Measurement-Based Analysis on Vehicle-to-Vehicle Connectivity in Tunnel Environment

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Introduction
Vehicular ad hoc network (VANET) is a subset of the mobile ad hoc network (MANET); it includes self-organizing vehicle and roadside infrastructure.In the VANET, vehicles and infrastructures are equipped with wireless communication equipment for real time information exchange, which is a fundamental of the efcient and safe intelligent transportation system [1][2][3].Particularly, the radio link connectivities of vehicle-to-infrastructure (V2I) and vehicle-tovehicle (V2V) is a key metric to assess the VANET communication performances in terms of both communication range and vehicle networking that strongly afect advanced techniques like cooperative communication and positioning.Terefore, it is a meaningful work to study the connectivity in the VANET.
Due to factors like propagation loss, the dynamic characteristic of VANET topology, high speed of vehicle movement, and the occlusion of other vehicles, the received signal strength of a specifc link changes rapidly, even drops down below the level of system sensitivity.As a result, established communication links may be interrupted or even lost.Over the past years, the connectivity of VANETs has been extensively studied, where the research focuses on the following aspects: (1) the infuence of trafc and weather factors on connectivity (e.g., the trafc fow, vehicle density, the distribution of vehicle and its speed, weather conditions, and trafc lights); (2) the efect of wireless propagation environments on connectivity (e.g., path loss (PL) model, Nakagami fading, Weibull fading, Rice fading, Rayleigh fading, and communication range); (3) diferent modelling approaches of connectivity; (4) performance enhancing method of connectivity, such as the deployment of roadside unit (RSU), and multihop communications.
Several research studies mainly concentrate on the impacts of various factors on VANETconnectivity.In [4,5], the infuences of communication range, vehicle speed, the safe distance, size of vehicle, trafc lights, and overtaking of vehicles on connectivity were analyzed.Te dependency between the connectivity of the VANET and mobility was studied in [6][7][8][9][10][11][12][13].In [11], authors analyzed the efects of vehicle mobility on the VANET connectivity probability based on a measurement data set.Simulation results showed that the connectivity probability decreases with a power-law decline if vehicle speed is greater than certain threshold, whereas the connectivity is not afected if the vehicle speed is below the threshold.In [14][15][16], the authors analyzed the impacts of transmission ranges of both base station (BS) and vehicle, vehicle density, and the distance between adjacent BSs on connectivity performance given diferent communication channel models, specifcally, unit disk (UD) model and log-normal shadowing model.When studying the factors that afect the connectivity of the VANET, the impact of user behavior on connectivity cannot be ignored.Te efect of user behavior on V2V and V2I connectivity was analyzed in [17,18].
Besides, wireless propagation environments have shown to have signifcant impact on the connectivity.In [17,[19][20][21][22][23][24][25][26], theoretical fading models (e.g., Rayleigh, Rice, Weibull, and Nakagami) were considered, based on which the authors presented analytical models.Moreover, the impacts of diferent channel parameters (e.g., shadow fading, PL exponent, and small scale fading parameters) on the link connectivity probability of the VANET were studied in these papers.In [27], the dual-slope PL model and its impact on the V2V connectivity are evaluated.Results showed that the connectivity probability is higher in the line-of-sight (LoS) environment than in the obstructed line-of-sight (OLoS) environment in shorter transmission distance; in longer transmission distance, the V2V connectivity probability is higher in the OLoS environment than that in the LoS environment.A larger path loss exponent may result in a smaller connectivity probability [28].In [24,25,29], the efect of lognormal shadowing on connectivity probability was analyzed.It is shown that lognormal shadowing can improve the VANET connectivity.
Te previous work on the connectivity performance analysis of the VANET mainly based on queuing theory [30][31][32][33][34] or geometric-assisted analytical models [35].Recently, a lot of analytical models were developed for analyzing the connectivity probability in the VANET.In [36], authors presented a connectivity model to estimate the downlink and uplink connectivity performances for the infrastructure-based VANET.Based on the path loss model, small-scale fading, and trafc fow model, a continuous connectivity model was developed [1,37].In [38,39], under the assumption that the entrance and exit of the highway are uniformly distributed and the vehicle arrival process can be viewed as obeying the Poisson distribution.In [40], the vehicular network was simplifed and modeled as geometric elements of lines and points.In addition, authors assumed that the arrival of vehicles obeys Poisson distribution, and analyzed the capacity and connectivity performance between two adjacent RSUs in the highway scenario.In [41], authors took into account the vehicles mobility and the large-scale channel fading, the moving vehicles were divided into clusters, and a clustering assumption based analytical model was proposed.Te proposed model includes two parts, namely, catch-up process and forwarding process.In [42], the cell transmission model was adopted to capture macroscopic trafc fow dynamics, and study the connectivity performance in the freeway environment.
