A Performance Modeling of Connectivity in Vehicular Ad Hoc ...lila/M2R/6.pdf2440 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 4, JULY 2008 A Performance Modeling of Connectivity

2440 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 4, JULY 2008

A Performance Modeling of Connectivity inVehicular Ad Hoc Networks

Mehdi Khabazian, Student Member, IEEE, and Mustafa K. Mehmet Ali, Member, IEEE

Abstract—In this paper, we study the statistical properties ofthe connectivity of Vehicular Ad hoc NETworks (VANETs) withuser mobility. It is assumed that the nodes travel along a multilanehighway that allows vehicles to pass each other. The nodes arriveat the highway through one of the traffic entry points according toa Poisson process and then travel in the same direction accordingto a user mobility model until they reach their exit points. Thenodes on the highway may be able to communicate with each other.We derive the probability distribution of the node population sizeon the highway and the node’s location distribution. Then, wedetermine the mean cluster size and the probability that the nodeswill form a single cluster. The analysis of this paper also applies toany path in a network of highways, as well as to two-way traffic.The numerical results show the significance of mobility on theconnectivity of VANETs. We also present simulation results thatconfirm the accuracy of the analysis. The results of this paper maybe used to study the routing algorithms, throughput, or delay inVANETs.

Index Terms—Ad hoc, connectivity, mobility, Vehicular Ad hocNETwork (VANET).

I. INTRODUCTION

MOBILE Ad hoc NETworks (MANETs) are comprisedof self-organizing mobile nodes that lack a network

infrastructure, such as base stations. This technology has beenproposed as a complementary to the fourth-generation wirelessnetworks [1]. It will be deployed in circumstances where theinfrastructure is not available or is insufficient. An emergingnew type of ad hoc network is the Vehicular Ad hoc NETwork(VANET), which envision intervehicle communications. In thistype of network, the vehicles are the mobile nodes of thenetwork. VANETs have mobility characteristics that distinguishthem from the other MANETs. They have a highly dynamictopology due to the fast motions of vehicles, and their motionsare restricted to a geographical pattern, such as a network ofhighways. Further, the nodes do not have power constraintsdue to their access to the electrical system of the vehicle. Theprospective applications of VANETs are categorized into twogroups as comfort and safety applications [2]. The first groupis expected to improve the passengers’ comfort and optimizetraffic efficiency, whereas the second one improves driving

Manuscript received July 10, 2006; revised April 23, 2007, July 2, 2007, andSeptember 12, 2007. The review of this paper was coordinated by Prof. B. Li.

The authors are with the Department of Electrical and Computer Engi-neering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.912161

safety. Examples of comfort applications are traffic informa-tion systems, alternative route selection, weather information,and mobile Internet access. Examples of safety applicationsare emergency warning systems, collision avoidance throughdriver assistance, road condition warning, and lane-changingassistance. In safety applications, two types of messages maybe exchanged in the network, i.e., periodic and event driven.The periodic message exchange (or beaconing) is preventive innature, and its objective is to avoid the occurrence of dangeroussituations. The beaconing messages may contain informationregarding the position, direction, and speed of the sendingvehicle. The event-driven messages may be generated as a resultof a dangerous situation, such as adjacent vehicles travelingin high speeds. In such a case, the received information mayeven be used to directly activate the actuator of a safety system.Successful communications in VANETs depend on the networkconnectivity of the vehicles.

In ad hoc networks, in addition to the nodes being sourcesor destinations of the data, they also cooperate with each otherto route the information within the network. As in VANETs,when the nodes are also mobile, the network may not have fullcommunications connectivity all the time, and the populationmay form several clusters, where a cluster refers to a set ofnodes that have direct or indirect communication with eachother. The connectivity impacts the main performance measuresof a network, such as throughput and delay.

The objective of this paper is to study the statistical propertiesof the connectivity of VANETs with user mobility and dynamicnode population. The mobility in VANETs is usually studiedusing a 1-D model [3], [4]. In the literature, the connectivityanalysis of ad hoc networks has been considered mostly underthe assumption of randomly distributed stationary nodes withinthe service area both for 1-D and 2-D cases [5]–[9]. Thesemodels fail to capture the effects of user mobility on theperformance of the network.

Next, we discuss the previous works on connectivity analy-sis, which includes user mobility. Reference [10] studies theconnectivity of a 1-D ad hoc network with a user mobil-ity model and derives a formula for the connectivity of asource–destination pair. References [11] and [12] analyzed theconnectivity of an ad hoc network with the random waypointmobility model. In [11], the motion of n nodes within a unitcircle has been considered, and an approximation for the prob-ability that the nodes forming a single cluster has been derived.Reference [12] studies the connectivity of a source–destinationpair in a 1-D network. All these studies assumed a constantnumber of nodes in their model, whereas, in a realistic situation,the population size of the nodes will be a random variable.

