggregate multicast throughput capacity of
vehicles. a VANET is of order O(f (n)) bits/second if there is a

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TABLE 1. Main notations.

FIGURE 1. A VANET with two multicast sessions, where D1,1 and D1,2
are the destinations of source vehicle S1 , D2,1 and D2,2 are the
destinations of source vehicle S2 . Any vehicle can be a relay node
in the multicast sessions.

deterministic constant c1 < +∞ such that

lim inf P(λ(n)) = c1 f (n) is feasible) = 0,

n→∞

and is of order 2(f (n)) bits per second if there is a determin-

istic constant 0 < c2 < c3 < +∞ such that

lim inf P(λ(n) = c2 f (n) is feasible) = 1,

n→∞
lim inf P(λ(n) = c3 f (n) is feasible) = 0.
n→∞

C. DELAY CONSTRAINT
FIGURE 2. The packets are delivered from the source to its destination
We say a successful transmission if the source node delivers successfully under the directional antennal model when the distance
packets to its p destinations within D consecutive time slots. between the source and the destination is no more than ri and the
antenna beam of two nodes will cover each other.

D. HITTING DISTANCE
At time slot t, the packet hits its destination if the distance and directional reception (DTDR) in VANETs. We approxi-
between the packet and its destination is less than or equal mate the directional antenna as a circular sector with angle θ,
to L [13]. the beam radius equals to the transmission/reception range r,
To facilitate the understanding, some important notations as shown in Fig. 2. We further assume the angle of sector
used in this paper are listed in Table I. approximates the beamwidth of the antenna [26]. In reality,
the directional antenna consists two portions: a mainlobe
III. SYSTEM MODEL which is in the transmission direction and several smaller
A. NETWORK MODEL backlobes in nontransmission direction arising from low effi-
As shown in Fig. 1, we consider the vehicular ad hoc networks ciency in antenna design. For the convenience of analysis, we
with n mobile vehicles (nodes) distributed in a unit square, ignore the impact of backlobes, i.e., the directional antenna
which is assumed to be a torus. In the multicast transmission gain is within specific angle θ and the backlobes gain is tend
model, we assume that there are ns multicast sessions and to zero.
each session has one source and p destinations. Each vehicle We consider that every vehicle is equipped with one direc-
is exactly the source of one session and the destination of tional antenna and each antenna can be steerable. That is to
another session. Moreover, every vehicle can serve as a relay say, the antenna beam can be placed towards any direction
node in any multicast session. Thus, we can obtain that n = at any time slot. Thus, the probability that the beam covers a
ns (1 + p) vehicles in the VANET. direction is θ/2π ∈ (0, 1).

B. DIRECTIONAL ANTENNA MODEL C. COMMUNICATION MODEL

In order to improve the capacity of VANETs, we adopt the In real VANETs, the radio signal propagation of a vehicle can
directional antenna model, which is directional transmission be interfered by many factors (obstacles, road geometry, etc).

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J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