Both vehicle movement and channel fading have an impact on vehicle connectivity [19-22, 27, 28, 41-46].Considering both the mobility of vehicles on the road and the characteristics of channel fading can make us have a deep understanding of the VANET connectivity performance.Some authors considered both vehicle mobility and channel fading to study the infuences of various parameters on connectivity probability [21,23].Ten, we aim to discuss the research on the infuence of small scale fading on vehicle connectivity in this paper.In [21], authors considered channel fading characteristics, and derived the connectivity probability of two consecutive vehicles in one dimensional (1D) VANET under channel randomness, specifcally, Rayleigh, Rice, and Weibull fading channels.In [22], authors presented the V2V connectivity probability of two consecutive cars in the 1D VANET under the Nakagami fading channel.Tese authors only considered the connectivity of two consecutive vehicles in the 1D VANET, they did not consider connectivity between any two vehicles.However, in the real VANET, sometimes two vehicles will be blocked by other vehicles or barriers.In this case, the density of vehicles and the degree of channel fading will be diferent, so the connectivity between two vehicles will also be diferent.Te study on connectivity probability between two consecutive vehicles is not suitable for the analysis of vehicle connectivity between any two vehicles.Hence, it is meaningful to study the connectivity between any two vehicles.However, there is little research on V2V connectivity probability of any two vehicles under the small-scale fading channel.Although authors proposed the probability of any two cars are connected under the Rayleigh channel in [23], Rice, Weibull, and Nakagami fading channels were not considered.It is very important to study the connectivity performance between any two vehicles under smallscale fading in OLoS and non-line-of-sight (NLoS) cases.
Table 1 summaries various connectivity models and corresponding parameters considered in the literature.Te research studies on vehicle connectivity mainly focus on urban [55,65], highway [47,51,66], and intersection scenarios [53,58,59].Tere are very few studies on vehicle connectivity for the tunnel scenario that is a critical use case in the trafc system.In [47], the minimal safe distance was considered, and the connectivity was analyzed in a highway tunnel, where, however, only the vehicle transmission range International Journal of Antennas and Propagation and vehicle density were considered without incorporating the channel fading efect.Furthermore, in realistic OLoS and NLoS scenarios, the fading of signal amplitude usually does not follow the ideal Rayleigh distribution due to the complex propagation environment.In outdoor environment, it has been reported that fading of signal amplitudes usually follow distributions with higher degree of freedom like Weibull and Nakagami distributions.However, there exist very limited research studies of the small-scale fading model for the tunnel environment.
In this work, we focus on flling the gap in analyzing the connectivity performance in a more generalized scenario for the tunnel environment, i.e., between any two vehicles (two vehicles are blocked by other vehicles) where the traditional connectivity model between two consecutive vehicles is not suitable anymore.As a key contribution, we take into account a more realistic small-scale fading model rather than the Rayleigh distribution to derive the connectivity probability.We extend the study in [22] from two adjacent vehicles to any two vehicles.Based on the channel measurement data in a typical tunnel scenario in Munich, Germany, we analyze the small-scale fading characteristics.Measurement results show that the Nakagami-m distribution is the best ftted fading model for the tunnel environment.Tereafter, the connectivity probability between any two vehicles under Nakagami-m and log-normal shadow fading channels for a 1D VANET is derived.Furthermore, we analyze the infuences of Nakagami-m fading, large-scale fading, and trafc parameters on V2V connectivity.
Te key contributions of this paper are summarized as follows: (i) A well calibrated channel measurement campaign was carried out in a typical tunnel environment in Munich, Germany.Te signal amplitude is found to experience the Nakagami-m fading rather than Rayleigh fading.(ii) Based on the small-scale fading characteristics obtained from the measurement data in the tunnel, we propose a closed-form connectivity probability between any two vehicles under the Nakagami-m channel and log-normal shadowing channel for a one-dimensional VANET.Furthermore, we also analyze the efects of the Nakagami fading factor, the neighbor order, and the threshold value of received SNR on connectivity performance.(iii) We study the infuences of large-scale fading parameters on VANET connectivity.We demonstrate that the shadowing fading has a positive impact on V2V connectivity.
Te rest of this paper is organized as follows: in Section 2, a channel measurement campaign in the tunnel is described.Tereafter, the PL and small-scale fading characteristics are studied in Section 3. Te V2V connectivity probability considering the PL model is presented, and the infuences of large-scale fading parameters on V2V connectivity are analyzed in Section 4. We propose the V2V connectivity probability model between any two vehicles under Nakagami fading for a one-dimensional VANET in Section 5.Moreover, we discuss the infuences of various parameters on V2V connectivity probability in this section.Finally, some concluding remarks are described in Section 6.