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KHABAZIAN AND ALI: PERFORMANCE MODELING OF CONNECTIVITY IN VEHICULAR AD HOC NETWORKS 2441

In [13], simulation is used to determine the minimum trans-mission range (MTR) that guarantees the network connectivityof VANETs at different traffic densities and road conditions.Reference [14] also studies the problem of critical MTR in amultidimensional region with a constant node population and auniformly distributed node location. The paper proposes someanalytical bounds for MTR.

Reference [15] considers the problem of automated schedul-ing of traffic in a city by constructing a simple mobility model.The authors derive a sufficient condition on the configurationof the vehicles on the road network and their routing, whichguarantees that a deadlock can be avoided. The objective ofthat work is to eliminate congestion from the roads, which isdifferent from ours.

In this paper, we assume a unidirectional highway that allowsvehicles to pass each other. In real scenarios, the number ofsimultaneous vehicle passings at any point is restricted bythe number of lanes on the highway; however, we assume norestriction on the number of passings for analytical simplicity.The accuracy of this assumption has been tested by simulation.

The arrival of nodes to the highway is assumed to follow aPoisson process. Following its arrival at an entry point, a nodestarts moving along the highway until it reaches its exit point,according to the user mobility model. A node that departs thehighway can no longer participate in communications. Thus,the node population on the highway is dynamically changing.We model each stream of nodes on the highway as an M/G/∞queue and determine the distribution of the node populationon the highway as well as the distribution of the distance ofa node from its arrival point at steady state. Then, we studythe statistical characteristics of a cluster seen by a new arrivalor a random node. We present the average cluster size and theprobability that all the nodes on the highway will form a singlecluster.

The analysis of this paper may also be applied to any pathin a network of highways. A user may be interested in findingthe traffic congestion information in different paths for makingroute selection. The analysis may also be applied to a systemthat allows the traffic flow in both directions. The integratednetwork may be helpful for collision avoidance. The results ofthis paper may be used in the study of routing algorithms [16],throughput, and delay in VANETs.

The remainder of this paper is organized as follows.Section II describes the system model. Section III presentsthe connectivity analysis of the network. Section IV describesthe simulation setup. Section V presents some numerical andsimulation results. Section VI presents the conclusions of thispaper.

II. SYSTEM MODEL

In this paper, we assume a unidirectional highway with mul-tiple lanes and length of R meters. The nodes will arrive at thehighway through one of the entry points according to a Poissonprocess, and then, they will begin to move along the highwayaccording to the assumed mobility model independent of allthe other nodes. The nodes depart from the highway throughone of the exit points. We are interested in the communications

Fig. 1. Highway model.

connectivity of the nodes, which are on the highway at steadystate. Next, we describe the highway and the nodes’ mobilitymodel, determine the population size of the nodes on thehighway, and then derive the probability density function (pdf)of the distance of a node from the beginning of the highway atsteady state.

A. Highway Model

We will assume that the highway is divided into I cells,which are numbered i = 1, 2, . . . , I . The length of each cellis equal to a node’s transmission range, which is d meters.We assume that the cells form K consecutive highway seg-ments. The kth segment begins at cell ik (i1 <, . . . , ik−1 <ik < ik+1, . . . , < I), where i1 = 1, and the length of thekth segment is given by rk = (ik+1 − ik)d units. Fig. 1 showsthe highway model.

The beginning of each segment is a service point that allowsthe entry and exit of the traffic from the highway. The end ofthe highway serves as the final exit point of the traffic. Thenew nodes arrive at the kth service point according to a Poissonprocess with the arrival rate λk. The old nodes arriving from thepreceding segment continue to the kth segment with probabilityαk or depart from the highway with probability 1 − αk. Wenote that a node cannot immediately depart from the servicepoint at which that it has just arrived. We let skj denote thestream of nodes with arrivals and departures at the servicepoint k and j, respectively. Letting λ̃kj denote the arrival rateof the skj th stream, it is then given by

λ̃kj = λk(1 − αj)j−1∏

�=k+1

α�. (1)

In the above, the product is assumed to have the value of 1 ifthe upper limit is less than the lower limit.

B. Mobility Model

In this section, we will determine the pdf of the distanceof a node from its arrival service point. We will assume acontinuous-time mobility model, which is similar to the modelin [17] but in 1-D. The movement of each node as a function oftime consists of a sequence of random intervals called mobilityepochs. The epoch durations of each node is independentidentically distributed (i.i.d.) with exponential distribution withparameter β. During each epoch, a node moves at a constantspeed independently chosen from a normal distribution withmean µ and variance σ2. Let ti−1 denote the beginning timeof the ith epoch and refer to it as the renewal epoch withthe assumption that t0 = 0. Since the epoch durations are


Fig. 2. Distance of a node from its arrival service point as a function of time.

exponentially distributed, the number of renewal epochs overany time interval has a Poisson distribution. We let Ti denotethe duration of the ith epoch and xi the amount of distancecovered during that epoch.

