To analyze the connectivity of VANETs in the presence of channel randomness, we rely on [4], in which the authors addressed the connectivity issues in one-dimensional ad hoc networks, from a queuing theoretic perspective. Authors exploited the relationship between coverage problems and infinite server queues, and by utilizing the results from an equivalent *G*/*G*/∞ queue, they addressed the connectivity properties of an ad hoc network. The authors also identified the equivalence between the following: (i) the busy period of an infinite server queue and the connectivity distance in an ad hoc network and (ii) the number of customers served during a busy period and the number of nodes in a connected cluster in the network. The following assumptions were utilized to obtain the results: (i) the inter-arrival times in the infinite server queue have the same distribution as the distance between successive nodes; and (ii) the service times have the same probability distribution as the transmission range of the nodes. In this paper, we study the connectivity properties of VANETs using the corresponding infinite server queuing model. For this, the probability distribution functions (PDF) of inter-vehicle distance and vehicle transmission range are required. We now present the system model, which includes the highway and mobility model, used for the connectivity analysis. A model to find the statistical characteristics of the transmission range for various fading models is then introduced.

*A*. Highway and mobility model

The highway and mobility model used for the connectivity analysis is based on [14] and is briefly described here. Assume that an observer stands at an arbitrary point of an uninterrupted highway (i.e., without traffic lights). Empirical studies have shown that Poisson distribution provides an excellent model for vehicle arrival process in free flow state [3]. Hence, it is assumed that the number of vehicles passing the observer per unit time is a Poisson process with rate *λ* vehicles/h. Thus, the inter-arrival times are exponentially distributed with parameter λ. Assume that there are *M* discrete levels of constant speed *v*_{
i
} , *i* = 1, 2,..., *M* where the speeds are i.i.d., and independent of the inter-arrival times. Let the arrival process of vehicles with speed *v*_{
i
} be Poisson with rate *λ*_{
i
} , *i* = 1, 2,..., *M*, and let {\Sigma}_{i=1}^{M}{\lambda}_{i}=\lambda. Further, it is assumed that these arrival processes are independent, and the probability of occurrence of each speed is *p*_{
i
} = λ_{i}/λ. Let *X*_{
n
} be the random variable representing the distance between *n* th closest vehicle to the observer and (*n* - 1)th closest vehicle to the observer. It has been proved in [14] that the inter-vehicle distances are i.i.d., and exponentially distributed with parameter {\rho}_{\mathsf{\text{av}}}={\Sigma}_{i=1}^{M}\frac{{\lambda}_{i}}{{v}_{i}}=\lambda {\Sigma}_{i=1}^{M}\frac{{p}_{i}}{{v}_{i}}. Specifically, the CDF of inter-vehicle distance *X*_{
n
} is given by

{F}_{Xn}\left(x\right)=1-{\mathsf{\text{e}}}^{-{\rho}_{\mathsf{\text{av}}}x},\phantom{\rule{1em}{0ex}}x\ge 0

(1)

In free flow state, the movement of a vehicle is independent of all other vehicles. Empirical studies have shown that the speeds of different vehicles in free flow state follow a Gaussian distribution [3]. We, therefore, assume that each vehicle is assigned a random speed chosen from a Gaussian distribution and that each vehicle maintains its randomly assigned speed while it is on the highway. To avoid dealing with negative speeds or speeds close to zero, we define two limits for the speed, i.e., *v*_{max} and *v*_{min} for the maximum and minimum levels of vehicle speed, respectively. For this, we use a truncated Gaussian probability density function (PDF), given by [14]

{g}_{V}\left(v\right)=\frac{{f}_{V}\left(v\right)}{{\int}_{{v}_{\text{min}}}^{{v}_{\text{max}}}{f}_{V}\left(u\right)\mathsf{\text{d}}u}

(2)

where {f}_{V}\left(v\right)=\frac{1}{{\sigma}_{v}\sqrt{2\pi}}\text{exp}\left(\frac{-{\left(v-{\mu}_{v}\right)}^{2}}{2{\sigma}_{v}^{2}}\right) is the Gaussian PDF, *μ*_{
v
} --average speed, *σ*_{
v
} --standard deviation of the vehicle speed, *v*_{max} = *μ*_{
v
} + 3*σ*_{
v
} the maximum speed and *v*_{min} = *μ*_{
v
} - 3*σ*_{
v
} the minimum speed [14]. Substituting for *f*_{
V
} (*v*) in (2), the truncated Gaussian PDF *g*_{
V
} (*v*) is given by