Channel Measurement Campaign
To study the channel propagation characteristics in tunnels, we conducted the channel measurement campaign in a tunnel, as shown in Figure 1. Figure 1 illustrates the measurement environment; the tunnel is located close to the city of Munich, Germany.During the measurement process, a Medav RUSK-DLR broadband channel sounder was used to measure the channel characteristics data, and obtain the channel frequency response (CFR).Te orthogonal frequency division multiplexing (OFDM) signal was emitted by the transmitter at a center frequency of 5.2 GHz.Te bandwidth of the signal is 120 MHz.Te receiver received the CFR H(t k , f z ), where t k � k • t j , t j is the measurement time grid, k denotes the time index of the measured data, k � 1, 2, 3, • • •, and f z is the frequency associated with the bin z, z � 0, 1, . . ., N − 1, where N stands for the subcarrier's number.We can acquire the complex channel impulse response (CIR) h(t k , τ z ) by taking the inverse Fourier transform of CFR H(t k , f z ), where τ z � z • τ ∆ � z/B, B is the bandwidth, τ ∆ denotes the delay resolution, and τ ∆ � 1/B.Table 2 summarizes the channel measurement parameters.
During the measurement, the transmitter was in one car and the receiver was in the other car; the transmitting and receiving antennas were mounted on the roof of two vehicles.Te rubidium atomic clock is widely used to achieve time synchronization between the receiver and the transmitter.Without loss of generality, we also made use of two rubidium atomic clocks on the transmitter and receiver side.In order to obtain the ground truth of distance between the receiver and transmitter, accurate position information are essential.However, due to the nature of the tunnel, signal blockage by concrete walls handicaps the successful receiving of global navigation satellite system (GNSS) signals.Terefore, ground-based augmentation approach is considered.Both transmitter and receiver were equipped with GNSS receivers, together with the inertial measurement unit (IMU) and LIDAR.Terefore, it is possible to combine all diferent sensor measurements to acquire the positions of the vehicle.
Two cars were travelling in the same direction in the tunnel.In addition, the speed and the distance between two vehicles vary with the actual density of the trafc fow during the channel measurement.For the LoS scenario, there exists a clear visual LoS path between the receive antenna (RX) and transmit antenna (TX).For the OLoS scenario, the LoS path between receive antenna and transmit antenna is partially blocked by other cars with small sizes.As for the NLoS scenario, there exist vehicles with large sizes (e.g., bus, van, and truck) located in between the RX and TX such that the visual LoS path is completely blocked.Te vehicle carrying the transmitter is called the transmitting vehicle and the   Te camera is located in the transmitting vehicle.Based on the video information, we can know whether there are other vehicles obscured between the TX and RX during the channel measurement and the size of the obscured vehicles.We determined the LoS, OLoS, and NLoS scenarios by manual calibration.During the measurement, the receiving vehicle is in front and the transmitting vehicle is behind.When analyzing the video information, if there are other vehicles obscured between the TX and the RX, draw a circle with a certain radius taking the receiving vehicle in the video as the center, and if the receiving vehicle in front can be seen, we consider it as an OLoS scenario.When the receiving vehicle in front cannot be seen, we consider it as the NLoS scenario.In other words, when transmitting and receiving vehicles are obscured by small cars between them, the receiving vehicle is not completely obscured; this scenario can be seen as the OLoS scenario.When the transmitting and receiving vehicles are obscured by large trucks, the receiving vehicle is completely obscured, this scenario can be seen as the NLoS scenario.When analyzing the video information, if there are no other vehicles obscured between the transmitter and the receiver, there is a LoS path; this scenario can be seen as the LoS scenario.Combining the real time of channel measurement campaign, we can estimate the real time of LoS, OLoS, and NLoS scenarios.Trough channel measurement, we can obtain CFR data and the time information of the CFR signal.Te obtained CFR data can be converted to CIR data.Each CIR has a corresponding real time information.Based on the obtained time information of LoS, OLoS, and NLoS, we can divide the collected CIR data into LoS, OLoS, and NLoS scenarios.

