RESEARCH Open Access Connectivity analysis of one ......properties in a fading channel. Keywords: connectivity distance, fading channels, highway, vehicle speed, vehicular ad hoc networks

RESEARCH Open Access

Connectivity analysis of one-dimensionalvehicular ad hoc networks in fading channelsNeelakantan Pattathil Chandrasekharamenon* and Babu AnchareV

Abstract

Vehicular ad hoc network (VANET) is a type of promising application-oriented network deployed along a highwayfor safety and emergency information delivery, entertainment, data collection, and communication. In this paper,we present an analytical model to investigate the connectivity properties of one-dimensional VANETs in thepresence of channel randomness, from a queuing theoretic perspective. Connectivity is one of the most importantissues in VANETs to ensure reliable dissemination of time-critical information. The effect of channel randomnesscaused by fading is incorporated into the analysis by modeling the transmission range of each vehicle as arandom variable. With exponentially distributed inter-vehicle distances, we use an equivalent M/G/∞ queue for theconnectivity analysis. Assuming that the network consists of a large number of finite clusters, we obtain analyticalexpressions for the average connectivity distance and the expected number of vehicles in a connected cluster,taking into account the underlying wireless channel. Three different fading models are considered for the analysis:Rayleigh, Rician and Weibull. The effect of log normal shadow fading is also analyzed. A distance-dependent powerlaw model is used to represent the path loss in the channel. Further, the speed of each vehicle on the highway isassumed to be a Gaussian distributed random variable. The analytical model is useful to assess VANET connectivityproperties in a fading channel.

Keywords: connectivity distance, fading channels, highway, vehicle speed, vehicular ad hoc networks

1. IntroductionVehicular Ad Hoc Networks (VANETs), which allowvehicles to form a self-organized network without therequirement of permanent infrastructures, are highlymobile wireless ad hoc networks targeted to support (i)vehicular safety-related applications such as emergencywarning systems, collision avoidance through driverassistance, road condition warning, lane-changing assis-tance and (ii) entertainment applications [1]. VANET is ahybrid wireless network that supports both infrastruc-ture-based and ad hoc communications. Specifically,vehicles on the road can communicate with each otherthrough a multi-hop ad hoc connection. They can alsoaccess the Internet and other broadband services throughthe roadside infrastructure, i.e., base stations (BSs) oraccess points (APs) along the road. These types of Vehi-cle to Vehicle (V2V) and Vehicle to Infrastructure (V2I)communications have recently received significant

interest from both academia and industry. The emergingtechnology for VANETs is Dedicated Short Range Com-munications (DSRC), for which in 1999, FCC has allo-cated 75 MHz of spectrum between 5,850 and 5,925MHz. DSRC is based on IEEE 802.11 technology and isproceeding toward standardization under the standardIEEE 802.11p, while the entire communication stack isbeing standardized by the IEEE 1609 working groupunder the name wireless access in vehicular environ-ments (WAVE) [1]. The goal of 802.11p standard is toprovide V2V and V2I communications over the dedicated5.9 GHz licensed frequency band and supports data ratesof 3 to 27 Mbps (3, 4.5, 6, 9, 12, 18, 24 and 27 Mbps) fora channel bandwidth of 10 MHz [1,2].Network connectivity is a fundamental performance

measure of ad hoc and sensor networks. Two nodes in anetwork are connected if they can exchange informationwith each other, either directly or indirectly. For VANETs,the connectivity is very important as a measure to ensurereliable dissemination of time-critical information to allvehicles in the network. Further, the connectivity of a

* Correspondence: [email protected] of Electronics and Communication Engineering, NationalInstitute of Technology, Calicut 673601, India

Chandrasekharamenon and AnchareV EURASIP Journal on Wireless Communicationsand Networking 2012, 2012:1http://jwcn.eurasipjournals.com/content/2012/1/1

© 2012 Chandrasekharamenon and Babu Anchare; licensee Springer. This is an Open Access article distributed under the terms of theCreative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

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VANET is directly related to the density of vehicles on theroad and their speed distribution. Unlike conventional adhoc wireless networks, a VANET may be required to dealwith different types of network densities. For example,VANETs on free-ways or urban areas are more likely toform highly dense networks during rush hour traffic, whilethese networks may experience frequent network frag-mentation in sparsely populated rural free-ways or duringlate night hours. If the vehicle density is very high, aVANET would almost surely be connected. The connec-tivity degrades, when the vehicle density is very low, andin this case, it might not be possible to transfer messagesto other vehicles because of disconnections. In traffic the-ory, this is known as the free flow state [3].In this paper, we investigate the connectivity properties

of one-dimensional VANET in the presence of channelrandomness. The presence of fading will cause thereceived signal power at a specific time instant to be a ran-dom variable. In this case, the transmission range of eachvehicle can no longer be a deterministic quantity but hasto be modeled as a random variable. Accordingly, weassume that each vehicle has a transmission range R, withcumulative distribution function (CDF), FR(a). To analyzethe connectivity, we use the results of Miorandi and Alt-man [4] that identified the equivalence between (i) thebusy period of an infinite server queue and the connectiv-ity distance in an ad hoc network and (ii) the number ofcustomers served during a busy period and the number ofnodes in a connected cluster in the network. With expo-nentially distributed inter-vehicle distances, we use anequivalent M/G/∞ queue for the connectivity analysis.The following metrics are used for our study: (i) connec-tivity distance, defined as the length of a connected pathfrom any given vehicle; and (ii) the number of vehicles in aconnected spatial cluster (platoon) or a connected pathfrom any given distance. Analytical expressions for theaverage connectivity distance and the expected number ofvehicles in a connected cluster are presented, taking intoaccount the effects of channel randomness. The connec-tivity distance is a very important metric since a large con-nectivity distance leads to a larger coverage area for safetymessage broadcast (recall that major applications ofVANET’s include broadcasting of safety messages). Pla-toon size implies how many vehicles are connected in acluster and thus are able to hear a vehicle in a broadcastapplication. This is also quite significant especially in abroadcast application scenario, where it is required toensure reliable dissemination of safety messages to asmany vehicles as possible.Realistic fading models are incorporated into the analysis

by considering different fading models such as Rayleigh,Rician and Weibull. The analysis provides a framework todetermine the impact of parameters such as vehicle den-sity, vehicle speed and various channel-dependent

parameters such as path loss exponent, Rician and Weibullfading parameters on VANET connectivity. Rest of thepaper is organized as follows: In Section 2, we describe therelated work. The system model is presented in Section 3.In Section 4, we present the connectivity analysis. Theresults are presented in Section 5. The paper is concludedin Section 6.