Fig. 2 shows the distance of a node from the service point atwhich it has arrived as a function of time. Then, we have thefollowing results regarding the statistics of renewal epochs andepoch durations.Lemma 1: Given that there are n epochs during the interval

(0, t), the joint pdf of renewal epochs ti, i = 1, . . . , n − 1, 0 ≤t1 ≤ · · · ≤ ti, . . . , tn−1 ≤ t is given by (n − 1)!/tn−1 [18].The duration of the ith epoch is given by

Ti ={

ti − ti−1, 1 ≤ i < n, t0 = 0t − tn−1, i = n.

(2)

Then, the jth moment of the random variable Ti is given by

E(T j

i |n epochs)

=(n − 1)!

tn

tn∫tn−1=0

· · ·ti+1∫

ti=0

· · ·t2∫

t1=0

× (ti − ti−1)jdt1, . . . , dti, . . . , dtn−1

1 ≤ i ≤ n, n ≥ 2, t0 = 0, tn = t. (3)

After evaluating the first two moments for the few initialvalues of the number of epochs, we conclude that

E(Ti|n epochs) =t

n, 1 ≤ i ≤ n (4)

E(T 2i |n epochs) =

2t2

n(n + 1), 1 ≤ i ≤ n. (5)

Let us define X(t) as the distance of a node as a functionof time from the service point at which it has arrived. X(t)will be denoted by Xn(t), given that there are n epochs duringthe time interval (0, t) and by Xn,Ti

(t) if the epoch durationsare also constant. Then, Xn,Ti

(t) will be given by Xn,Ti(t) =∑n

i=1 xi =∑n

i=1 ViTi, where Vi is the speed of a node duringthe ith epoch. Since Xn,Ti

(t) is given by a linear sum of i.i.d.normal random variables with constant coefficients, it will havea normal distribution with mean and variance given by

εxn,Ti(t) =E [Xn,Ti

(t)] = µ

n∑i=1

Ti (6)

θxn,Ti(t) = var [Xn,Ti

(t)] = σ2n∑

i=1

T 2i . (7)

The mean and variance of Xn(t) will be given by

εxn(t) =µn∑

i=1

E[Ti] (8)

θxn(t) =σ2n∑

i=1

E[T 2i ]. (9)

Where the corresponding expected values are given by (4)and (5)

εxn(t) = µt (10)

θxn(t) =2σ2t2

(n + 1). (11)

Finally, the mean of X(t) is given by

εx(t) =∞∑

n=1

εxn(t) Pr (n epochs in (0, t))

=∞∑

n=1

µt(βt)n−1e−βt

(n − 1)!(12)

or

εx(t) = µt (13)

and its variance by

θx(t) =∞∑

n=1

θxn(t) Pr (n epochs in the interval (0, t)) . (14)

Substituting (11) in (14), we have

θx(t) =∞∑

n=1

2σ2t2

(n + 1)(βt)n−1e−βt

(n − 1)!

=2σ2t2

β2t2

∞∑n=1

n(βt)n+1e−βt

(n + 1)!

=2σ2

β2

( ∞∑m=0

(m − 1)(βt)me−βt

m!+ e−βt

)= a(βt − 1 + e−βt) (15)

where a = 2σ2/β2.We will assume that the random variable X(t) will still have

a normal distribution with the mean and variance given by(13) and (15), respectively. Letting bx(t)(r) and Bx(t)(r) denotethe pdf and the cumulative distribution function (cdf) of X(t),we have

bx(t)(r) =1√

2πθx(t)

e−(r−εx(t))2/2θx(t) (16)

Bx(t)(r) = Pr(X(t) ≤ r) =

r∫0

bx(t)(y)dy. (17)

From (13) and (15), the mean and variance increase as thetime increases. As a result, the pdf shifts to the right, andits maximum value shrinks and gradually becomes more flat.Thus, the pdf of the distance of a node from its arrival servicepoint approaches a uniform distribution as the time increases.In Table I, we give a summary of the pdfs and cdfs defined inthis paper.


TABLE ILIST OF PROBABILITY FUNCTIONS

C. Steady-State Distribution of a Node’s Location

Next, we will determine the probability distribution of thedistance of a node from its arrival service point at steady state.Let the pdf and cdf of the distance of a node that belongs tothe skj th stream from its arrival service point be denoted bygxkj(t)(r) and Gxkj(t)(r), respectively. Since the nodes arriveat a service point according to a Poisson process, the arrivaltime of each node will be uniformly distributed over the timeinterval (0, t). Given that a node has arrived at time τ , then

gxkj(t)(r) =1t

t∫0

bx(t−τ)(r)dτ (18)

or

gxkj(t)(r) =1t

t∫0

bx(τ)(r)dτ (19)

Gxkj(t)(r) =1t

t∫0

Bx(τ)(r)dτ (20)

where bx(τ)(r) and Bx(τ)(r) are given by (16) and (17).Next, we let g̃xkj

(r) denote the pdf of the distance of a nodefrom the skj th stream at steady state. Then

g̃xkj(r) = lim

t→∞gxkj(t)(r) (21)

where gxkj(t)(r) is given by (19). From the remark following(16) and (17), the steady-state pdf approaches a uniform distri-bution, and g̃xkj

(r) will be given by

g̃xkj(r) =

1Rkj

, for 0 ≤ r ≤ Rkj (22)

where Rkj =∑j

�=k r�. We note that the simulation results alsoconfirm this approximation.