Martinez et al. [24] propose a new propagation model, which

is called Real Attenuation and Visibility (RAV). However,
for the convenience of analysis, in this paper we adopt the
protocol model introduced in [5] to analyze the impact of
interference on capacity in VANETs. For simplicity, denote
ri as the transmission radius of node i. We assume that all
vehicles have the same transmission radius and common
transmission power P. If node i transmits to node j success-
fully, then two following conditions need to be satisfied:
(1) The position of receiving node is within the transmis-
sion range of the transmitter, i.e.,
|Xi (t) − Xj (t)| ≤ ri .
(2) Other transmitters Xk delivering packets at the same
time slot does not interfere the receiving node j, i.e.,
FIGURE 4. One-dimensional random walk model.
|Xk (t) − Xj (t)| ≥ (1 + 1)ri . the multicast sessions ns are both even numbers. Then, let
Here 1 > 0 denotes the guard zone, which is a constant that n/2 nodes, named H-nodes, move on the horizontal lines; and
doesn’t depend on n. Xi (t) not only denotes the location of the other n/2 nodes, named V-nodes, move on the vertical
a node but refers to the node itself at time slot t. According lines. We further assume that sources and destinations are
to the above two conditions, when node i transmits to node j the same type of nodes. Next, each vehicle’s horizontal
successfully if and only if the antenna beams of i and j cover (or vertical) orbit will be divided into 1/S intervals. At the
each other. We further assume that each transmitter-receiver beginning of time slot, a node will randomly and uniformly
pair can deliver W bits/s in a successful transmission. move to one of two adjacent intervals or stay at the current
interval, as shown in Fig. 4. Finally, the probability for each
D. MOBILITY MODEL interval to be selected as the destination interval is 1/3.
In this paper, we focus on the following mobility models. Two above mobility models can be very practical in real
VANETs. For example, one vehicle is at two lane intersection
in a single time slot. The vehicle will have two choices in the
next time slot, one of which is moving across the intersection.
The other is turning around and moving on the opposite
lane. The trajectory of vehicle resembles the one-dimensional
random walk model. However, the real trajectory of vehicles
may have some differences with theoretical random walk
mobility model, we find that these theoretical mathematical
models can capture the essential mobility features of vehicles.
We consider two time scales of vehicular mobility in this
paper. One is the fast mobility, i.e., the mobility of vehicles
is at the same time scale as the packets transmission. So, the
packets delivered to destinations can be finished only by one-
hop in a single time slot and W is a constant independent of n.
Fast mobility can simulate the scenario that the vehicles move
FIGURE 3. Two-dimensional random walk model.
in the uncrowded road condition. The other is slow mobility,
i.e., the mobile speed of vehicles is much slower than the
(1) Two-dimensional random walk model: Consider that packets transmission. That is, the packets transmission can
all vehicles are in a unit square network, which is fur- be achieved by multihop fashion within a single time slot
ther divided into 1/S 2 smaller squares of the same size. and W = (n). Thus, the packet size can be considered as
We assume that S is an integer. Each smaller square will be W /h(n) for h(n) = O(n) to guarantee h(n)-hops transmission
called a cell. Two cells are said to be adjacent if they share in a time slot. Slow mobility can simulate the scenario that
a common point, as shown in Fig. 3. At initial time slot, a the vehicles move in the traffic jam condition.
vehicular node will independently and uniformly select one of
nine adjacent cells (including itself cell) and stay there in the IV. THE UPPER BOUND CAPACITY ANALYSIS
next time slot. So the probability for each cell to be selected A. TWO-DIMENSIONAL RANDOM WALK FAST
as the destination cell is 1/9. MOBILITY MODEL
(2) One-dimensional random walk model: First, In this section, we investigate the upper bound on the multi-
we assume that the total number of mobile vehicles n and cast capacity of VANETs with the two-dimensional random

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 J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

walk fast mobility model under the conditions of the direc- minimum transmission distance between source node si and
tional antenna and a delay constraint D. First, we introduce its nearest destination denoted by D(si , t), i.e.,
some notations, which will be used in the following proof.
D(si , t) = min dist(si (t), di,j (t)).
• λd (T ): The number of bits that are successfully transmit- 1≤j≤p
ted to destinations in [0,T], when the packets are directly
Inspired by the definition of hitting distance, we know a
transmitted to their destinations.
successful transmission at a time slot can be achieved if
• λr (T ): The number of bits that are successfully trans-
and only if the transmission radius ri of the source node si
mitted to destinations in [0,T], when the packets are
needs to be at least L. Hence, we assume all nodes utilize
delivered to their destinations by relays.
a common transmission radius L, i.e., ri = L. The packet
• λ(T ): λ(T ) = λd (T ) + λr (T ), which denotes the total
can hit its one of destinations with probability θ4ππL
2 2
2 S 2 . Then,
number of bits that are successfully transmitted to desti-
the probability that a packet can hit all of its destinations is
nations in [0,T].
1 − (1 − θ4ππL
2 2
p
• B: Index of a bit of packet stored in the VANET. 2 S 2 ) . Based on the properties of random walk
• αB : The transmission radius used to deliver bit B. model in [13], we obtain that
Then, before deriving the aggregate multicast throughput
capacity of a VANET, we give some important lemmas, which θ 2 π L 2 99 S 2 p
are useful in the following analysis. P(D(si , t) ≤ L, covered by si ) ≤ 1 − (1 − ) 10
4π 2 S 2
Lemma 1: Assume that all nodes are equipped with direc-
tional antennas under the protocol model, the following 99 2 2
≤ θ L p ≤ θ 2 L 2 p,
inequalities hold: 40π
λd (T ) ≤ ns pWT , (1) which implies
λr (T ) ≤ ns (p + 1)WT ,
" T n #
(2) XX s
E 1D(si ,t)≤L,covered by si ≤ θ 2 L 2 Tns p.
X]
B[T
12 2WT
(αB )2 ≤ . (3) t=1 i=1
4 θ
B=1 A source node can send at most W bits in each successful
Inequality (1) holds since the number of bits direct delivered transmission, then we have
in T time slots cannot exceed ns pWT . Inequality (2) holds "B[T ] # "B[T ] #
since the maximum number of bits transmitted by relays can-
X X
E[B[T ]] = E 1αB ≤L + E 1αB >L
not exceed ns (p + 1)WT in T time slots. This is because that B=1 B=1
the source vehicle can serve as a relay node. Inequality (3)
ns
T X
" #
holds since the probability that the directional antenna covers X
a direction is θ/2π and the disk of radius 1αB /2 times the ≤ WE 1D(si ,t)≤L,covered by si
length of transmission range centered at receivers should be t=1 i=1
disjoint under the protocol model. "B[T ] #
X
Lemma 2: Consider the protocol model and the random +E 1αB >L
walk mobility model with the directional antenna, there exists B=1
k > 0, the following inequality holds [22]: "B[T ]
X
#