2. Related workA number of studies concerning ad hoc network connec-tivity, modeling and analysis have been reported in theliterature [4-12]. Most of these works study the problemin static ad hoc networks or networks with low mobility.Well-known mobility models such as random way pointmodels are also used for analysis. However, these resultsare not directly applicable to VANETs because of the fol-lowing fundamental characteristics exhibited by thesenetworks. First, the vehicular movement in a VANET isrestricted to a predetermined traffic network, but themobile nodes in MANETs have multiple degrees of free-dom. Second, the mobility of the nodes in a VANET isaffected by the traffic density, which is determined by theroad capacity and the underlying driver behavior, such asunexpected acceleration or deceleration. Lastly, the con-nectivity of a VANET is influenced by factors such asenvironmental conditions, traffic headway and vehiclemobility.Recently, there were many attempts by the research

community to address the connectivity properties ofVANETs as well [13-23]. The connectivity analysis ofVANETs for both highway and simple road configura-tions presented in [13] proposed that a fixed transmissionrange does not adapt to the frequent topology changes inVANETs; but a dynamic transmission range is alwaysrequired. In [14], authors presented a way to improve theconnectivity in VANET by adding extra nodes known asmobile base stations. The connectivity properties of amobile linear network with high-speed mobile nodes andstrict delay constraints were investigated in [15]. VANETconnectivity analysis based on a comprehensive mobilitymodel was presented in [16] by considering the arrivaland departure of nodes at predefined entry and exitpoints along a highway. A new analytical mobility modelfor VANETs based on product-form queuing networkshas been proposed in [17]. Authors of [18] presentedconnectivity analysis of both one-way and two-way high-way scenarios assuming that all vehicles maintain a con-stant speed. In [19], authors developed an analyticalmodel of multi-hop connectivity of an inter-vehicle com-munication system. An analytical characterization of theconnectivity of VANETs on freeway segments wasderived in [20]. In [21], authors investigated the coverageand access probability of the vehicular networks withfixed roadside infrastructure. In [22], authors presented


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the connectivity of message propagation in the two-dimensional VANETs, for highway and city scenarios. In[23], authors investigated how intersections and two-dimensional road topology affect the connectivity ofVANETs in urban areas.A major limitation of the above-mentioned works is

that they rely on a simplistic model of radio wave pro-pagation, where vehicles communicate to each other ifand only if their separation distance is smaller than agiven value. Further, the analysis assumes that all thevehicles in the network have the same transmissionrange. The effect of randomness inherently present inthe radio communication channel is not considered forthe analysis. In this paper, we analyze the connectivitycharacteristics of one-dimensional VANET from a queu-ing theoretic perspective, taking into account the effectof channel randomness. The presence of fading willresult in randomness in the received signal power, mak-ing the transmission range of each vehicle, a randomvariable. It may be noted that the impact of fading onthe connectivity and related characteristics of static adhoc networks was extensively analyzed in the literature(e.g.,[9-12]). On the other hand, to the best of theseauthors’ knowledge, the impact of channel randomnesson the connectivity properties of VANETs has not beenanalyzed in the literature so far.Recently, many researchers have paid much attention

to V2V channel measurements, for understanding theunderlying physical phenomenon in V2V propagationenvironments (ex:[24-33]). Analysis of probability densityfunction (PDF) of received signal amplitude was reportedin [24-26] for V2V systems. In [24], the authors consid-ered different V2V communication contexts at 5.9 GHz,which include express-way, urban canyon and suburbanstreet, and modeled the PDF of received signal amplitudeas either Rayleigh or Rician, with the help of empiricalmeasurements. When the distance between transmitterand receiver is less than 5 m, the fading follows Rician,tending toward Rayleigh at larger distances. When thedistance exceeds 70-100 m, the fading was observed to beworse than Rayleigh, due to the intermittent loss of LOScomponent at larger distances. In [25], it was reportedthat, for suburban driving environments, the PDF of thereceived signal in a V2V system with a carrier frequencyof 5.9 GHz gradually transits from near-Rician to Ray-leigh as the vehicle separation increases. When LOScomponent is intermittently lost at large distances, thechannel fading becomes more severe than Rayleigh. In[26], the following V2V settings were considered: urban,with antennas outside the cars; urban, with antennasinside the cars; small cities; and open areas (highways)with either high or low traffic densities. It was observedthat Weibull PDF provides the best fit for the PDF of thereceived signal amplitude. An extensive survey of the

state-of-the-art in V2V channel measurements and mod-eling was presented in [27-29], justifying the above mod-els for V2V channels. In general, V2V communicationconsists of LOS along with some multi-path components,arising out of reflections of mobile scatterers (e.g., mov-ing cars), and static scatterers (e.g., building and roadsigns located on the roadside). The amount of multi-pathcomponent depends on the surroundings of the highway,i.e., presence of obstacles and reflectors and the numberof moving (vehicles) obstacles on the road. In rural high-ways, the number of obstacles could be less, so the com-munication can be modeled as purely LOS in nature, forwhich Rician fading model is more appropriate. But incongested city roads, the multi-path component becomesmore significant. For this case, Rayleigh fading model ismore suitable. Hence, for V2V communication, differentfading models may be applicable depending on the nat-ure of surrounding environment and the vehicle density.In [30], empirical results and analytical models were