D. Distribution of Population Size

Next, we will determine the steady-state distribution of thenumber of nodes from each stream on the highway. Each streamof nodes may be modeled as an M/G/∞ queue with the servicetime of a node being the amount of time that it spends on

the highway. The service time of a node terminates with itsdeparture from the highway. Let Nkj(t) denote the numberof nodes from the skj th stream on the highway at time t.Following the analysis of the M/G/∞ queue in [19] and [20],we have

Pk(t) = Pr [Nkj(t) = k]

=∞∑

n=k

Pr [Nkj(t) = k|n arrivals in(0, t)]e−λ̃kjt(λ̃kjt)n

n!.

(23)

Defining ps(t) as the probability that a node is still on thehighway at time t, then

ps(t) = Gxkj(t)(Rkj). (24)

The probability that a node has departed from the highwayby time t is given by

pf (t) = 1 − Gxkj(t)(Rkj). (25)

Since each node may be on the highway at time t accordingto i.i.d. Bernoulli trials, we have

Pr [Nkj(t)=k|n arrivals in(0, t)]=(

n

k

)[ps(t)]

k [pf (t)]n−k .

(26)

Substituting (24) and (25) in (26), and following some sim-plifications, we have

Pk(t) = e−λ̃kj

t∫0

Bx(τ)(Rkj)dτ

[λ̃kj

t∫0

Bx(τ)(Rkj)dτ

]k

k!. (27)

Thus, the probability distribution of the number of nodesfrom the skj th stream on the highway at time t is Poisson withthe parameter λ̃kj

∫ t

0 Bx(τ)(Rkj)dτ . The steady-state distribu-tion of the node population is also Poisson with the parameter

φkj = limt→∞

λ̃kj

t∫0

Bx(τ)(Rkj)dτ

= limt→∞

λ̃kj

t∫0

Rkj∫0

1√2πθx(τ)

e− (y−µτ)2

2θx(τ) dydτ. (28)

The above integral may be numerically evaluated, includingfor large values of t.

E. Distribution of the Node Population Size Within a Cell

Let Ni denote the node population within cell i at steadystate. Defining k as the segment where cell i is located, then


k = {max(1, . . . , j, . . . , K) ∀ij < i}. Ni also has a Poissondistribution with the parameter φi, which is determined from

φi =�k∑

�=1

K∑n=�k+1

φ̃�n(i) (29)

where

φ̃�n(i) = φ�nP�n(i) (30)

and P�n(i) = Prob (a node from the s�nth stream is located incell i). From (22)

P�n(i) =d

R�n. (31)

Then, the probability distribution of the number of nodeswithin cell i at steady state is given by

Pr( Ni = n) = e−�φi

φni

n!. (32)

Clearly, the probability distribution of the number of nodeswithin the transmission range of a randomly chosen node willalso be given by the above Poisson distribution. These are thenodes that may interfere with the transmission of the randomnode. Clearly, this result will be useful in the design of effectivemedium access protocols (MAC) for the VANETs.

III. ANALYSIS OF NETWORK CONNECTIVITY

In this section, we will determine the network connectivity ofa new node arriving at the beginning of the highway and a ran-dom node on the highway at steady state. A new arrival or therandom node will be chosen as the cluster head, and the clusterseen by this node will be determined. All the nodes within thecluster will have direct or indirect communications with thecluster head. It will be assumed that two nodes will be able todirectly communicate if L < d, where L is the distance betweenthem, and d is the constant transmission range of a node. Asbefore, we will assume that the length of the highway R is aninteger multiple of a node’s transmission range d. Clearly, allthe nodes within a cell are able to directly communicate as theyare within each other’s transmission range.

A. Derivation of the Connectivity Performance Measuresfor a New Arriving Node

The new arriving node will see the equilibrium distribution ofthe population size, and it will be able to directly communicatewith all the nodes within the first cell. Next, we will determinethe direct communication probability of two nodes located atconsecutive cells i and i + 1. Let us define pi = Pr (L ≤ d |two nodes are located at consecutive cells i and i + 1).Lemma 2: The direct communication probability of two

nodes located within two consecutive cells has a constant valueof 1/2.