International Journal of Antennas and Propagation
Figure 2 shows the LoS scenario in the tunnel.Figure 3 depicts the OLoS scenario in the tunnel, where the LoS path is obstructed by small cars.Figure 4 illustrates a NLoS scenario in the tunnel.Te transmitter and the receiver were obstructed by a blue van as shown in Figure 4, where the receiver was travelling in front of the blue van.

Measurement-Based Results
In this section, we aim to study the large-scale and smallscale fading characteristics based on the measured data in the tunnel.

Path Loss and Shadow
Fading.Te log-distance PL model in [67] is a widely used propagation model to describe the power loss during transmission; it is defned as follows: where d stands for the distance between RX and TX, P L (d) stands for the PL at distance d, ε represents a constant reference value, n represents the PL exponent and is the shadowing fading, which is usually a random variable that follows a zero-mean Gaussian distribution (in dB scale), and its standard deviation is σ.
In combination with the video information obtained during the channel measurements, the measurement data can be split into three independent data sets, namely, LoS condition, OLoS condition, and NLoS condition.We estimate the parameters (i.e., n, ε, and σ in equation (1) of the path loss model using the least square method.Te measurement PL and theoretical PL model for diferent scenarios are depicted in Figures 5-7.
Generally, shadow fading in the dB scale usually follows a zero-mean Gaussian distribution [68,69].Te shadow fading values are extracted from measurement data, and ftted with a Gaussian distribution.Te extracted PL model parameters for diferent environments are listed in Table 3.It illustrates the PL exponent n in LoS cases, and it is smaller than the n in OLoS and NLoS cases.Because of the waveguide efect in the tunnel, the obtained PL exponent n in the tunnel is less than the value in free space.Tis conclusion is consistent with theory in [70].

Small-Scale Amplitude Fading Modelling.
To mitigate the infuence of large-scale fading, a sliding window of 50 wavelengths in length was utilized to average and normalize the power.We estimate the distribution of the received signal amplitudes.We statistically modeled the received amplitudes distribution using six diferent distributions, which are widely utilized in V2V scenario, such as Rayleigh distribution, Rice distribution, normal distribution, lognormal distribution, Nakagami-m distribution, and Weibull distribution.Te Kolmogorov-Smirnov (KS) test with a confdence level 95% is applied to identify the best ft       International Journal of Antennas and Propagation Table 4 illustrates the obtained GoF values of various distributions.It can be seen that the Nakagami-m distribution has the smallest GoF value in the LoS, OLoS, and NLoS cases.Results showed that the Nakagami-m distribution is suitable for describing channels' randomness in the tunnel environment.Tis fnding is consistent with the results in [72,73].

Large-Scale Fading Model and Connectivity
In the VANET, the vehicles transmit information through multihop communication, therefore the connectivity of the VANET is very important in determining VANET communication capacity [11].Moreover, the connectivity of the vehicular network is correlated with positioning accuracy of the vehicle and Cramer-Rao lower bound of localization in VANETs [74], improvement of connectivity probability is helpful to improve vehicles' positioning accuracy.Terefore, VANET connectivity is a meaningful topic, and it is widely studied.In this section, we frstly discuss modelling approach of V2V connectivity based on log-distance path loss model.Secondly, based on simulations, we present the infuence of large-scale fading on V2V connectivity.

Single-Link Connectivity Probability.
In VANETs, each vehicle can be seen as a transmission node.Te distance between two vehicles denotes d.We defne that these two vehicles are connected when the PL is less than the PL threshold value P t , saying the received power is still above the sensitivity level of a system so that the signal can be detected and retrieved.Given the log-distance PL model in (1), the V2V connectivity probability P c (d) of any two vehicles at a distance d is defned as follows: where n denotes the path loss exponent, ε denotes a constant reference value, d stands for the distance between the receiver and transmitter, σ denotes the standard derivation of shadowing fading, and P t is the path loss threshold value.When the P L is larger than path loss threshold P t , the received signal is too weak to be detected.As a consequence, the radio connection is lost between these two vehicles.erfc(•) is a complementary error function, which is written as follows: Due to the monotonicity of the function erfc(•), it indicates that the PL exponent has a negative efect on connectivity performance.Te PL threshold and shadow fading have a positive impact on V2V connectivity performance.