E[λ(T )] ≤ θ L Tns pW + E
2 2
1αB >L .
16kWT B=1
≤ 5k log(θ ns p)E[B[T ]] + p(log p) log(θ ns p).
12 Using Cauchy-Schwarz inequality and inequality (3), we
Now, we consider the scenario that the source vehicles trans- have
mit packets directly to the destinations with directional anten- !2
X] X]
B[T B[T
! B[T ] !
X
nas without delay constraints. αB ≤ 1 (αB ) 2
Lemma 3: Under the conditions of the protocol model and B=1 B=1 B=1
DTDR model, we consider the two-dimensional random walk
8WT
fast mobility, when the packets are directly transmitted from ≤ B[T ] .
the source vehicle to their destinations, then we have θ 12
√ which implies
4 π WT √
 
E[λd (T )] ≤ 5k log(θ ns p) ns p
1(1 − θ/2π )
"B[T ] # r !
X 8WT hp i
16kWT E αB ≤ E B[T ] . (4)
+ p(log p) log(θ ns p). θ 12
12 B=1

Proof: Let si denote the source of multicast session i, and This inequality shows the upper bound on the expected dis-
let di,j denote the jth destination in the multicast session i, the tance travelled of all bits. Furthermore, according to Jensen’s

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J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

inequality and inequality (4), we obtain that Furthermore, we know that the total direct transmission
r "r # distance that all bits are travelled is less or equal to the
8WT p 8WT
E[B[T ]] ≥ E B[T ] transmission distance by multihop transmissions. Hence,
θ 12 θ 12 we have
"B[T ] #
X X]
B[T
αB
X
≥E 1αB ≤L ≤ 1H (B)≤L ,
B=1 B=1 B∈3r [T ]
"B[T ] #
and
X
≥ LE 1αB >L
"B[T ] # "B[T ] #
B=1 X X
≥ L(E[B[T ]] − θ 2 L 2 Tns pW ), E[B[T ]] = E 1αB ≤L + E 1αB >L
q B=1 B=1
E[B[T ]]
then, let L = 2πθ WTns p , and substituting L into above   "B[T ] #
inequality, we have X X
√ ≤ E 1H (B)≤L  + E 1αB >L
4 π WT √ B∈3r [T ] B=1
ns p ≥ E[B[T ]].
1(1 − θ/2π ) "B[T ] #
By submitting into the bound on λ(T ) in Lemma 2, we have X
√ ≤ ns (p + 1)WT θ L pD + E 2 2
1αB >L .
4 π WT √
 
d B=1
E[λ (T )] ≤ 5k log(θ ns p) ns p
1(1 − θ/2π ) Then, using the Cauchy-Schwarz inequality, we have
16kWT
+ p(log p) log(θ ns p).  !2
12 X] X]
B[T B[T
! B[T ] !
X
In the following analysis, we consider the data packets trans- αB ≤ 1 (αB ) 2