Proof: Let the random variable xi denote the distance ofa node located at cell i from the beginning of the highway and

gxi(r) the corresponding pdf of this variable. As before, the

cell i is assumed to be located in segment k; thus, a node in thiscell may belong to any of the first k streams. Thus, gxi

(r) willbe determined by the weighted average of the pdfs of distancesof the nodes from different streams given by g̃xkj

(r) in (22). Wealso have to take into consideration that the pdf of the distancefor each stream is relative to its service point. Thus

gxi(r) =

1φi

�k∑�=1

K∑n=�k+1

φ̃�n(i)

× g̃x�n(r − R1�)

G̃x�n(id − R1�) − G̃x�n

[(i − 1)d − R1�]

=1φi

�k∑�=1

K∑n=�k+1

φ̃�n(i)1

(id − R1�) − [(i − 1)d − R1�]

gxi(r) =

1d

(33)

where (i − 1)d ≤ xi ≤ id, 1 ≤ i ≤ I − 1, and G̃xkj(r) is the

cdf of g̃xkj(r).

The direct communication probability of the two nodes lo-cated at consecutive cells i and i + 1, respectively, is given by

pi = Pr(xi+1 − xi < d)

=∫ ∫

ri,ri+1

gxi(ri)gxi+1(ri+1)dridri+1. (34)

Substituting (33) in (34) and integrating over appropriateintervals, we determine that pi = 1/2, which completes theproof. �

Next, we will determine the probability that the populationof cell i + 1 will have connectivity with that of cell i, given thatthe latter has nonzero population for 0 < i < I . Let us considerpairs of nodes, where the two nodes are not located in the samecell. The populations of two cells will have connectivity if thereis at least a single pair of nodes that have direct communicationswith each other. Let us define qi+1 = Pr (a node located in celli + 1 will not have direct communications with any of the nodesin cell i), and Ci+1 = Pr (node population in cell i + 1 willhave connectivity with node population in cell i).

Let us assume that the population size of cell i is nonzero,Ni > 0, i.e.,

qi+1 =1

1 − Pr( Ni = 0)

∞∑j=1

(1 − pi)j Pr( Ni = j). (35)

Since the node population in cell i has a Poisson distributionwith the parameter φi, we have

qi+1 = (e−0.5�φi − e−�φi)/(1 − e−

�φi) (36)

Ci+1 = 1 −∞∑

j=0

(qi+1)j Pr( Ni+1 = j). (37)


The summation above corresponds to the probability thatthere is not even a single pair of nodes that have direct com-munication with each other. We note that if cell i + 1 is notpopulated, then it will also not have connectivity with cell i.From (32), (36), and (37), we have

Ci+1 = 1 − e−�φi+1(1−qi+1), for 0 < i < I. (38)

Let us also define C1 = Pr (node population in cell one willhave connectivity with the new arrival)

C1 = 1 − Pr( N1 = 0) = 1 − e−�φ1 . (39)

Next, we will derive the statistics of the cluster seen by a newarrival. Let us assume that the cluster consists of the first j cells.This means that the population of the j + 1th cell does not haveconnectivity with that of the jth cell. Defining ωj = Pr (thecluster consists of the first j cells), then

ωj = (1 − Cj+1)j∏

i=1

Ci, for 1 ≤ j ≤ I (40)

where Ci is given by (38) and (39), and we define CI+1 = 0.We note that ωI corresponds to the probability that a new

arriving node will see a single cluster.Next, we will determine the probability distribution of the

cluster size. Let us define Q(m) = Pr (cluster consists ofm nodes).

Then, it is given by

Q(m) =I∑

j=1

ωjQ(m|cluster consists of j cells). (41)

Since the number of nodes in each cell has a Poisson distribu-tion given by (32), then

Q(m) =I∑

j=1

ωje−

j∑i=1

�φi

(j∑

i=1

φi

)m

[1 −

j∑i=1

φi

]m!

. (42)

Defining m as the average cluster size, then (from the above)

m =I∑

j=1

ωjmj (43)

where

mj =j∑

i=1

φi

1 − e−�φi

.

Defining ξ as the normalized mean cluster size, then

ξ = m/ψ (44)

where ψ is the mean node population on the highway and isgiven by

ψ =I∑

i=1

φi. (45)

B. Derivation of the Connectivity Performance Measuresfor a Random Node

Next, the above analysis will be extended to the determina-tion of a cluster seen by a random node. The random node willbe chosen as the clusterhead, letting fr denote the probabilitythat the random node is located in cell r, i.e.,

fr =φr

ψ, r = 1, 2, . . . , I. (46)

The cells that belong to the cluster form an uninterruptedchain. We assume that all the cells in the range of � ≤ i ≤ hbelong to the cluster, where � and h denote the two cells at theopposite ends of the chain. Let us define q′i−1 = Pr (a node lo-cated in cell i − 1 will not have any direct communications withnode population in cell i), and C ′

i−1 = Pr (node population incell i − 1 will have connectivity with node population in cell i).