{g}_{V}\left(v\right)=\frac{2{f}_{V}\left(v\right)}{\mathsf{\text{erf}}\left(\frac{{v}_{\text{max}}-{\mu}_{v}}{{\sigma}_{v}\sqrt{2}}\right)-\mathsf{\text{erf}}\left(\frac{{v}_{\text{min}}-{\mu}_{v}}{{\sigma}_{v}\sqrt{2}}\right)},\phantom{\rule{1em}{0ex}}{v}_{\text{min}}\le v\le {v}_{\text{max}}

(3)

where erf(.) is the error function [34]. Since the inter-vehicle distance *X*_{
n
} is exponentially distributed with parameter *ρ*_{av}, the average vehicle density on the highway is given by

{\rho}_{\mathsf{\text{av}}}=\frac{1}{E\left[X\right]}=\lambda \sum _{i=1}^{N}\frac{{p}_{i}}{{v}_{i}}=\lambda E\left[\frac{1}{V}\right]

(4)

where *E*[.] is the expectation operator and *V* is the random variable representing the vehicle speed. When the vehicle speed follows truncated Gaussian PDF, the average vehicle density is computed as follows:

{\rho}_{\mathsf{\text{av}}}=\frac{2\lambda /\sqrt{2\pi {\sigma}_{v}}}{\mathsf{\text{erf}}\left(\frac{{v}_{\text{max}}-{\mu}_{v}}{{\sigma}_{v}\sqrt{2}}\right)-\mathsf{\text{erf}}\left(\frac{{v}_{\text{min}}-{\mu}_{v}}{{\sigma}_{v}\sqrt{2}}\right)}\underset{{v}_{\text{min}}}{\overset{{v}_{\text{max}}}{\int}}\frac{1}{v}\text{exp}\left(\frac{-{\left(v-{\mu}_{v}\right)}^{2}}{2{\sigma}_{v}^{2}}\right)\mathsf{\text{d}}v

(5)

It may be noted that the average vehicle density given in (5) does not have a closed-form solution but has to be evaluated by numerical integration. Numerical and Simulation results for *ρ*_{av} are presented in Section 5. It is observed that the parameters *μ*_{
v
} and *σ*_{
v
} have significant impact on *ρ*_{av}. Since each vehicle enters the highway with a random speed, the number of vehicles on the highway segment of length *L* is also a random variable. The average number of vehicles on the highway is then given by *N*_{av} = *Lρ*_{av}. Next, we present a model to find the statistical characteristics of transmission range for various fading models.

*B*. Statistical characteristics of transmission range

The effect of randomness caused by fading is incorporated into the analysis by assuming the transmission range *R* to be a random variable with CDF *F*_{
R
} (*a*). Let *Z* be the random variable representing the received signal envelope and let *l* be the distance between transmitting and receiving nodes. Further we assume that "good long codes" are used, so that probability of successful reception, as a function of the signal-to-noise ratio (SNR) approaches a step function, whose threshold is denoted by *ψ*[4]. Additive Gaussian noise of power *W* watts is assumed to be present at the receiver. The received power is then given by *P*_{
rx
} = *P*_{
tx
}*z*^{2}*K*/*l*^{α} where *P*_{
tx
} is the transmit power, *α* is the path loss exponent and *K* is a constant associated with the path loss model. Here, *K* = *G*_{
T
}*G*_{
R
}*C*^{2}/(4*π f*_{
c
} )^{2}, where *G*_{
T
} and *G*_{
R
} , respectively, represent the transmit and receive antenna gains, *C* is the speed of light and *f*_{
c
} is the carrier frequency [18, 35, 36]. In this paper, we assume that the antennas are omni directional (*G*_{
T
} = *G*_{
R
} = 1), and the carrier frequency *f*_{
c
} = 5.9 GHz. The thermal noise power is given by *W* = *FkT*_{
o
}*B* where *F* is the receiver noise figure, *k* = 1.38 × 10^{-23} J/K is the Boltzmann constant, *T*_{
o
} is the room temperature (*T*_{
o
} = 300° K) and *B* is the transmission bandwidth (*B* = 10 MHz for 802.11p). The received SNR is computed as *γ* = *P*_{
tx
}*Z*^{2}*K*/*l*^{α}*W*. Assuming that *E*[*Z*^{2}] = 1, the average received SNR is \stackrel{\u0304}{\gamma}={P}_{tx}K/{l}^{\alpha}W. In our model, the transmitted message can be correctly decoded if and only if the received SNR *γ* is greater than a given threshold *ψ*. In the remaining part of this section, we find the statistics of the transmission range for various fading models. For Rayleigh fading, these results were reported in [4]. We extend the analysis to Rician and Weibull fading models. We also consider the combined effect of lognormal shadow fading and small-scale fading models.