mitted from relays to destinations, i.e., the packets will be B=1 B=1 B=1
transmitted in multihop fashion within a delay constraint D. 8WT
We will calculate the upper bound on the expected number of ≤ B[T ] ,
θ 12
bits under the relaying scheme.
which implies that
Lemma 4: Under the conditions of the protocol model and r "r #
DTDR model, we consider the two-dimensional random walk 8WT p 8WT
fast mobility, when packets have to be transmitted from relays E[B[T ]] ≥ E B[T ]
θ 12 θ 12
to their destinations with a delay constraint D, then we have
 √ "B[T ]
4 π WT (p + 1) p
 #
r
X
E[λ (T )] ≤ 5k log(θ ns p) ns D ≥E αB
1(1 − θ/2π )
B=1
16kWT
+ p(log p) log(θ ns p). "B[T ]
12
#
X
Proof: In the proof of this Lemma, we assume that every ≥ LE 1αB >L
B=1
vehicle can be a relay node. So we can use inequality (2) to
bound the maximum number of bits at relay nodes. Let H (B) ≥ L(E[B[T ]] − ns (p + 1)WT θ 2 L 2 pD),
denote the minimum distance between the relay node carry- √
ing bit B and one of the p destinations in a multicast session since the inequality holds for L ∈ [0, S/ π ), and choose
under the conditions of the directional antenna beamwidth θ
q
E[B[T ]]
L = 2πθWTn s p(p+1)D
, we have
and a delay constraint D. Follow the analysis in Lemma 3, √
we calculate the probability that a packet can hit all of its 4 π WT p
ns p(p + 1)D ≥ E[B[T ]],
destinations in one of D time slots is 1 − (1 − θ4ππL
2 2
Dp , for
√ 2S2 ) 1(1 − θ/2π )
any L ∈ [0, S/ π ) and based on the properties of random we know that n = ns (1 + p) in the multicast transmission
walk in [13], we have model, thus, the above inequality can be further simplified as
 99 S 2 pD √
θ 2 π L 2 10

P(H (B) ≤ L) ≤ 1 − 1 − 4 π WT p
4π 2 S 2 npD ≥ E[B[T ]].
1(1 − θ/2π )
99 2 2
≤ θ L p ≤ θ 2 L 2 pD, Submitting to Lemma 2, we can finally have
40π √
4 π WT p
 
which implies r
E[λ (T )] ≤ 5k log(θ ns p) npD
  1(1 − θ/2π )
X
E 1H (B)≤L  ≤ ns (p + 1)WT θ 2 L 2 pD. 16kWT
+ p(log p) log(θ ns p). 
B∈λr (T ) 12
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 J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

Theorem 1: Under the two-dimensional random walk fast TABLE 2. Simulation parameters.
mobility model, the aggregate multicast capacity of VANETs
with directional antennas and a delay constraint D is
 
log p log(θ ns p) p √
λ(n) = O ( npD + ns p) .
1 − θ/2π

FIGURE 5. Multicast capacity (p > 1) with the directional antenna under FIGURE 6. Unicast capacity (p → 1) with the directional antenna under
the 2D random walk fast mobility model vs. the multicast capacity with the 2D random walk fast mobility model vs. the unicast capacity with the
the omnidirectional antenna under the 2D-i.i.d fast mobility model. omnidirectional antenna under the 2D random walk fast mobility model.

directional transmission and directional reception (DTDR)

Remark 1: From the Lemma 3 and Lemma 4, we have model.
obtained the aggregate multicast throughput capacity of a
VANET when all vehicular nodes follow the two-dimensional B. TWO-DIMENSIONAL RANDOM WALK SLOW
random walk fast mobility model. We can find that the MOBILITY MODEL
throughput capacity by using relays dominates the throughput In this section, we investigate the upper bound on multicast
by direct transmissions. Furthermore, if D is large enough, capacity of the two-dimensional random walk slow mobility
the aggregate multicast throughput capacity can be simpli- model. From Section II, we know that the mobility of nodes
p log(θ ns p) √ is much slower than the transmission of packets under slow
 
fied as O log1−θ/2π npD . Since we know that there
mobility model. Hence, in a time slot, the packets delivery can
are n vehicles,q the maximum throughput per vehicle is use multihop transmissions. Let hB denote the number of hops
O( log1−θ/2π
p log(θ ns p) pD
n ) under the fast mobility model. bit B travels to destinations in the time slot. The Euclidean
Specially, when the beamwidth θ tends to 2π and the distance bit B travels in a time slot denoted by L(B). And in
multicast destination p tends to 1,qwe find that the multicast hop h for 1 ≤ h ≤ hB , the packets transmission radius is αBh .
capacity of pre-node trends to O( Dn ), which means the uni- Then, we have the following lemmas.
cast capacity with omnidirectional antenna model under the Lemma 5: For any vehicular mobility model, the following
random walk model in [13] can be regarded as the special case inequalities hold under the directional antenna model and the
of our result. The numerical simulations are shown in Fig. 5 protocol model:
and Fig. 6. We compare the obtained throughput capacity of
X] X
B[T hB
pre-node with the existing results. In Table II, we list the 1 ≤ ns (p + 1)WT , (5)
essential numerical simulation parameters. B=1 h=1
The first case that when the number of destinations X] X
B[T hB
12 h 2 2WT
p > 1 is plotted in Fig. 5, we find that the directional antenna (α ) ≤ . (6)
can improve the multicast capacity compared with [25]. 16 B θ
B=1 h=1
In addition, Fig. 5 shows that the multicast capacity increases Lemma 6: Under the conditions of the protocol model
as the beamwith increases. This can be explained as a larger and DTDR model, we consider the two-dimensional random
beamwidth of a directional antenna equipped in the vehi- walk slow mobility model, when the packets are directly
cle can lead to an increase in the transmission probability transmitted from source vehicle to their destinations, then
between transmitter-receiver pairs. Hence, the upper bound  2
on multicast capacity can be improved. The second case that d 8π ns 3 1
E[λ (T )] ≤ 5k log(θ ns p) WT (p(p+1)) 3
when the number of destinations trends to 1 is shown in Fig. 6, 1(1 − θ/2π)
which can be approximately considered as the unicast trans- 16kWT
mission. We find our result is better than [13] by adopting the + p(log p) log(θ ns p).
12
3964 VOLUME 5, 2017
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Proof: Using Cauchy-Schwarz inequality and inequality