We note that

q′i−1 = qi+1, i ≥ 2 (47)

C ′i−1 = 1 − e−

�φi−1(1−q′i−1), i ≥ 2. (48)

Let us define ω′�,r,h as the probability that the cluster of a

random node located in cell r consists of all the cells i such that� ≤ i ≤ h. Then, ω′

�,r,h is given by

ω′�,r,h = (1 − C ′

�−1)(1 − Ch+1)r−1∏k=�

C ′k

h∏j=r+1

Cj

1 ≤ � ≤ r ≤ h ≤ I. (49)

In the above, we assume that C ′k = 0 for k ≤ 0, Cj = 0

for j > I , and, as before, we follow the convention that∏jk=i Ak = 1 if i > j.Next, let us define m′

�,r,h as the average cluster size for thecase under consideration. Then

m′�,r,h =

h∑k=�

φk

1 − Pr( Nk = 0)=

h∑k=�

φk

1 − e−�φk

. (50)

Let m′ and ω′I denote the unconditional average cluster size

and the probability that a random node will see the entire nodepopulation in a single cluster. Then

ω′I =

I∑r=1

I∑h=r+1

r−1∑�=1

frω′�,r,h (51)

m′ =I∑

r=1

I∑h=r+1

r−1∑�=1

frm′�,r,hω′

�,r,h. (52)


Fig. 3. Simple parallel highway network.

As before, the normalized average number of nodes in thecluster is given by

ξ′ = m′/ψ. (53)

C. Application of Connectivity Analysis to a Network ofHighways With Two-Way Traffic

The previous analysis may be applied to any path in a net-work of highways. A path will consist of consecutive highwaysegments. The traffic load of each segment may be determinedby the simultaneous solution of the traffic equations. As anexample, in Fig. 3, we consider a network with two parallelhighways, which are joined at the two ends. The highways arereferred to as A and B, which have three and four segments,respectively.

We will be interested in the network connectivity on each ofthe highways seen by a new arrival to the beginning of the twohighways. The user may be interested in finding out the trafficcongestion information on these highways to make a routeselection. Clearly, the analysis of this paper will be applicableto each of these highways. Numerical results regarding thisexample will be given in the next section.

In a network of highways, identical VANETs will serve tothe users traveling in the opposite directions. The system mayallow the vehicles to be part of a single communication system,independent of the direction of their travels. In the integratedsystem, the size of the node population within a cell is stillgiven by the Poisson distribution in (31) with the appropriateparameter values. As before, the pdf of the location of a nodewithin a cell is given by a uniform distribution; therefore, themean cluster size and the probability of a single cluster calcu-lation remain the same, as in (43) and (40), respectively. Theintegrated system will achieve a higher network connectivity;however, the cluster size of the integrated system will changemore frequently compared to a 1-D system carrying an equiv-alent amount of traffic. The integrated system will be useful incollision avoidance and when the network connectivity breaksdown during a response to a query. In the latter case, theresponse message may be stored in vehicles traveling in theopposite direction until network connectivity is re-established.

IV. SIMULATION SETUP

In this section, we describe the simulation software that isused to verify the analysis introduced in the previous sections.This is an event-driven simulation program written in theMATLAB software environment. We have performed multiple

Fig. 4. Flowchart of the simulation.

independent simulation runs, and each run terminates after aspecified number of nodes goes through the network. In eachrun, the statistics have been collected after a warm-up period toguarantee that the system has reached steady state. At the end,the collected statistics have been averaged over multiple runs.The arrival of the nodes in each stream is according to a Poissonprocess, and each node travels through the network accordingto the mobility model.

Fig. 4 presents the flowchart of the simulation algorithm. Inthis algorithm, an array called Event_scheduler keeps tracksof all future events. Two types of events have been defined.Type 1 corresponds to the arrival of a new node, and Type 2corresponds to the termination of an epoch. The node informa-tion is stored in an array called Node_array. This informationconsists of node and stream IDs, present epoch duration, nodespeed, and last recorded location of the node. Whenever a type 1event happens, a new entry for the new node will be made inthe Node_array, whereas whenever a type 2 event happens, thelocation for the specific node will be updated in the Node_array.

V. NUMERICAL AND SIMULATION RESULTS

In this section, we present some numerical results regardingthe analysis in this paper, together with simulation results, toconfirm the accuracy of the analysis.


Fig. 5. PDF of the distance of a node on the highway g̃xkj (r) at threedifferent time values of t = 50, 150, and 350 s from the analysis and simulationfor a mean node speed of 25 m/s (90 km/h).

First, we present the results for a single highway, as shownin Fig. 1. We assume that the length of the highway is R =10.0 km. As stated before, the transmission range of a node d ischosen such that there will be an integer number of cells on thehighway. We assume that there are three service points that arelocated at distances of 1.0, 3.0, and 6.0 km from the beginningof the highway. Unless otherwise stated, the following trafficarrival and departure rates have been assumed at the servicepoints: λ1 = 0.5, λ2 = 0.1, λ3 = 0.2, λ4 = 0.1 nodes/s, andα2 = α3 = α4 = 0.7. The epoch rate and the standard devia-tion of the nodes’ speed are set to constant values of β = 1 andσ = 3, respectively. Fig. 5 shows the pdf of the distance of anode at three instances following its arrival at the highway.