Infuence of Large-Scale Fading on Connectivity.
Based on the measurement results, we discuss the infuence of large-scale fading parameters on vehicle connectivity performance.In this subsection, we set simulation parameters based on the estimated parameters of the PL model in the NLoS scenario.We set ε � 50.595, the shadow fading standard deviation σ � 2.385, path loss exponent n � 1.785, PL threshold value P t ranges from 60 dB to 100 dB, and transmission distance ranges from 1 m to 1000 m. Figure 8 presents the impacts of the path loss threshold value and transmission distance on connectivity probability.It indicates that as the distance between RX and TX gradually increases, the V2V connectivity probability gradually decreases.If the PL threshold value increases, the V2V connectivity probability increases.
Figure 9 depicts the infuence of the PL exponent n on V2V connectivity, in which the simulation parameters are given as follows: ε � 50.595, the standard deviation of shadow fading 2.385, intervehicle distance d � 200 m, and the PL exponent n ranges from 1 to 2.5.Te setting of the path loss threshold value is related to many factors, such as transmit power, the system bandwidth, and system requirements.In this paper, we set the path loss threshold value P t � 90 dB.It can been that connectivity probability decreases when n increases.
Figure 10 illustrates the infuence of shadowing parameter σ on V2V connectivity, in which the path loss exponent n � 1.785 and the shadowing parameter σ ranges from 0.1 dB to 10 dB.It shows that the larger the shadow fading standard deviation σ is, the higher the V2V connectivity probability becomes.Terefore, it is concluded that shadow fading standard deviation σ can improve connectivity performance.Tis conclusion is in agreement with the earlier fnding, in which the greater shadow fading standard deviation σ leads to higher connectivity performance of VANETs [20,24].

Small-Scale Fading Model and Connectivity
Based on the measured data, results show that the channel in tunnel experiences the Nakagami-m distribution.However, the analysis of V2V connectivity between any two vehicles under the Nakagami channel in tunnels has not yet been studied.In this section, frstly, we derive a closed-form connectivity probability between any two vehicles under the Nakagami-m channel for a one-dimensional VANET, which has not been given in the literature.Secondly, based on the proposed connectivity model, we discuss the infuences of Nakagami-m fading parameters on V2V connectivity.

Trafc Flow Model.
We take into account the 1D VANET, the vehicles are in a free-fow state in the network, so the vehicle movements are independent of each other.For the free-fow trafc model, the number of vehicles passing through the observation point obeys the Poisson distribution.Te arrival time of adjacent vehicles obeys the exponential distribution.Te distance between two consecutive 8 International Journal of Antennas and Propagation vehicles obeys an exponential distribution, we defne it as the adjacent vehicle distance.In the VANET, because two vehicles may be obscured by other vehicles, the distance of any two vehicles has a diferent meaning than the distance of two adjacent vehicles, and we need to distinguish between these two cases.So, the distance between any vehicles can be defned as intervehicle distance.In this paper, X j is a random variable representing the distance between the j-th car and the (j + 1)-th car.Te probability density function (PDF) of X j is as follows: where ρ denotes average vehicle density.Under the free-fow state, it is assumed that the vehicle speed is a discrete-time stochastic process, which obeys the Gaussian distribution.We focus on the tunnel environment where speed limitation is usually applied due to safety reasons.Terefore, in this paper, we assumed that the vehicle speed obeys a truncated Gaussian PDF, which can be calculated by [12]: where v min denotes the lower bound of velocity and v max stands for the upper bound of velocity.In equation (5), f(v) can be defned as follows: where σ v stands for the standard deviation of velocity, μ v represents the average velocity, v max � μ v + 3σ v , and v min � μ v − 3σ v .Ten, equation ( 5) can be written as follows: where v min ≤ v ≤ v max , erf(•) represents the error function, and v denotes the vehicle speed.We defne the distance between one vehicle and its q-th neighbor as the intervehicle distance Z q .Generally, by calculating the sum of q adjacent vehicle distances X i , we could obtain the intervehicle distance Z q .As a result, the intervehicle distance can be calculated as Z q ≜  q i�1 X i .Te PDF of Z q is given by [23]:   International Journal of Antennas and Propagation where ρ stands for the average vehicle density, q represents the neighbor order, and (•)! stands for the factorial operator.