(4) and (5), we have
!2
X] X X] X
B[T hB B[T hB
! B[T ] h !
XX B
αB
h
≤ 1 h 2
(αB )
B=1 h=1 B=1 h=1 B=1 h=1
X] X
B[T hB
≤ ns (p + 1)WT (αBh )2
B=1 h=1
32WT
≤ ns (p + 1)WT ,
θ 12
Ph B
h=1 αB
And h ≥ L(B), we have
√ r B[T ]
4 2WT ns (p + 1) X
≥ L(B).
1 θ FIGURE 7. The multicast capacity of per-node under the 2D fast mobility
B=1 model vs. the multicast capacity of per-node under the 2D slow mobility
model.
Similar to Lemma 3, we have
"B[T ] # "B[T ] #
X X
E[B[T ]] = E 1αB ≤L + E 1αB >L Combining Lemma 6 and Lemma 7, we can obtain the
B=1 B=1 following Theorem.
Theorem 2: Under the two-dimensional random walk slow
"B[T ] #
X
≤ θ L Tns pW + E
2 2
1αB >L . mobility model, the aggregate multicast capacity of VANETs
B=1 with directional antennas and a delay constraint D is
applying the Jensen’s inequality, we obtain that 
ns (p + 1) 2 1 1

" √ λ(n) = O log p log(θ ns p)( ) 3 (p 3 + (Dp) 3 ) .
1 − θ/2π
r # "B[T ] #
4 2WT ns (p + 1) X
E ≥E L(B)
1 θ Remark 2: Since we know that n = ns (p + 1) and the
B=1
"B[T ] # throughput capacity by using relays dominates the through-
X put by direct transmissions,
 the throughput capacity can
≥ LE 1L(B)>L n 2 1

B=1
be further simplified as O log p log(θ ns p)( 1−θ/2π ) 3 (Dp) 3 .
≥ L(E[B[T ]] − θ 2 L 2 Tns pW ), Hence, under the slow mobility model, q the maximum
q throughput per-node is O( log p log(θ ns2p) 3 Dp
n ). As shown in
E[B[T ]]
let L = 2πθ WTns p , we obtain that (1−θ/2π) 3
Fig. 7, it compares the multicast capacity in Theorem 1
 2 with the result in Theorem 2, which indicates the multi-
8π ns 3 1
WT (p(p + 1)) 3 ≥ E[B[T ]], cast capacity of the two-dimensional slow mobility becomes
1(1 − θ/2π ) smaller than the two-dimensional fast mobility with the same
submitting to Lemma 2, we can finally have delay constraint and the directional beamwidth. This can be
 2 explained as follows. Assume that the speed of a vehicle in
8π ns 3 1
fast mobility is Vf and the speed of a vehicle in slow mobility
E[λd (T )] ≤ 5k log(θ ns p) WT (p(p+1)) 3
1(1 − θ/2π ) is Vs , 0 < Vs  Vf . For the same transmission radius, the
16kWT time required for the vehicle with the speed of Vf moving
+ p(log p) log(θ ns p). 
12 to the receiver and transmitting the packets is much smaller
Lemma 7: Under the conditions of the protocol model and than a given delay D. In this case, the multicast capacity of
DTDR model, we consider the two-dimensional random walk a vehicle is denoted by λf (n). For the slow mobility model,
slow mobility model, when the packets have to be transmitted the time required for the vehicle with the speed of Vs is much
from relays to their destinations with a delay constraint D, larger than a given delay D. Hence, some packets can’t be
then we have delivered to the destinations within the delay D. In this case,
 2 the multicast capacity is denoted by λs (n). When the speed of
8π ns (p + 1) 3 1
a vehicle is Vs < V < Vf , the multicast capacity of per-node
E[λr (T )] ≤ 5k log(θ ns p) WT (Dp) 3
1(1 − θ/2π ) is between λf (n) and λs (n), as shown in Fig. 8.
16kWT
+ p(log p) log(θ ns p).
12 C. ONE-DIMENSIONAL RANDOM WALK FAST
Proof: The proof process is similar with the packets MOBILITY MODEL
transmitted from relays to destinations in the In this section, we will study the one-dimensional random
two-dimensional random walk fast mobility model.  walk mobility model with fast mobiles. We first give two