presented for the path loss, considering four differentV2V environments: highway, rural, urban and suburban.For the rural scenario, the path loss was modeled by atwo-ray model. For the highway, urban and suburbanscenarios, a classical power law model was found to besuitable. Similar results were reported by Kunisch andPamp [31], who used a power law model for highway andurban environments; but found a two-ray model best sui-ted for rural environments. The measurements of Chenget al. [25,32] suggested that a break point model is suita-ble to describe the V2V path loss. The results in [33],obtained from the empirical measurements of the IEEE802.11p communications channel, under normal drivingconditions in rural, urban and highway scenarios justifiedthe use of classical power law model for V2V path loss.To incorporate realistic V2V channel model into the con-nectivity analysis, we consider different small-scale fadingmodels such as Rayleigh, Rician and Weibull for our ana-lysis. For the path loss, the classical power law model isemployed. In the next section, we describe the systemmodel employed for the connectivity analysis.

3. System modelTo analyze the connectivity of VANETs in the presenceof channel randomness, we rely on [4], in which theauthors addressed the connectivity issues in one-dimen-sional ad hoc networks, from a queuing theoretic per-spective. Authors exploited the relationship betweencoverage problems and infinite server queues, and by uti-lizing the results from an equivalent G/G/∞ queue, theyaddressed the connectivity properties of an ad hoc net-work. The authors also identified the equivalencebetween the following: (i) the busy period of an infiniteserver queue and the connectivity distance in an ad hocnetwork and (ii) the number of customers served during


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a busy period and the number of nodes in a connectedcluster in the network. The following assumptions wereutilized to obtain the results: (i) the inter-arrival times inthe infinite server queue have the same distribution asthe distance between successive nodes; and (ii) the ser-vice times have the same probability distribution as thetransmission range of the nodes. In this paper, we studythe connectivity properties of VANETs using the corre-sponding infinite server queuing model. For this, theprobability distribution functions (PDF) of inter-vehicledistance and vehicle transmission range are required. Wenow present the system model, which includes the high-way and mobility model, used for the connectivity analy-sis. A model to find the statistical characteristics of thetransmission range for various fading models is thenintroduced.

A. Highway and mobility modelThe highway and mobility model used for the connec-tivity analysis is based on [14] and is briefly describedhere. Assume that an observer stands at an arbitrarypoint of an uninterrupted highway (i.e., without trafficlights). Empirical studies have shown that Poisson distri-bution provides an excellent model for vehicle arrivalprocess in free flow state [3]. Hence, it is assumed thatthe number of vehicles passing the observer per unittime is a Poisson process with rate l vehicles/h. Thus,the inter-arrival times are exponentially distributed withparameter l. Assume that there are M discrete levels ofconstant speed vi, i = 1, 2,..., M where the speeds are i.i.d., and independent of the inter-arrival times. Let thearrival process of vehicles with speed vi be Poisson with

rate li, i = 1, 2,..., M, and let �Mi=1λi = λ . Further, it is

assumed that these arrival processes are independent,and the probability of occurrence of each speed is pi =li/l. Let Xn be the random variable representing the dis-tance between nth closest vehicle to the observer and (n- 1)th closest vehicle to the observer. It has been provedin [14] that the inter-vehicle distances are i.i.d., andexponentially distributed with parameter

ρav = �Mi=1

λivi= λ�M

i=1pivi. Specifically, the CDF of inter-

vehicle distance Xn is given by

FXn(x) = 1 − e−ρavx, x ≥ 0 (1)

In free flow state, the movement of a vehicle is inde-pendent of all other vehicles. Empirical studies haveshown that the speeds of different vehicles in free flowstate follow a Gaussian distribution [3]. We, therefore,assume that each vehicle is assigned a random speedchosen from a Gaussian distribution and that each vehi-cle maintains its randomly assigned speed while it is onthe highway. To avoid dealing with negative speeds or

speeds close to zero, we define two limits for the speed,i.e., vmax and vmin for the maximum and minimum levelsof vehicle speed, respectively. For this, we use a trun-cated Gaussian probability density function (PDF), givenby [14]

gV(v) =fV(v)∫ vmax

vminfV(u)du

(2)

where fV(v) = 1σv

√2π

exp(−(v−μv)

2

2σ 2v

)is the Gaussian

PDF, μv–average speed, sv–standard deviation of thevehicle speed, vmax = μv + 3sv the maximum speed andvmin = μv - 3sv the minimum speed [14]. Substitutingfor fV(v) in (2), the truncated Gaussian PDF gV(v) isgiven by

gV(v) =2fV(v)

erf(vmax−μv

σv√2

)− erf

(vmin−μv

σv√2

) , vmin ≤ v ≤ vmax (3)

where erf(.) is the error function [34]. Since the inter-vehicle distance Xn is exponentially distributed withparameter rav, the average vehicle density on the high-way is given by

ρav =1

E[X]= λ

N∑i=1

pivi

= λE[1V

](4)

where E[.] is the expectation operator and V is therandom variable representing the vehicle speed. Whenthe vehicle speed follows truncated Gaussian PDF, theaverage vehicle density is computed as follows:

ρav =2λ/

√2πσv

erf(vmax−μv

σv√2

)− erf

(vmin−μv

σv√2

)vmax∫

vmin

1vexp

(−(v − μv)

2

2σ 2v

)dv (5)

It may be noted that the average vehicle density givenin (5) does not have a closed-form solution but has tobe evaluated by numerical integration. Numerical andSimulation results for rav are presented in Section 5. Itis observed that the parameters μv and sv have signifi-cant impact on rav. Since each vehicle enters the high-way with a random speed, the number of vehicles onthe highway segment of length L is also a random vari-able. The average number of vehicles on the highway isthen given by Nav = Lrav. Next, we present a model tofind the statistical characteristics of transmission rangefor various fading models.