Connectivity in Nakagami-m Fading Channels.
According to the measurement results in Section 3, it is found that the Nakagami-m distribution can describe the fast fading well in the tunnel.In addition, it has been proven that compared to other considered fading models (i.e., Rayleigh, Weibull, and Rice), the Nakagami-m distribution is the most suitable model to represent the V2V channel randomness in both NLoS and LoS VANET environments [71].Terefore, the analysis of the connectivity performance under Nakagami-m fading is meaningful for the design of Internet of vehicles system in the tunnel.In the following, the mathematical representation of V2V connectivity probability under the Nakagami-m channel is presented.
Te probability of two nodes correctly send and receive signals can be represented as a function of the received SNR.When the distance between the receiver and transmitter is d, the SNR at the receiver is calculated by [23] where c represents SNR, n denotes the PL exponent, P noise denotes the total additive noise power, and P T represents the transmit power; It is assumed that E W 2   � 1, the received average SNR � c with distance d can be written as [21]: Given the condition that the received signal magnitude follows the Nakagami distribution.Te PDF of the received signal magnitude w is defned as [71]: where w ≥ 0 denotes the received signal magnitude and m is the shape factor, which determines the severity of channel fade.When m is equal to 1, the distribution yields to the Rayleigh distribution.When < 1, the distribution yields more severe fading.Ω is the scale parameter and Γ(•) represents the Gamma function.Furthermore, the PDF of the received SNR considering the Nakagami fading model can be written as [25]: where m denotes the shape factor, c represents the received SNR, and � c denotes the average SNR.
When the received SNR c is more than the given SNR threshold Ψ, the vehicles are connected, and the transmitted message can be decoded correctly.Terefore, the probability of accurately received transmitted information at a distance d is as follows: where Γ(m, mΨ/� c) represents the incomplete Gamma function: If m is a positive integer, Γ(m) � (m − 1)!.Γ(m, a) is calculated by [75]: Together with equation ( 15), equation ( 13) can be further modifed and represented as follows: where β is a constant value, P T denotes the transmit power, m stands for the shape factor of the Nakagami distribution, n denotes the PL exponent, Ψ stands for the SNR threshold value, and P noise represents the noise power.
In [22], authors considered the vehicle connectivity performance of two consecutive cars, however, the V2V connectivity between any two vehicles is not analyzed.Inspired by [22], we extend the two adjacent vehicles to any two vehicles to analyze the vehicle connectivity in the 1D VANET.In this paper, we consider that a wireless radio link is established if SNR c is larger than the given SNR threshold value Ψ. Te probability that the vehicle and its q-th neighboring vehicle are connected is given by 10 International Journal of Antennas and Propagation After calculation, equation (17) becomes where Tere is an integral term in equation ( 18), and the integral term is complicated to calculate.In this paper, we consider the derivation in [22] to calculate the integral in equation (18).Te integral is given by where G c,l r,j is the Meijers G function, t and u are greater than 0, s is also bigger than 0. It should be noted that equation ( 20) is valid when b is a positive integer value.Taking equation (20) into equation ( 18), we obtain with where β is a constant value, n denotes the PL exponent, and P T stands for the transmit power.ρ denotes average vehicle density, m is the shape factor of the Nakagami distribution, Ψ denotes the SNR threshold value, and P noise represents the noise power.Terefore, for integer values of n and m, there exists the closed-form solution to the connectivity probability given by equation (21).However, for noninteger values of n and m, it is very hard to acquire a closed-loop solution to equation (17).Equation ( 21) is also the innovation point of this paper.Te closed-loop expression of the V2V connectivity probability between any two vehicles under Nakagami-m fading is not given by others.
A special case would be with two consecutive vehicles, where the consecutive vehicle distance obeys an exponential distribution; its PDF is f X i (x).As a result, the probability of two consecutive vehicles connected is calculated as [22] where n denotes the PL exponent, P T denotes the transmit power, P noise is the noise power, ρ denotes average vehicle density, and Ψ denotes the SNR threshold value.β is a constant value and For integer values of n, taking equation ( 20) into equation ( 23) results in [22] Terefore, in this special case the closed-form solution to the V2V connectivity probability of two consecutive vehicles in equation ( 23) is written as equation (24).