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 J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

we further obtain,
"B[T ] # "B[T ] #
X X
E[B[T ]] = E 1αB ≤L + E 1αB >L
B=1 B=1
"B[T ] #
θ 2 Lp X
≤ ns TW + E 1αB >L .
2
B=1

Then, using Cauchy-Schwarz inequality, we have

!2
X] X]
B[T B[T
! B[T ] !
X
αB ≤ 1 (αB ) 2

B=1 B=1 B=1

8WT
≤ B[T ] ,
FIGURE 8. The multicast capacity λf (n) of a vehicle when the speed is Vf θ 12
vs. the multicast capacity λs (n) of a vehicle when the speed is Vs .
and applying Jensen’s inequality,
r "r #
lemmas on the direct transmission and the packets transmit- 8WT p 8WT
ted by relays. Then, we will obtain the maximum through- E[B[T ]] ≥ E B[T ]
θ 12 θ 12
put capacity with the delay constraint. Furthermore, we "B[T ] #
find the throughput capacity in this case is larger than the X
two-dimensional fast mobility model. ≥E αB
Lemma 8: Under the conditions of the protocol model and B=1
"B[T ] #
DTDR model, we consider the one-dimensional random walk X
fast mobility model, when the packets are directly transmitted ≥ LE 1αB >L
from sources to their destinations, then we have B=1

√ !2 θ 2 Lp
3 ≥ L(E[B[T ]] − ns TW ),
1 2 2π ns p 2
E[λd (T )] ≤ 5k log(θ ns p)WT θ 3
1(1 − θ/2π ) E[B[T ]]
let L = πθ ns pTW , we have
16kWT
+ p(log p) log(θ ns p). √ !2
12 1 2 2π ns p 3
WT θ 3 ≥ E[B[T ]].
Proof: In this case, each node moves along a straight 1(1 − θ/2π)
line. If the mobile trajectory of sources and destinations are
vertical to each other, then D(si , t) < L holds only if the two Finally, we obtain that
vehicles are in the same square with side length 2L at some √ !2
3
time slot t. Thus, 2L < 1, we can have 1 2 2πns p
E[λd (T )] ≤ 5k log(θns p)WT θ 3
1(1 − θ/2π )
θ 2 4L 2 33 2
P(D(si , t) ≤ L) ≤ 1 − (1 − ) 10 S p . 16kWT
4π 2 S 2 + p(log p) log(θ ns p). 
12
If the mobile trajectory of sources and destinations are paral-
lel to each other, we have Lemma 9: Under the conditions of the protocol model and
DTDR model, we consider the one-dimensional random walk
θ 2 2L 33 Sp fast mobility model, when the packets have to be transmitted
P(D(si , t) ≤ L) ≤ 1 − (1 − ) 10 .
4π 2 S from relays to their destinations with a delay constraint D,
Thus, we have assumed that sources and destinations are the then we have
same type nodes in Section II, for L ≤ 1/2, we conclude √ !2
3
1 2 2π ns p(p + 1)D
that E[λr (T )] ≤ 5k log(θ ns p)WT θ 3
1(1 − θ/2π )
θ 2 2L 33 Sp θ 2 Lp
P(D(si , t) ≤ L) ≤ 1 − (1 − ) 10 ≤ . 16kWT
4π 2 S 2 + p(log p) log(θ ns p).
12
which implies Proof: The proof of this Lemma is similar to Lemma 4
" T n # and Lemma 8. 
XX s
θ 2 Lp
E 1D(si ,t)≤L,covered by si ≤ ns T , Theorem 3: Under the one-dimensional random walk
2 fast mobility model, the aggregate multicast capacity of
t=1 i=1

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J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