B. Statistical characteristics of transmission rangeThe effect of randomness caused by fading is incorpo-rated into the analysis by assuming the transmissionrange R to be a random variable with CDF FR(a). Let Zbe the random variable representing the received signal


Page 4 of 16

envelope and let l be the distance between transmittingand receiving nodes. Further we assume that “good longcodes” are used, so that probability of successful recep-tion, as a function of the signal-to-noise ratio (SNR)approaches a step function, whose threshold is denotedby ψ [4]. Additive Gaussian noise of power W watts isassumed to be present at the receiver. The receivedpower is then given by Prx = Ptxz

2K/la where Ptx is thetransmit power, a is the path loss exponent and K is aconstant associated with the path loss model. Here, K =GTGRC

2/(4π fc)2, where GT and GR, respectively, repre-

sent the transmit and receive antenna gains, C is thespeed of light and fc is the carrier frequency [18,35,36].In this paper, we assume that the antennas are omnidirectional (GT = GR = 1), and the carrier frequency fc =5.9 GHz. The thermal noise power is given by W =FkToB where F is the receiver noise figure, k = 1.38 ×10-23 J/K is the Boltzmann constant, To is the roomtemperature (To = 300° K) and B is the transmissionbandwidth (B = 10 MHz for 802.11p). The receivedSNR is computed as g = PtxZ

2K/laW. Assuming that E[Z2] = 1, the average received SNR is γ̄ = PtxK/lαW . Inour model, the transmitted message can be correctlydecoded if and only if the received SNR g is greaterthan a given threshold ψ. In the remaining part of thissection, we find the statistics of the transmission rangefor various fading models. For Rayleigh fading, theseresults were reported in [4]. We extend the analysis toRician and Weibull fading models. We also consider thecombined effect of lognormal shadow fading and small-scale fading models.1) Rayleigh fadingAssume that the received signal amplitude in V2V chan-nel follows Rayleigh PDF. The Rayleigh distribution isfrequently used to model multi-path fading with nodirect line-of-sight (LOS) path. It has been reported inthe literature that, in V2V communication as the separa-tion between source and destination vehicles increases,the LOS component may be lost and hence the PDF ofthe received signal amplitude gradually transits fromnear-Rician to Rayleigh [24-26]. Further, the multi-pathcomponent becomes more significant when comparedto the LOS component in congested city roads, andhence the Rayleigh fading model is more suitable todescribe the PDF of the received signal amplitude insuch scenarios. It is also assumed that the fading is con-stant over the transmission of a frame and subsequentfading states are i.i.d. (block-fading) [33]. The receivedSNR has exponential distribution given by [35]

f (γ ) =1γ̄e−γ /γ̄ , γ ≥ 0

where γ̄ is the average SNR. The probability that themessage is correctly decoded at a distance l is given by

P[γ (l) ≥ ψ] = e−ψ/γ̄ = e−lαWψ/PtxK (7)

The CDF of the transmission range is then computedas follows [4]:

FR(a) = P(R ≤ a) = 1 − P(R > a) = 1 − P(γ (a) ≥ ψ)

= 1 − e−ψaαW/KPtx

The average transmission range is given by [4]:

E(R) =∫ ∞

0(1 − FR(a))da

=(1/α)

α

(PtxK

ψW

)1/α (9)

where Γ(.) is the Gamma function [34].2) Rayleigh fading with superimposed lognormalshadowingLet Y be the random variable representing shadow fad-

ing. Its PDF is given by f (y) = 1√2πσ y

e− (ln(y)−ln(Kl−α))2

2σ 2 ,

where s is the standard deviation of shadow fading pro-cess [35] and l is the transmitter to receiver separation.For the superimposed lognormal shadowing and Ray-leigh fading scenario, the CDF of the transmission rangecan be computed as follows [4]:

FR(a) = 1−P(R > a) = 1−P(γ (a) > ψ) = 1−∞∫

ψ

e−ψWKPtxx

1√2πσ y

e−

(ln(y) − ln(Ka−α)

)22σ 2 (10)

The average transmission range is then given by [4]

E[R] =

∞∫0

(1 − FR(a))da

=(1/α)

αeσ 2/2α2

(PtxK

ψW

)1/α(11)

3) Rician fadingThe Rician fading is used to model propagation pathsconsisting of one strong LOS component and many ran-dom weaker components. In rural highways, the multipath components may be weak, so the communicationcan be modeled as purely LOS in nature, for whichRician fading model is more appropriate. Empirical stu-dies for different V2V communication contexts at 5.9GHz, which include express-way, urban canyon and sub-urban street, have predicted the PDF of received signalamplitude to be either Rayleigh or Rician [24]. Whenthe distance between transmitter and receiver is less


Page 5 of 16

than 5 m, the fading follows Rician, which is character-ized by the Rician factor � (defined as the ratio ofenergy in the LOS path to the energy in the scatteredpath). The PDF of the received SNR in a Rician fadedchannel is given as follows [35]:

f (γ ) =1 + κ

γ̄e−κe−(κ+1)γ /γ̄ I0(2

√κ(κ + 1)γ /γ̄ ) (12)

where � is the Rician factor, and I0(.) represents themodified Bessel function of the zeroth order and firstkind. Now I0(x) can be expanded as

I0(x) =∑∞

t=01

t!(t+1)

( x2

)2t. For integer values of t, Γ(t +

1) = Γ(t) = t!, so that I0(x) =∑∞

t=01

(t!)2( x2

)2t. Hence, the

PDF of received SNR becomes:

f (γ ) =1 + κ

γ̄e−κ

∞∑t=0

κ t

(t!)2

((κ + 1)γ

γ̄

)t

e−(κ+1)γ

γ̄ (13)

The CDF of transmission range is calculated as fol-lows:

FR(a) = 1 − P(γ (a) > ψ) = 1 −∞∫

ψ

f (γ )dγ (14)

Substituting (13) in (14) and simplifying, we get

FR(a) = e−(κ+1)ψ

γ̄ e−κ

∞∑t=0

κ t

t!

t∑l=0

((κ + 1)ψ)l

γ̄ ll!(15)

where γ̄ = PtxK/aαW . The expected value of thetransmission range under the Rician fading model isgiven by

E[R] =

∞∫0

(1 − FR(a))da

=

∞∫0

e−(κ+1)ψaαW

KPtx e−κ

∞∑t=0

κ t

t!

t∑l=0

1l!