Impact of Nakagami-m Fading on V2V Connectivity.
VANET connectivity is a function of vehicle mobility, vehicle density, SNR threshold, transmit power, and channel fading parameters.Tese parameters are directly correlated with the connectivity.In the following, we will discuss the infuences of these parameters on V2V connectivity.
In this section, we set β � 2.1 * 10 − 5 , average vehicle density ρ � 0.02 vehicles/m, the transmit power P T � 37 dBm, PL exponent n � 1.785, and the shape factor of the Nakagami distribution m ranges from 1 to 10. Figure 11 shows the efect of Nakagami fading parameter m on the proposed V2V connectivity probability.Simulation results showed that as m increases, the V2V connectivity probability increases.Terefore, the Nakagami fading parameter m has a positive efect on V2V connectivity performance.
Figure 12 shows the infuence of the SNR threshold Ψ on connectivity probability.Results showed that the SNR threshold has a signifcant impact on V2V connectivity.A larger SNR threshold Ψ results in a smaller V2V connectivity probability.Tis fnding is consistent with the earlier conclusion in [21].
Furthermore, the impact of average vehicle density on the V2V connectivity under Nakagami fading is illustrated in Figure 13, where ρ ranges from 0.001 to 0.1 vehicles/m.Te result shows that the variations in vehicle density impact the communications' connectivity of the VANET.In free trafc state, the V2V connectivity probability increases with increasing vehicle density.
International Journal of Antennas and Propagation Figure 14 illustrates the results of relation between the connectivity probability and the transmit power P T , where we can conclude that increasing the transmit power enhances the connectivity probability.Figure 15 shows the dependency between single-link connectivity probability and the neighbors' order q in the Nakagami fading channel.
Te result shows that the connectivity is signifcantly infuenced by the neighbors' order.Te farther neighbors' results in lower connectivity probability.