*1)* Rayleigh fading

Assume that the received signal amplitude in V2V channel follows Rayleigh PDF. The Rayleigh distribution is frequently used to model multi-path fading with no direct line-of-sight (LOS) path. It has been reported in the literature that, in V2V communication as the separation between source and destination vehicles increases, the LOS component may be lost and hence the PDF of the received signal amplitude gradually transits from near-Rician to Rayleigh [24–26]. Further, the multi-path component becomes more significant when compared to the LOS component in congested city roads, and hence the Rayleigh fading model is more suitable to describe the PDF of the received signal amplitude in such scenarios. It is also assumed that the fading is constant over the transmission of a frame and subsequent fading states are i.i.d. (block-fading) [33]. The received SNR has exponential distribution given by [35]

f\left(\gamma \right)=\frac{1}{\stackrel{\u0304}{\gamma}}{\mathsf{\text{e}}}^{-\gamma /\stackrel{\u0304}{\gamma}},\phantom{\rule{1em}{0ex}}\gamma \ge 0

where \stackrel{\u0304}{\gamma} is the average SNR. The probability that the message is correctly decoded at a distance *l* is given by

P\left[\gamma \left(l\right)\ge \psi \right]={\mathsf{\text{e}}}^{-\psi /\stackrel{\u0304}{\gamma}}={\mathsf{\text{e}}}^{-{l}^{\alpha}W\psi /{P}_{tx}K}

(7)

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

\begin{array}{ll}\hfill {F}_{R}\left(a\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}P\left(R\le a\right)=1-P\left(R>a\right)=1-P\left(\gamma \left(a\right)\ge \psi \right)\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}1-{\mathsf{\text{e}}}^{-\psi {a}^{\alpha}W/K{P}_{tx}}\phantom{\rule{2em}{0ex}}\end{array}

The average transmission range is given by [4]:

\begin{array}{ll}\hfill E\left(R\right)\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}{\int}_{0}^{\infty}\left(1-{F}_{R}\left(a\right)\right)\mathsf{\text{d}}a\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}\frac{\mathrm{\Gamma}\left(1/\alpha \right)}{\alpha}{\left(\frac{{P}_{tx}K}{\psi W}\right)}^{1/\alpha}\phantom{\rule{2em}{0ex}}\end{array}

(9)

where Γ(.) is the Gamma function [34].

*2)* Rayleigh fading with superimposed lognormal shadowing

Let *Y* be the random variable representing shadow fading. Its PDF is given by f\left(y\right)=\frac{1}{\sqrt{2\pi}\sigma y}{\mathsf{\text{e}}}^{-\frac{{\left(\text{ln}\left(y\right)-\text{ln}\left(K{l}^{-\alpha}\right)\right)}^{2}}{2{\sigma}^{2}}}, where *σ* is the standard deviation of shadow fading process [35] and *l* is the transmitter to receiver separation. For the superimposed lognormal shadowing and Rayleigh fading scenario, the CDF of the transmission range can be computed as follows [4]:

{F}_{R}\left(a\right)=1-P\left(R>a\right)=1-P\left(\gamma \left(a\right)>\psi \right)=1-\underset{\psi}{\overset{\infty}{\int}}{\mathsf{\text{e}}}^{\frac{-\psi W}{K{P}_{tx}x}}\frac{1}{\sqrt{2\pi}\sigma y}{\mathsf{\text{e}}}^{-\frac{{\left(\text{ln}\left(y\right)-\text{ln}\left(K{a}^{-\alpha}\right)\right)}^{2}}{2{\sigma}^{2}}}

(10)

The average transmission range is then given by [4]

\begin{array}{ll}\hfill E\left[R\right]\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\underset{0}{\overset{\infty}{\int}}\left(1-{F}_{R}\left(a\right)\right)\mathsf{\text{d}}a\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}\frac{\mathrm{\Gamma}\left(1/\alpha \right)}{\alpha}{\mathsf{\text{e}}}^{{\sigma}^{2}/2{\alpha}^{2}}{\left(\frac{{P}_{tx}K}{\psi W}\right)}^{1/\alpha}\phantom{\rule{2em}{0ex}}\end{array}