((κ + 1)ψaαW

KPtx

)l(16)

On simplifying (16), the average transmission range isobtained as:

E(R) =1α

(KPtx

ψW(κ + 1)

)1/α

e−κ

∞∑t=0

κ t

t!

t∑l=0

( 1

α+ l

)l!

(17)

4) Rician fading with superimposed lognormal shadowingIn this case, the CDF of the transmission range can becomputed as follows:

FR(a) = 1 − P(γ (a) > ψ)

= 1 −∞∫

ψ

dx1√2πσ x

e− (ln(x)−ln(Kl−α))2

2σ 2

[e

−(κ+1)ψγ̄ e−κ

∞∑t=0

κ t

t!

t∑l=0

((κ + 1)ψ)l

γ̄ ll!

](18)

The mean transmission range is then determined asfollows:

E[R] =

∞∫0

(1 − FR(a))da

=1α

(KPtx

ψW(κ + 1)

)1/α

e2σ 2

α2 e−κ

∞∑t=0

κ t

t!

t∑l=0

1( 1α+ l)

l!

(19)

5) Weibull fadingThe Weibull distribution is often found to be very suita-ble to fit empirical non-LOS V2V fading channel mea-surements [25,26]. In [26], the authors reported severefading in multiple V2V settings based upon measure-ments in the 5 GHz band and found that Weibull distri-bution can be used to approximate measured severefading conditions. It may be noted that Weibull fadingis capable of representing both LOS and non-LOS cases.The PDF of the received SNR under Weibull fading isgiven by [37]:

f (γ ) =c2

((1 + 2/c)

γ̄

)c/2

γ c/2−1 exp[−

((1 + 2/c)γ

γ̄

)]c/2

(20)

where c is the Weibull fading parameter which rangesfrom zero to infinity and Γ(.) is the Gamma function.The CDF of the received SNR under Weibull fading isgiven by [37]:

F(γ ) = 1 − exp

[−

((1 + 2/c)γ

γ̄

)c/2]

(21)

The CDF of the transmission range is given by

FR(a) = 1 − P(γ (a) > ψ) = e−

((1+2/c)ψ

γ̄

)c/2

(22)

where γ̄ = PtxK/aαW . The average transmission rangeis computed as follows:

E[R] =

∞∫0

(1 − FR(a))da

=

∞∫0

e−

((1+2/c)ψ

γ̄

)c/2

dx

=(

KPtxψW(1 + 2/c)

)1/α (

1(αc/2)

)(αc/2

(23)

With these preliminary results, we present the connec-tivity analysis in the next section.

4. Connectivity analysisIn this section, we present an analytical procedure forfinding connectivity-related parameters of a one-


Page 6 of 16

dimensional VANET, a network formed by wirelessequipped vehicles on the highway. Assume that thehighway and the mobility specifications are described inSection 3, with inter-vehicle distances modeled as i.i.d.random variables having exponential distribution withparameter rav. Consider a pair of consecutive vehicles inthe network. These two vehicles will communicate witheach other, if the inter-vehicle distance is less than orequal to the vehicle’s transmission range R. Accordingto the results reported in [6], since the vehicle density l< ∞, the probability for a broken link to occur betweenany pair of consecutive nodes is strictly positive, what-ever be the value of l and R. Further, the resulting net-work will be disconnected almost surely, and hence thenetwork will be almost surely divided into an infinitenumber of finite clusters, between which no communi-cation is possible [6]. To study the connectivity charac-teristics, we select an equivalent queuing model for thenetwork. Since the inter-vehicle distances are exponen-tially distributed and the vehicle transmission range hasgeneral probability distribution FR(.), an equivalent M/G/∞ queue is used for analyzing the connectivity. Theconnectivity properties of the network depend on clusterstatistics. A spatial cluster in the network corresponds toa busy period in the queuing system. Accordingly, thelength of the connected component corresponds to thebusy period duration and number of vehicles in a clustercorresponds to the number of customers served duringa busy period [4].Let D be the random variable representing the con-

nectivity distance, which is defined as the length of theconnected path from a given vehicle. The Laplace trans-form of the connectivity distance D is then given by[14,38]:

hD(s) = 1 +s

λE[ 1V

] − 1

λE[ 1V

]p∗(s)

(24)

where λE[ 1V

]is the average vehicle density and p*(s)

is the Laplace transform of P0(t) which is defined as.