Conclusion
In this paper, we studied the V2V connectivity models considering the log-distance PL model and Nakagami-m fading model based on measurement results.First, we analyzed the channel characteristics in the tunnel, and found that compared to Rice, Rayleigh, Weibull, normal, and lognormal distributions, the Nakagami-m distribution is the best ft distribution for the signal amplitude fading in the tunnel scenario.Based on this fnding, for the integer path loss exponent and fading factor, we derived a closed-form solution for the V2V connectivity probability of any two vehicles under the Nakagami channel.In addition, we studied the infuences of large-scale and small-scale fading parameters on V2V connectivity performance.A bigger shape factor of Nakagami fading will result in a larger connectivity probability.Te neighbor order also afect the vehicle connectivity in the VANET.Te higher the neighbor order, the smaller the connectivity probability.If the PL exponent increases, the V2V connectivity probability    decreases.Te shadow fading has a positive impact on connectivity probability.When the path loss threshold increases, the V2V connectivity probability increases, whereas if the SNR threshold increases, the connectivity probability decreases in Nakagami fading.Furthermore, we analyzed the infuences of transmit power, transmission distance, and trafc parameters on V2V connectivity.Results showed that the higher the transmit power, the greater the connectivity probability.A larger transmission distance will bring smaller V2V connectivity probability.In free trafc fow, the V2V connectivity probability increases with increasing vehicle density.
Communication range, vehicle speed, the length of vehicle, the safe distance, and trafc lights Urban road with intersections No [5, 18, 47] Te vehicle speed follows Gaussian distribution and the arrival time of vehicles obeys an exponential distribution Communication range, vehicle speed, density and arrival rate, the ratio of active cars, the vehicle safety distance, user behavior, the ratio of cars transmit signals, and the distance between two adjacent RSUs A two-way lane and an one-wayone-lane highway No [6] A unit disk graph radio connectivity model Vehicle movements, transmission range, and the number of neighbors Highway Yes [7, 38-40, 48] Te arrival of vehicles obeys the Poisson distribution Vehicle movements, transmission range, velocity, the probability of a car passing through the entrance or exit, and vehicle arrival rate Highway No [8] Te arrival of vehicles obeys a Poisson distribution and the adjacent vehicle arrival time and the adjacent vehicle distance follow the exponential distribution; generalized speed factor system model Te number of road lanes, vehicle density and speed, transmission range, the vehicular mobility, and the vehicle arrival rate Highway No [11] A simple unit disk model Te vehicular speed, communication range, and the component speed and size Urban Yes [14, 15, 49] A Poisson distribution, a UD model; log-normal shadowing model Te coverage range of BSs and vehicles, vehicle density, inter-BS distance, multihop communication, average length of vehicles, minimum safety headway, the distribution of vehicles, and one-hop transmission range A road No [16] A multipriority Markov model Te coverage of RSUs and vehicles, trafc density, the platoon ratio of VANET, and the distance of adjacent RSUs One-way road and two-way road No [17] Te arrival of vehicles obeys a homogeneous Poisson process; Nakagami distribution User behavior, active vehicle ratio, trafc fow, and the channel fading factors Highway No [50] Te arrival of vehicles follows homogenous Poisson process; Nakagami-m distribution Nakagami fading parameter, transmission power, PL exponent, and signal-to-noise ratio (SNR) threshold value A road No [51] In free-fowing trafc and congestion conditions, the headway distance follows the exponential distribution and the Gaussian unitary ensemble distribution, respectively; Nakagami-m model Headway distance and vehicle density Highway Yes [52, 53] Path loss model Adverse weather conditions, trafc density, communication range, and path loss threshold value A two-lane road; intersection Yes [19-25] A queuing theoretic model, path loss model, Nakagami, Rayleigh, Weibull, and rice fading model PL exponent, shadow fading, fading factors, vehicle density, vehicle speed, vehicle arrival rate, transmission range, and received SNR threshold Highway No [1, 37] Trafc fow model, path loss model, and small-scale fading model Diferent trafc fows and efective communication coverage Highway Yes [27, 28] Te dual-slope path loss and trafc fow model Path loss exponent, trafc fow, and intervehicle distance Urban; highway No [30, 31] An equivalent trafc model based on queuing theory Transmission range, trafc parameters, and transmission range Unidirectional lane road; highway No 4 International Journal of Antennas and Propagation Two mobility models Te mobility pattern, transmission range, bus routes, trafc lights, and background trafc Streets No [36, 54] Te Poisson trafc model, the UD model, and log-normal shadowing model Distance between adjacent BSs, radio coverage range of vehicle, trafc density, the number of roadside infrastructure, data sinks, the maximum number of hops in a propagation path, and intervehicle distance Te road between two adjacent BSs No [41] Te cluster-based analytical model, the UD model, and log-normal shadowing model Propagation distance, vehicle density, the distribution of vehicle, one-hop transmission range, and the minimum safety distance of two adjacent cars Highway No [55] Dynamic clustering model, the adjacent vehicle distances follow the exponential distribution, and the distribution of vehicles obeys the homogenous Poisson distribution Vehicle driving, vehicle speed, and the trafc environment features Urban No [56] Te arrival time of vehicles obeys exponential distribution and the vehicle speed follows the Gaussian distribution Vehicle density, vehicles communication range, and the minimum safety distance A two-lane road No [57] Te exponential distribution model and generalized extreme value (GEV) distribution Intervehicle spacing distribution and vehicle density Highway No [42] Te cell transmission model and Rayleigh distribution Te vehicle fow, the message size, the communication radius, the path loss exponent, and the data rate Te transmission range, vehicle density, and street width Intersection No [60] BA-realtime, BA-realtime + TL, and BA-fullroad Vehicle mobility model, vehicle density, vehicle speed, transmission range, and the position and number of RSUs A road No [61] Microscopic mobility and lane changing decision model Network metric data delivery rate, vehicle density, velocity and arrival rate, deceleration or acceleration, and the safety gap Highway No [62] Microscopic models Communication range, vehicular density, the number of highway lanes, and the speed and specifc daytime Te bond percolation model and Bollobas model Vehicle density and transmission range -No International Journal of Antennas and Propagation vehicle carrying the receiver is called the receiving vehicle.

Figure 7 :
Figure 7: Te path loss model for NLoS scenario in the tunnel.

Figure 8 :Figure 9 :
Figure 8: Te impacts of path loss threshold value and transmission distance on connectivity probability in NLoS scenarios.

Figure 10 :
Figure 10: Te impact of σ on connectivity probability.
denotes a constant value, where c represents the speed of light; G R and G T stand for the RX and TX gains, respectively.During the measurement, omnidirectional antennas were employed, therefore, we set G R � 1 and G T � 1. f o presents the carrier frequency.In equation(9), the noise power P noise is defned as P noise � K b TB, where T stands for temperature in K, B denotes the transmission bandwidth, and K b � 1.38 * 10 (− 23) J/K denotes the Boltzmann constant.

Figure 14 :Figure 15 :
Figure 14: Te infuence of P T on connectivity probability.
h |F H (h) − F(h)|, where sup h stands for the supremum operator and F(h) and F H (h) denote the theoretical and empirical cumulative distribution functions (CDFs) of the measurement data h, respectively.Te lower GoF value indicates a better ft for the model.

Table 3 :
Parameters for the PL model.