The figure presents both simulation and numerical resultsfrom (16) at steady state. For simplicity, in this figure, we haveassumed that the arrival and departure rates at the service pointsare zero. As may be seen, even at t = 50 s (the smallest timeinstant), the peak of the pdf is less than 5 × 10−4. As the timeincreases, the pdf becomes more flat, approaching to a constantvalue of 10−4. This constant value is the inverse of the lengthof the highway. These results confirm the assumption that thepdf of the distance of a node approaches a uniform distributiongiven by (22).

Fig. 6 shows both the analytical and simulation results fora mean node population within each cell for a transmissionrange of d = 200 m. As expected, there are discontinuities atthe beginning of each segment due to arrivals and departuresat service points, and the node densities in the first and thirdsegments are higher than the other segments. As may be seen,the numerical and simulation results are close to each other.The variations in node density impact the communicationsconnectivity of the network.

Fig. 7 shows the probability distribution of the cluster sizeseen by a new arriving node to the beginning of the highwayfrom (42). The results are given for three transmission ranges(i.e., d = 100, 200, and 500 m) and a mean node speed of25 m/s. It may be seen that, for the low transmission range, the

Fig. 6. Mean node population within cells for a transmission range ofd = 200 m and a mean node speed of 25 m/s (90 km/h).

Fig. 7. Probability that the cluster seen by a new arrival consists of n nodes forthe three values of transmission ranges (d = 100, 200, and 500 m) and a meannode speed of 25 m/s (90 km/h). (a) d = 100 m. (b) d = 200 m. (c) d = 500 m.

peak of the distribution is close to zero and sharply drops; thus,there is only a very small number of nodes in the cluster. Forthe high transmission range, the peak of the distribution occursat a large cluster size. The middle transmission range results ina large spread of the cluster size.

Figs. 8 and 9 present the simulation and numerical resultsfor a normalized mean cluster size from (44) and (53) seenby a new arriving node and a random node, respectively. Theresults are plotted as a function of the transmission range with


Fig. 8. Normalized mean cluster size seen by a new arrival as a function oftransmission range with the mean node speed as a parameter.

Fig. 9. Normalized mean cluster size seen by a random node as a function oftransmission range with the mean node speed as a parameter.

the mean node speed as a parameter. It may be seen that as thetransmission range increases, the normalized mean cluster sizeincreases. The increase is steeper at lower speeds because therewill be a larger node population on the highway. Finally, thenumerical and simulation results are close to each other, whichjustifies the approximations in the analysis. A comparison ofthe corresponding curves from Figs. 8 and 9 shows that, forany given transmission range, the mean cluster size is slightlyhigher for a random node than for a new arriving node. This isbecause the latter cluster is always anchored at the beginning ofthe highway.

Figs. 10 and 11 show the probability that an arriving and arandom node will see the entire node population on the highwayas a single cluster from (40) for j = I and (51), respectively.The plots assume the same system parameters as those inFigs. 8 and 9, and similar observations apply to these curves.As expected, the probability that a new arriving or a randomnode will see a single cluster is the same since in both cases,

Fig. 10. Probability that a new arrival will see a single cluster as a function oftransmission range with the mean node speed as a parameter.

Fig. 11. Probability that a random node will see a single cluster as a functionof transmission range with the mean node speed as a parameter.

all the nodes will be included in the cluster. It may be seenthat, at higher speeds, a node has to significantly increase itstransmission range to achieve a high probability of a singlecluster. When the single cluster probability is high, since thelocations of the nodes are uniformly distributed, the nodes willprobably be scattered all over the highway. As a result, thenodes will be able to receive traffic information from any partof the highway.

Next, we present the numerical results for the two parallelhighway networks shown in Fig. 3. It is assumed that bothhighways have the same length of R = 10 km. In Fig. 3,highway A has two service points located at distances of 3.0and 6.0 km, and highway B has three service points located atdistances of 1.0, 3.0, and 6.0 km from their beginnings. Thearrival rates of the nodes are given by λ2 = λ3 = λ5 = λ6 =λ7 = 0.1 at the service points and by λ1 = λ4 = 0.5 nodes/s attheir beginnings. It is assumed that the probability that a nodecontinuing on the next highway segment is same at all service


Fig. 12. Normalized mean cluster size seen in each of the highways by anew arrival as a function of transmission range with the mean node speed asa parameter.

Fig. 13. Probability that a new arrival will see a single cluster on each highwayas a function of transmission range with the mean node speed as a parameter.

points and is given by αi = 0.7, i = 2, 3, 5, 6, 7. The resultshave been plotted for two mean node speeds of 15 and 25 m/s.