P0(t) = e−λE

[ 1V

] t∫0(1−FR(a))

da(25)

The average connectivity distance is given by [14,38]

E[D] =1

λE[ 1V

]P0

− 1

λE[ 1V

] (26)

where P0 = limt→∞P0(t) = e−λE

[ 1V

]E[R]Hence, E(D) is

computed as follows:

E(D) =1 − e

−λE[ 1V

]E[R]

λE[ 1V

]e−λE

[ 1V

]E[R] (27)

Let N be the random variable representing the num-ber of vehicles in a cluster (also known as cluster size orplatoon size). The average platoon size E[N] is com-

puted as E[N] = ρavx̄ = λE[ 1V

]x̄ , where x̄ = 1

λE[ 1V

]P0

is

the average distance between beginning of two consecu-tive platoons [39]. Substituting for P0, E[N] is given by

E[N] =1P0

= eλE

[ 1V

]E[R](28)

For Rayleigh fading, E[D] and E[N] are obtained bycombining (27), (28) and (9). These expressions aregiven as follows:

E[D] =1 − e

−λE[ 1V ](1/α)

α(PtxKψW )

1α

λE[ 1V

]e−λE[ 1V ]

(1/α)α

(PtxKψW

) 1α

(29)

E[N] = eλE[ 1V ]

(1/α)α

(PtxKψW )

1α (30)

With superimposed lognormal shadowing and Ray-leigh fading, the E[D] and E[N] are obtained by combin-ing (27), (28) and (11). These quantities are given by

E[D] =1 − e−λE[ 1V ]

(1/α)α

eσ2 /2α2(PtxKψW )

1α

λE[1/V]e−λE[1/V]

(1/α)eσ 2

αeσ2/2α2 (

PtxKψW

)

1α

(31)

E[N] = eλE[ 1V ]

(1/α)α

eσ2/2α2

(PtxKψW )

1α (32)

For Rician fading channel, the connectivity parametersE[D] and E[N] are obtained by combining (27), (28) and(17) and are given by

E(D) =

[exp

(λE[ 1V ]

1α

(PtxK

(κ+1)ψW

) 1α e−κ

∑∞t=0

κ t

t!

∑tl=0

( 1α+l)

l!

)]− 1

λE[ 1V ]

(33)

E(N) = exp

⎛⎝λE[

1V]1α

(KPtx

(κ + 1)ψW

) 1α

e−κ

∞∑t=0

κ t

t!

t∑l=0

( 1α+ l)

l!

⎞⎠ (34)


Page 7 of 16

For Rician fading with shadowing, E[D] and E[N] areobtained by combining (27), (28) and (19) and are com-puted as follows:

E(D) =

[exp

(λE[ 1V ]

1α

(KPtx

ψW(κ+1)

)1/αe2σ 2

α2 e−κ∑∞

t=0κ t

t!

∑tl=0

(1/α+l)l!

)]− 1

λE[ 1V ](35)

E(N) = exp

(λE

[1V

]1α

(KPtx

ψW(κ + 1)

)1/α

e2σ 2

α2 e−κ

∞∑t=0

κ t

t!

t∑l=0

( 1α+ l)

l!

)(36)

With Weibull fading, the connectivity parameters E[D]and E[N] are obtained by combining (27), (28) and (23).The corresponding expressions are given by

E(D) =

[exp

(λE[ 1V ]

(KPtx

ψW(1+2/c)

)1/α ( 1(αc/2)

)(αc/2)

)]− 1

λE[ 1V ]

(37)

E(N) = exp

⎛⎝λE

[1V

](KPtx

ψW(1 + 2/c)

)1/α (

1(αc/2)

)(αc/2)

⎞⎠ (38)

The proposed analytical model can be used to find outthe impact of channel randomness on connectivity char-acteristics of VANETS. In the next section, we providethe numerical and simulation results.

5. Numerical and simulation resultsIn this section, we present analytical and simulationresults for average connectivity distance and average pla-toon size. Both the analytical and the simulation resultsare obtained using Matlab. The analytical results corre-spond to the mathematical models presented in Sections3 and 4. As mentioned before, in the free flow state, thevehicle speed and traffic flow are independent andhence there are no significant interactions between indi-vidual vehicles. Hence, we use Matlab to simulate anuninterrupted highway. We consider highway of length:L = 10 km. Vehicles are generated from a Poisson pro-cess with arrival rates l veh/s. Each vehicle is assigned arandom speed chosen from a truncated Gaussian distri-bution. Table 1 shows typical values for the mean (μvkm/h) and standard deviation (sv km/h) of the vehiclespeed on the highway [14]. To find the simulation

results, we proceed as follows: We consider one snapshot of the highway at the arrival instant of each vehicleand find the inter-vehicle distance values from the simu-lation. If there are N vehicles on the highway, there willbe (N - 1) links and hence (N - 1) values for the inter-vehicle distances. For each link, we then find the averageSNR, γ̄ (d) = KPtx/dαW , corresponding to the measuredvalue of inter-vehicle distance, d, of that link. AssumingRayleigh fading environment, we then generate an expo-nentially distributed random variable, representing thereceived SNR over that link, with average value γ̄ (d) . Ifthe received SNR is greater than the threshold value ψ,the link is considered to be connected. The same pro-cess is repeated for all the (N - 1) links with their corre-sponding inter-vehicle distance values. If all the links ina snap shot are connected, the network is considered tobe connected. The connectivity distance is then evalu-ated as the length of connected path from a vehicle.The connectivity distance evaluation process is thenrepeated 10,000 times. The average values of connectiv-ity distance and platoon size are calculated from these10,000 sample values. For Rician, Weibull and othermodels, we follow the same procedure and find theaverage values of connectivity distance and platoon size.First, we provide analytical and simulation results for

the average vehicle density rav. We select a highway oflength L = 10 Km, and vehicle arrival rate is kept as l =0.14 veh/s. Figure 1 shows the variation of rav againstthe standard deviation of vehicle speed sv for variousvalues of mean vehicle speed μv. Here, the analyticalresults are obtained using (5). It can be observed that,for fixed μv, the average vehicle density rav increases assv increases. As sv varies from 3 to 21 km/h for μv = 70km/h, rav varies from 5.15 to 5.8 veh/km. Further, forany given value of sv, rav decreases as μv increases.Thus, the results show that both μv and sv have signifi-cant impact on rav.Next, we present the connectivity results for various

channel models. For getting analytical and simulationresults, we fix various parameters as follows: Path lossconstant K = 16.37 × 10-6, received SNR threshold ψ =10 dB and the total additive noise power W = 1.65 × 10-13 Watts. Figures 2 and 3, respectively, show the impactof standard deviation of shadow fading s on averageconnectivity distance E[D] and average platoon size E[N]. Rayleigh fading with superimposed log normal sha-dowing is considered. The figures also depict the impactof mean μv and standard deviation sv of vehicle speedon connectivity distance. To get the results, we choosevehicle arrival rate l = 0.1 veh/s, transmit power Ptx =33 dBm and path loss exponent a = 2.5. The analyticalresults for E[D] are obtained from (31), while the resultsfor E[N] are obtained from (32). The figures show that