(11)

*3)* Rician fading

The Rician fading is used to model propagation paths consisting of one strong LOS component and many random weaker components. In rural highways, the multi path components may be weak, so the communication can be modeled as purely LOS in nature, for which Rician fading model is more appropriate. Empirical studies for different V2V communication contexts at 5.9 GHz, which include express-way, urban canyon and suburban street, have predicted the PDF of received signal amplitude to be either Rayleigh or Rician [24]. When the distance between transmitter and receiver is less than 5 m, the fading follows Rician, which is characterized by the Rician factor *κ* (defined as the ratio of energy in the LOS path to the energy in the scattered path). The PDF of the received SNR in a Rician faded channel is given as follows [35]:

f\left(\gamma \right)=\frac{1+\kappa}{\stackrel{\u0304}{\gamma}}{\mathsf{\text{e}}}^{-\kappa}{\mathsf{\text{e}}}^{-\left(\kappa +1\right)\gamma /\stackrel{\u0304}{\gamma}}{I}_{0}\left(2\sqrt{\kappa \left(\kappa +1\right)\gamma /\stackrel{\u0304}{\gamma}}\right)

(12)

where *κ* is the Rician factor, and *I*_{0}(.) represents the modified Bessel function of the zeroth order and first kind. Now *I*_{0}(*x*) can be expanded as {I}_{0}\left(x\right)={\sum}_{t=0}^{\infty}{\frac{1}{t!\mathrm{\Gamma}\left(t+1\right)}\left(\frac{x}{2}\right)}^{2t}. For integer values of *t*, Γ(*t* + 1) = Γ(*t*) = *t*!, so that {I}_{0}\left(x\right)={\sum}_{t=0}^{\infty}\frac{1}{{\left(t!\right)}^{2}}{\left(\frac{x}{2}\right)}^{2t}. Hence, the PDF of received SNR becomes:

f\left(\gamma \right)=\frac{1+\kappa}{\stackrel{\u0304}{\gamma}}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{{\left(t!\right)}^{2}}{\left(\frac{\left(\kappa +1\right)\gamma}{\stackrel{\u0304}{\gamma}}\right)}^{t}{\mathsf{\text{e}}}^{\frac{-\left(\kappa +1\right)\gamma}{\stackrel{\u0304}{\gamma}}}

(13)

The CDF of transmission range is calculated as follows:

{F}_{R}\left(a\right)=1-P\left(\gamma \left(a\right)>\psi \right)=1-\underset{\psi}{\overset{\infty}{\int}}f\left(\gamma \right)\mathsf{\text{d}}\gamma

(14)

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

{F}_{R}\left(a\right)={\mathsf{\text{e}}}^{\frac{-\left(\kappa +1\right)\psi}{\stackrel{\u0304}{\gamma}}}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{t!}\sum _{l=0}^{t}\frac{{\left(\left(\kappa +1\right)\psi \right)}^{l}}{{\stackrel{\u0304}{\gamma}}^{l}l!}

(15)

where \stackrel{\u0304}{\gamma}={P}_{tx}K/{a}^{\alpha}W. The expected value of the transmission range under the Rician fading model is given by

\begin{array}{ll}\hfill E\left[R\right]\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}\left(1-{F}_{R}\left(a\right)\right)\mathsf{\text{d}}a\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}{\mathsf{\text{e}}}^{\frac{-\left(\kappa +1\right)\psi {a}^{\alpha}W}{K{P}_{tx}}}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{t!}\sum _{l=0}^{t}\frac{1}{l!}{\left(\frac{\left(\kappa +1\right)\psi {a}^{\alpha}W}{K{P}_{tx}}\right)}^{l}\phantom{\rule{2em}{0ex}}\end{array}

(16)

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

E\left(R\right)=\frac{1}{\alpha}{\left(\frac{K{P}_{tx}}{\psi W\left(\kappa +1\right)}\right)}^{1/\alpha}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{t!}\sum _{l=0}^{t}\frac{\mathrm{\Gamma}\left(\frac{1}{\alpha}+l\right)}{l!}

(17)