Fig. 12 shows the normalized mean cluster size seen oneach highway by a new arrival to their beginnings, and Fig. 13shows the probability that a new arrival will see the entire nodepopulation on each highway as a single cluster. The resultsare plotted as a function of the transmission range. A newarriving node may be interested in finding out informationregarding traffic congestion on the two highways to make aroute selection. As may be seen, for the selected parametervalues, the normalized mean cluster size and the probability ofa single cluster are slightly higher for A than B.

Next, we present the simulation results about the effect of afinite number of lanes on highways, which limits the number ofvehicles passing. First, we have measured the average numberof passing vehicles per second experienced by a random nodeduring the simulation. The measured value of this average is

Fig. 14. Normalized mean cluster size seen by a new arrival as a functionof transmission range from simulation for a two-lane highway and from thenumerical result.

0.31 passes/s when the mean node speed is 20 m/s. The averagevalue increases to 0.72 passes/s when the stream arrival rates aredoubled (λ1 = 1, λ2 = 0.2, λ3 = 0.4, λ4 = 0.2) while keepingthe mean node speed constant. As may be seen, this average isnot high. Next, we have assumed that the number of lanes ona highway is limited to M . We have modified the simulationsuch that the number of passes experienced by a node is at mostM during an epoch duration. Fig. 14 presents the simulationresults for a normalized mean cluster size seen by a new arrivingnode for M = 2. The results are plotted as a function of thetransmission range with node speed as a parameter. In addition,the analytical results given before in Fig. 8 are also shown inthis figure, which correspond to an unlimited number of laneson a highway. As may be seen, the simulation and analyticalresults are close to each other, except for some divergenceat high speeds and in smaller transmission ranges. However,this divergence sharply drops at a higher number of lanes ornode arrival rates. We also note that a congested network isnot interesting from the connectivity point of view since thenetwork will have full connectivity. Thus, these results justifythe assumption of no limitation on the number times a vehiclemay make a pass in the analysis.

The conclusions of this section remain the same with longerhighways and different parameter values. We also note the closeagreement between the numerical and simulation results, whichvalidates the approximations in the analysis.

VI. CONCLUSION

In this paper, we have studied the network connectivity ofmobile nodes in VANETs. The network consists of a singlehighway with multiple lanes that allow vehicles to pass eachother. We assume that the flow of traffic is unidirectional andthat the highway has several traffic entry/exit points. The nodesarrive at the highway through one of the entry points andthen travel along the highway following the assumed mobilitymodel until they depart from the system through one of the exit


points. We determine the distribution of a node’s location on thehighway. Then, we derive the distributions of node populationsizes on the highway as well as within the transmission rangeof a random node at steady state. The latter distribution willbe useful in designing effective MAC protocols for VANETs.Afterward, we derive the connectivity performance measures,such as the mean cluster size and the probability that a newarriving node or a random node will see the entire node popula-tion in a single cluster. It is seen that the probability of a singlecluster is lower at higher node speeds than at lower speedsfor constant arrival rates. As a result, the nodes may have toincrease their transmission ranges at higher speeds to achieve ahigh probability of network connectivity.

It is shown that the analysis may be applied to paths in anetwork of highways. This will enable a user to find out trafficcongestion information in different paths before making a routeselection.

In a network of highways, identical VANETs will serve tousers traveling in opposite directions. The system may allowthe vehicles to be part of a single communication system,independent of the direction they are traveling. The resultsin this paper may be combined to give the mean cluster sizeand the probability of a single cluster in such a system. Theintegrated system will achieve a higher network connectivity,and such a system will be needed for collision avoidance.However, the cluster stability of the integrated system will belower compared to a 1-D system carrying an equivalent amountof traffic.

The results of this paper may be used to study the perfor-mance of different routing algorithms, throughput, or delay inVANETs.

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Mehdi Khabazian (S’05) received the B.S. degreein electronic engineering from Iran University ofScience and Technology (IUST), Tehran, Iran, inSeptember 1998 and the M.S. degree in telecommu-nications engineering from Shiraz University, Shiraz,Iran, in January 2002. He is currently working to-ward the Ph.D. degree in telecommunications atConcordia University, Montreal, QC, Canada.

From 2001 to 2003, he was a Researcher with theIran Telecommunication Research Center (ITRC),Tehran. His research interests include wireless com-

munications, performance analysis, mobile ad hoc networks, sensor networks,and quality of service.

Mustafa K. Mehmet Ali (M’88) received the B.Sc.and M.Sc. degrees in electrical engineering fromBogazichi University, Istanbul, Turkey, in 1977 and1979, respectively, and the Ph.D. degree in electricalengineering from Carleton University, Ottawa, ON,Canada, in 1983.

Until the end of 1984, he was a Research Engi-neer with Telesat Canada. Since 1985, he has beenwith the Department of Electrical and ComputerEngineering, Concordia University, Montreal, QC,Canada, where he is currently a Professor. His cur-

rent research interest is the performance modeling of wireless networks.

A Performance Modeling of Connectivity in Vehicular Ad Hoc ...lila/M2R/6.pdf2440 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 4, JULY 2008 A Performance Modeling of Connectivity

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