VANETs with directional antennas and a delay constraint Proof: Recall that L(B) denotes the Euclidean distance
D is that bit B travels in time slot t and αBh denotes the the packets
transmission radius in hop h. Assume that the mobile trajec-
λ(n)
tory of sources and destinations are parallel to each other,
1
!
log p log(θns p)θ 3  2 2
 which is same as the one-dimensional fast mobility model.
=O 2
(ns p) 3 + (ns p(p + 1)D) 3 . Then, we can obtain that
(1 − θ/2π ) 3
θ 2 2L 33 Sp θ 2 Lp
Remark 3: Since the throughput by using relays dominates P(L(B) ≤ L) ≤ 1 − (1 − ) 10 ≤ .
the throughput by direct transmissions, the aggregate 4π 2 S 2
throughput capacity under the one-dimensional fast mobility Using Cauchy-Schwarz inequality and Lemma 5, we have
  1
log p log(θns p)θ 3 2
model can be further simplified as O (npD) 3 . !2
X] X X] X
2 B[T hB B[T hB
! B[T ] h !
B
(1−θ/2π) 3 XX
And 
the maximumqmulticast
 throughput capacity of per-node αBh
≤ 1 h 2
(αB )
log p log(θ ns p) 3 θ p2 D2 B=1 h=1 B=1 h=1 B=1 h=1
is O 2 n .
(1−θ/2π) 3 X] X
B[T hB
As shown in Fig. 9, it compares the multicast capac- ≤ ns (p + 1)WT (αBh )2
ity in Theorem 1 with the numerical result in Theorem 3, B=1 h=1
which indicates the multicast throughput capacity in the 32WT
≤ ns (p + 1)WT ,
one-dimensional fast mobility model is larger than the θ 12
two-dimensional fast mobility model with the same delay Ph B
h=1 αB
and h ≥ L(B), we have
constraint and the directional beamwidth. In the order sense,
the maximum multicast capacity of per-node in the two- √ r B[T ]
dimensional fast model 4 2WT ns (p + 1) X
q and the one-dimensional fast model ≥ L(B).
1 θ
q
3 D2 p2
is O( Dp n ) and O( n ), respectively. Both of them are B=1
nondecreasing functions, which means the multicast capacity Similar with the proof of Lemma 6, we have
is larger in the one-dimensional fast model under the given
"B[T ] # "B[T ] #
delay D. Moreover, for the one-dimensional random walk X X
mobility network, the location of destinations that the source E[B[T ]] = E 1αB ≤L + E 1αB >L
can choose becomes less compared with the two-dimensional B=1
"B[T ]
B=1
#
mobility network. In other words, the probability that the θ 2 Lp X
vehicle selects the destination locations in the next time slot ≤ ns TW + E 1αB >L .
2
becomes larger in the one-dimensional fast mobility model B=1
than the two-dimensional fast mobility model. In brief, the Then, using Jensen’s inequality, we have
figure suggests that our theoretical results coincide with the " √ r # "B[T ] #
intuitive thinking accurately. 4 2WT ns (p + 1) X
E ≥E L(B)
D. ONE-DIMENSIONAL RANDOM WALK SLOW
1 θ
B=1
MOBILITY MODEL
"B[T ] #
X
In this section, we will investigate multicast capacity of the ≥ LE 1L(B)>L
one-dimensional random walk slow mobility model. Similar B=1
to above deducing process, we will give two lemmas on θ 2 Lp
the multicast throughput capacity under the conditions that ≥ L(E[B[T ]] − ns TW ),
2
packets transmitted from the source to destinations directly
E[B[T ]]
and from relays to destinations, respectively. Then, we will by choosing L = πθ ns pTW , we have
derive the maximum aggregate multicast throughput capacity √
of a VANET with a delay constraint. The multicast capacity 4 2π p
1 1 3
WT (θ (p + 1)) ( 4 ) 2 ns4 ≥ E[B[T ]].
in this case becomes comparatively smaller as a result of the 1(1 − θ/2π )
low rate of the packets transmission. By submitting to Lemma 2, we finally have
Lemma 10: Under the conditions of the protocol model √
and DTDR model, we consider the one-dimensional random d 4 2π p 1 1 3
E[λ (T )] ≤ 5k log(θ ns p)WT (θ (p+1)) ( ) 2 ns4
4
walk slow mobility model, when the packets are directly 1(1−θ/2π )
transmitted from sources to their destinations, then we have 16kWT
√ + p(log p) log(θ ns p). 
1 4 2π p 1 3 12
d
E[λ (T )] ≤ 5k log(θ ns p)WT (θ (p +1)) 4 ( ) 2 ns4
1(1−θ/2π ) Lemma 11: Under the conditions of the protocol model and
16kWT DTDR model, we consider the one-dimensional random walk
+ p(log p) log(θ ns p).
12 slow mobility model, when the packets have to be transmitted
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 J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

TABLE 3. Comparing multicast capacity of per node with unicast capacity under different random walk mobility models.