Table 1 Normal-vehicle speed statistics [14]

μv (km/h) sv(km/h)

70 21

90 27

110 33

130 39

150 45


Page 8 of 16

0 5 10 15 20 25 30 35 40 452

2.5

3

3.5

4

4.5

5

5.5

6

Standard Deviation of Vehicle Speed, σv (km/hr)

Ave

rage

Veh

icle

Den

sity

,ρav

(veh

/km

)μv=70 km/hr −Analysis

μv=70 km/hr −Simulation

μv=90 km/hr−Analysis

μv=90 km/hr−Simulation







Figure 1 Average vehicle density versus standard deviation of vehicle speed.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

0.5

1

1.5

2

2.5

3x 104

Standard Deviation of Shadow Fading, σ

Ave

rage

Con

nect

ivity

Dis

tanc

e E

(D) (

m)

μv= 150 km/hr, σv= 25 km/hr − analytical

μv= 150 km/hr, σv= 25 km/hr − simulation

μv= 110 km/hr, σv= 10 km/hr− analytical

μv= 110 km/hr, σv= 10 km/hr− simulation


μv= 110 km/hr, σv= 25 km/hr−simulation

Figure 2 Average connectivity distance versus standard deviation of shadow fading s, (a = 2.5, Ptx = 33 dBm, l = 0.1 veh/s).


Page 9 of 16

shadow fading standard deviation s has positive impacton both E[D] and E[N], improving the connectivity per-formance of VANETs. Similar results were reported in[9] for the case of static ad hoc wireless networks. It isalso observed that that both μv and sv have significantimpact on E[D] and E[N ]. For fixed μv, as depicted inFigure 1, rav increases as sv increases; this improves theaverage values of connectivity distance and platoon size(shown in Figures 2, 3). As shown in Figure 1, for agiven value of sv , rav decreases as μv increases, result-ing in degradation of E[D] and E[N ], and the corre-sponding results are depicted in Figures 2 and 3.Figures 4 and 5, respectively, show the impact of path

loss exponent a on average connectivity distance E[D]and average platoon size E[N]. Once again, we considerRayleigh fading with superimposed log normal shadow-ing. Here, various parameters are selected as follows:Transmit power Ptx = 33 dBm and the shadow fadingstandard deviation s = 2. The analytical results for E[D]are obtained from (31), while the results for E[N] areobtained from (32). The results show that both E[D]and E[N] degrade as the path loss exponent a increases.As mentioned before in Section 2, empirical studieshave shown that for highway, urban, and suburban con-ditions, V2V channels are in general characterized bylow values for a ranging from 1.8 to 2.5 [30,31]. Ourresults show that, for such small values of a, both E[D]

and E[N] are high. In rural scenario for which a two-raymodel is suitable (higher a) [30,31], both E[D] and E[N]are observed to be less.Figures 6 and 7, respectively, show the impact of vehi-

cle density rav on average connectivity distance E[D]and average platoon size E[N], assuming Rayleigh fadingwith superimposed log normal shadowing. Here, variousparameters are selected as follows: Path loss exponent a= 2.5, transmit power Ptx = 33 dBm and two differentvalues are considered for the shadow fading standarddeviation s = 2 and 2.5. The analytical results for E[D]are obtained from (31), while the results for E[N] areobtained from (32). The results show that both E[D]and E[N] increases as the average vehicle density ravincreases. Further, as shadow fading standard deviations increases, both E[D] and E[N] increases, which meansthat the average vehicle density required to satisfy agiven value of average connectivity distance decreases,as s increases.Figures 8 and 9, respectively, show the impact of

Rician factor � on average connectivity distance E[D]and average platoon size E[N]. Further, the impact ofWeibull fading parameter c on the connectivity metricsE[D] and E[N] is depicted in Figures 10 and 11 respec-tively. Here, various parameters are selected as follows:Path loss exponent a = 2.5 and transmit power Ptx = 33dBm. For Rician fading, the analytical results for E[D]

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

10

20

30

40

50

60

70

80

90

100


Ave

rage

Pla

toon

Siz

e E

(N)







Figure 3 Average platoon size versus standard deviation of shadow fading s, (a = 2.5, Ptx = 33 dBm, l = 0.1 veh/s).


Page 10 of 16

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 30

0.5

1

1.5

2

2.5x 104

Path loss exponent, α

Ave

rage

Con

nect

ivity

Dis

tanc

e E

(D) (

m)







Figure 4 Average connectivity distance versus path loss exponent a, (s = 2, Ptx = 33 dBm, l = 0.1 veh/s).

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 30

10

20

30

40

50

60

70

80


Ave

rage

Pla

toon

Siz

e E

(N)







Figure 5 Average platoon size versus path loss exponent a, (s = 2, Ptx = 33 dBm, l = 0.1 veh/s).


Page 11 of 16

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4x 104

Average Vehicle Density (Veh/km)

Ave

rage

Con

nect

ivity

Dis

tanc

e E

(D) (

m)

Rayleigh(analytical)Rayleigh(simulation)Rayleigh with Lognormal, σ= 2(analytical)Rayleigh with Lognormal, σ= 2(simulation)Rayleigh with Lognormal, σ= 2.5(analytical)Rayleigh with Lognormal, σ= 2.5(simulation)

Figure 6 Average connectivity distance versus vehicle density, (a = 2.5, Ptx = 33 dBm).