*4)* Rician fading with superimposed lognormal shadowing

In this case, the CDF of the transmission range can be computed as follows:

\begin{array}{ll}\hfill {F}_{R}\left(a\right)\phantom{\rule{1em}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}1-P\left(\gamma \left(a\right)>\psi \right)\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}1-\underset{\psi}{\overset{\infty}{\int}}\mathsf{\text{d}}x\frac{1}{\sqrt{2\pi}\sigma x}{\mathsf{\text{e}}}^{-\frac{{\left(\text{ln}\left(x\right)-\text{ln}\left(K{l}^{-\alpha}\right)\right)}^{2}}{2{\sigma}^{2}}}\left[{\mathsf{\text{e}}}^{\frac{-\left(\kappa +1\right)\psi}{\stackrel{\u0304}{\gamma}}}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{t!}\sum _{l=0}^{t}\frac{{\left(\left(\kappa +1\right)\psi \right)}^{l}}{{\stackrel{\u0304}{\gamma}}^{l}l!}\right]\phantom{\rule{2em}{0ex}}\end{array}

(18)

The mean transmission range is then determined as follows:

\begin{array}{ll}\hfill E\left[R\right]\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\underset{0}{\overset{\infty}{\int}}\left(1-{F}_{R}\left(a\right)\right)\mathsf{\text{d}}a\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{1em}{0ex}}\frac{1}{\alpha}{\left(\frac{K{P}_{tx}}{\psi W\left(\kappa +1\right)}\right)}^{1/\alpha}{\mathsf{\text{e}}}^{\frac{2{\sigma}^{2}}{{\alpha}^{2}}}{\mathsf{\text{e}}}^{-\kappa}\sum _{t=0}^{\infty}\frac{{\kappa}^{t}}{t!}\sum _{l=0}^{t}\frac{{\mathrm{\Gamma}}^{1}\left(\frac{1}{\alpha}+l\right)}{l!}\phantom{\rule{2em}{0ex}}\end{array}

(19)

*5)* Weibull fading

The Weibull distribution is often found to be very suitable to fit empirical non-LOS V2V fading channel measurements [25, 26]. In [26], the authors reported severe fading in multiple V2V settings based upon measurements in the 5 GHz band and found that Weibull distribution can be used to approximate measured severe fading conditions. It may be noted that Weibull fading is capable of representing both LOS and non-LOS cases. The PDF of the received SNR under Weibull fading is given by [37]:

f\left(\gamma \right)=\frac{c}{2}{\left(\frac{\mathrm{\Gamma}\left(1+2/c\right)}{\stackrel{\u0304}{\gamma}}\right)}^{c/2}{\gamma}^{c/2-1}\text{exp}{\left[-\left(\frac{\mathrm{\Gamma}\left(1+2/c\right)\gamma}{\stackrel{\u0304}{\gamma}}\right)\right]}^{c/2}

(20)

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

F\left(\gamma \right)=1-\text{exp}\left[-{\left(\frac{\mathrm{\Gamma}\left(1+2/c\right)\gamma}{\stackrel{\u0304}{\gamma}}\right)}^{c/2}\right]

(21)

The CDF of the transmission range is given by

{F}_{R}\left(a\right)=1-P\left(\gamma \left(a\right)>\psi \right)={{\mathsf{\text{e}}}^{-\left(\frac{\mathrm{\Gamma}\left(1+2/c\right)\psi}{\stackrel{\u0304}{\gamma}}\right)}}^{{}^{{}^{c/2}}}

(22)

where \stackrel{\u0304}{\gamma}={P}_{tx}K/{a}^{\alpha}W. The average transmission range is computed as follows:

\begin{array}{c}E[R]\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{\displaystyle \underset{0}{\overset{\infty}{\int}}(1-{F}_{R}(a))\text{d}a}\\ =\phantom{\rule{0.5em}{0ex}}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\text{e}}^{-{\left({\scriptscriptstyle \frac{\mathrm{\Gamma}(1+2/c)\psi}{\overline{\gamma}}}\right)}^{c/2}}\text{d}x}\\ =\phantom{\rule{0.5em}{0ex}}{\left(\frac{K{P}_{tx}}{\psi W\mathrm{\Gamma}(1+2/c)}\right)}^{1/\alpha}\frac{\mathrm{\Gamma}\left({\scriptscriptstyle \frac{1}{(\alpha c/2)}}\right)}{(\alpha c/2}\end{array}

(23)

With these preliminary results, we present the connectivity analysis in the next section.