FIGURE 9. The multicast capacity of per-node under the 2D fast mobility

model vs. the multicast capacity of per-node under the 1D fast mobility
model.
FIGURE 11. The multicast capacity of per-node under the 2D slow
mobility model vs. the multicast capacity of per-node under the 1D slow
mobility model.

FIGURE 10. The multicast capacity of per-node under the 1D fast mobility
model vs. the multicast capacity of per-node under the 1D slow mobility
model.
FIGURE 12. The delay D vs. the multicast capacity of per-node under four
different mobility models.
from relays to their destinations with a delay constraint D,
then we have with directional antennas and a delay constraint D is
1
!
√ log p log(θ ns p)θ 4 1 3

4 2π Dp 1 λ(n) = O 1
(D(p + 1)) (ns (p + 1))
2 4 .
(1 − θ/2π ) 2
1 3
E[λ (T )] ≤ 5k log(θ ns p)WT θ 4 (
r
) 2 (ns (1+p)) 4
1(1−θ/2π )
16kWT Remark 4: In this case, the q
maximum throughput capacity
+ p(log p) log(θ ns p). θ (p+1)2 D2
of per-node is O( log p log(θ ns1p)
4
12 n ), which shows the
(1−θ/2π) 2
throughput is smaller than the one-dimensional fast mobility
Proof: The proof of this Lemma is similar to the two- with the same delay constraint.
dimensional random walk slow mobility model.  Fig. 10 compares the multicast capacity in Theorem 3
Theorem 4: Under the one-dimensional random walk slow with the result in Theorem 4. It also can be seen that the
mobility model, the aggregate multicast capacity of VANETs multicast capacity of one-dimensional fast mobility is larger

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J. Ren et al.: Multicast Capacity for VANETs With Directional Antenna and Delay Constraint

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oriented overview,’’ IEEE Access, vol. 4, pp. 9293–9307, 2016.
and information engineering from Huaiyin Normal
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University in 2014. He is currently pursuing the
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Apr. 2001, pp. 1360–1369. and Technology, Donghua University, Shanghai,
[7] X. Lin, G. Sharma, R. R. Mazumdar, and N. B. Shroff, ‘‘Degen- China. His research interests include the capacity
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Jun. 2006.

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GUANGLIN ZHANG received the B.S. degree DEMIN LI received the Ph.D. degree in electronic
in applied mathematics from Shandong and computer engineering from the Nanjing Uni-
Normal University in 2003, the M.S. degree versity of Science and Technology, China, in 1998.
in operational research and control theory from He is currently a Professor of the Department
Shanghai University in 2006, and the Ph.D. degree of Telecommunication and Electronic Engineer-
in electronic engineering from Shanghai Jiao Tong ing, College of Information Science and Technol-
University in 2012. From 2013 to 2014, he was a ogy, Donghua University, Shanghai, China. From
Post-Doctoral Research Associate with the Insti- 2002 to 2005, he was a Research Scientist with
tute of Network Coding, The Chinese University Nanjing University, China. He has authored over
of Hong Kong. He is currently an Associate Pro- 60 research papers in journals and conferences.
fessor and also the Department Chair of the Department of Communication His recent research interests include telecommunication system engineering,
Engineering, Donghua University. His research interests include the capacity wireless mobile networking, mobile decision theory, and mobile decision
scaling of wireless networks, vehicular networks, smart micro-grid, and support systems. He is currently as Associate Chairman of the Circuits and
energy management of data centers. He serves as a Technical Program Systems Committee, Shanghai.
Committee Member of the IEEE Globecom from 2016 to 2017, the Mobile
and Wireless Networks Symposium of the IEEE ICC 2015–2017, the Signal
Processing for Communications Symposium of the IEEE ICC 2014–2017,
the IEEEVTC2017-Fall, the Wireless Networking and Multimedia Sympo-
sium of the IEEE/CIC ICCC 2014, and the Future Networking Symposium
and Emerging Areas in Wireless Communications of WCSP in 2014, APCC
2013, and WASA 2012. He serves as an Associate Editor of the IEEE Access
and as an Editor on the Editorial Board of China Communications. He is the
Local Arrangement Chair of the ACM TURC 2017.

3970 VOLUME 5, 2017