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400


Ave

rage

Pla

toon

Siz

e E

(N)

Rayleigh(analytical)Rayleigh(simulation)Rayleigh with Lognormal, σ= 2(analytical)Rayleigh with Lognormal, σ= 2(simulation)Rayleigh with Lognormal, σ= 2.5(analytical)Rayleigh with Lognormal, σ= 2.5(simulation)

Figure 7 Average platoon size versus vehicle density, (a = 2.5, Ptx = 33 dBm).


Page 12 of 16

1 2 3 4 5 6 7 8 9 100

500

1000

1500

2000

2500

3000

3500

4000

4500

5000


Ave

rage

Con

nect

ivity

Dis

tanc

e E

(D)(

m)

Rician factor κ= 2(analytical)Rician factor κ= 2(simulation)Rician factor κ= 4(analytical)Rician factor κ= 4(simulation)


1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50


Ave

rage

Pla

toon

Siz

e E

(N)

Rician factor κ= 2(analytical)Rician factor κ= 2(simulation)Rician factor κ= 4(analytical)Rician factor κ= 4(simulation)



Page 13 of 16

1 2 3 4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000


Ave

rage

Con

nect

ivity

Dis

tanc

e E

(D) (

m)

Weibull parameter c = 1.6(analytical)Weibull parameter c = 1.6(simulation)Weibull parameter c = 3(analytical)Weibull parameter c = 3(simulation)Weibull parameter c = 5(analytical)Weibull parameter c = 5(simulation)


1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60


Ave

rage

Pla

toon

Siz

e E

(N)

Weibull parameter c = 1.6(analytical)Weibull parameter c = 1.6(simulation)Weibull parameter c = 3(analytical)Weibull parameter c = 3(simulation)Weibull parameter c = 5(analytical)Weibull parameter c = 5(simulation)



Page 14 of 16

are obtained from (33) and (35), while the results for E[N] are obtained from (34) and (36). For Weibull fading,we use (37) and (38) to find the analytical results. Asdetailed in Section 2, Rician fading is used to statisticallydescribe the V2V communication in urban, suburbanand highway environments, when the distance betweencommunicating vehicles is less and a strong LOS com-ponent is present. As the vehicle separation increases,the fading gradually transits from Rician to Rayleigh.When the distance exceeds 70-100 m, the fadingbecomes worse than Rayleigh, modeled using WeibullPDF. The Weibull fading parameter c controls the sever-ity of fading. When c = 2, Weibull distribution isequivalent to Rayleigh, and for c >2, the distribution isanalogous to Rician and represents rural scenario withsignificant LOS components. Values of c <2, correspondto worse than Rayleigh fading, representing city environ-ments with significant non-LOS components. When thepropagation distance is less, observed values of c rangefrom 2.4 to 5.1, while for larger distances, c values rangefrom 1.6 to 2 [26]. Our results show that the Rician fad-ing factor � has a positive influence on the connectivity.In the Weibull case, the connectivity distance getsdegraded significantly when the Weibull parameter cgoes below 2 (urban highway with strong multi-path),and this corresponds to worse than Rayleigh fading. Theconnectivity probability gets improved when c >2 (ruralhighway with strong LOS).Since VANETs are targeted to support applications

such as safety and emergency information delivery,entertainment, data collection, reliable data dissemina-tion would be one of the critical requirements of suchnetworks. For the delivery of safety and emergencyinformation, such networks have to be operated in thebroadcast mode, while for comfort applications, the net-work must support unicast as well. For broadcast appli-cations, the connectivity distance is equivalent tocoverage area for a transmitted message, while for com-fort applications, this metric decides the accessibility toroadside units for accessing the Internet. Similarly, if thenumber of vehicles in a connected path is quite large(larger cluster size), a message that is sent by a taggednode in the cluster immediately gets delivered to allthese vehicles. This paper has extensively analyzed thesetwo important parameters and the results are useful tofind out the impact of various traffic-dependent andchannel-dependent parameters on these metrics.

6. ConclusionIn this paper, we have presented an analytical model tofind the connectivity characteristics of a VANET in afading channel from a queuing theoretic perspective. Inparticular, we have analytically characterized the effect

of channel randomness on the average connectivity dis-tance and average platoon size. To perform the connec-tivity analysis, we have used results from an equivalentM/G/∞ queue. Three different fading models were con-sidered for the analysis: Rayleigh, Rician and Weibull.The impact of physical layer parameters such as pathloss exponent, shadow fading standard deviation andfading factors was analyzed. By assuming vehicle speedto be a random variable with truncated Gaussian prob-ability distribution, we presented the dependence ofvehicle speed statistics (such as its mean and standarddeviation) and average vehicle density on the connectiv-ity characteristics. The analytical model and the resultspresented in this paper would be useful for a networkdesigner developing a self organizing vehicular ad hocnetwork for intelligent transport applications. The paperprovides information regarding the influence of signifi-cant system parameters, such as vehicle arrival rate,vehicle density, mean and standard deviation of vehiclespeed and physical layer parameters on VANET connec-tivity. Extensive simulations were carried out to validatethe analytical model findings. It was observed that thesimulation results agree closely with the theoreticalresults.

Competing interestsThe authors declare that they have no competing interests.

Received: 7 February 2011 Accepted: 2 January 2012Published: 2 January 2012

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doi:10.1186/1687-1499-2012-1Cite this article as: Chandrasekharamenon and AnchareV: Connectivityanalysis of one-dimensional vehicular ad hoc networks in fadingchannels. EURASIP Journal on Wireless Communicationsand Networking 2012 2012:1.

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