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

Authors: Suying Jiang; Wei Wang; Ibrahim Rashdan; Paul Unterhuber; Peng Peng
Journal: International Journal of Antennas and Propagation
 (2023)

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## Abstract

Vehicular ad hoc network (VANET) brings an excellent solution to ensure road safety and transportation efficiency in critical environment like tunnel. Particularly, radio link connectivity of vehicle-to-vehicle (V2V) significantly influences the performance of VANETs. The communication range of the radio systems is a random variable in reality due to the channel fading effect. Therefore, the connectivity model between vehicles in realistic environment is a key for accurate evaluation of system performances. In this paper, we study the V2V connectivity performance in the presence of channel randomness for tunnel environment. Firstly, based on channel measurement campaign, empirical path loss (PL) and small-scale fading channel models are established. Secondly, we study the influence of large-scale fading parameters on V2V connectivity. Thirdly, based on real small-scale fading characteristics, we derive the V2V connectivity probability between any two vehicles under Nakagami fading channel for one-dimensional VANET, and give the closed-form of V2V connectivity probability. Finally, we study the influences of various parameters (i.e., Nakagami fading factor, vehicle density, and neighbor order) on V2V connectivity performance. Results show that with the Nakagami fading shape factor increases, the connectivity probability increases. The shadowing fading can improve connectivity in the VANET; the path loss exponent, transmission distance, and signal-to-noise ratio (SNR) threshold have a negative impact on connectivity probability. The transmit power, vehicle density, and path loss threshold value have a positive impact on connectivity.


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## Body

## 1. 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 efficient and safe intelligent transportation system [1–3]. Particularly, the radio link connectivities of vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) is a key metric to assess the VANET communication performances in terms of both communication range and vehicle networking that strongly affect advanced techniques like cooperative communication and positioning. Therefore, 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 specific 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 influence of traffic and weather factors on connectivity (e.g., the traffic flow, vehicle density, the distribution of vehicle and its speed, weather conditions, and traffic lights); (2) the effect of wireless propagation environments on connectivity (e.g., path loss (PL) model, Nakagami fading, Weibull fading, Rice fading, Rayleigh fading, and communication range); (3) different 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 VANET connectivity. In [4, 5], the influences of communication range, vehicle speed, the safe distance, size of vehicle, traffic lights, and overtaking of vehicles on connectivity were analyzed. The dependency between the connectivity of the VANET and mobility was studied in [6–13]. In [11], authors analyzed the effects 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 affected if the vehicle speed is below the threshold. In [14–16], 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 different communication channel models, specifically, unit disk (UD) model and log-normal shadowing model. When studying the factors that affect the connectivity of the VANET, the impact of user behavior on connectivity cannot be ignored. The effect of user behavior on V2V and V2I connectivity was analyzed in [17, 18].Besides, wireless propagation environments have shown to have significant impact on the connectivity. In [17, 19–26], theoretical fading models (e.g., Rayleigh, Rice, Weibull, and Nakagami) were considered, based on which the authors presented analytical models. Moreover, the impacts of different 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 effect of lognormal shadowing on connectivity probability was analyzed. It is shown that lognormal shadowing can improve the VANET connectivity.The previous work on the connectivity performance analysis of the VANET mainly based on queuing theory [30–34] 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 traffic flow 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 simplified 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. The proposed model includes two parts, namely, catch-up process and forwarding process. In [42], the cell transmission model was adopted to capture macroscopic traffic flow 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 influences of various parameters on connectivity probability [21, 23]. Then, we aim to discuss the research on the influence 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, specifically, 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. These 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 different, so the connectivity between two vehicles will also be different. The 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 small-scale fading in OLoS and non-line-of-sight (NLoS) cases.Table1 summaries various connectivity models and corresponding parameters considered in the literature. The research studies on vehicle connectivity mainly focus on urban [55, 65], highway [47, 51, 66], and intersection scenarios [53, 58, 59]. There are very few studies on vehicle connectivity for the tunnel scenario that is a critical use case in the traffic 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 and vehicle density were considered without incorporating the channel fading effect. 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.Table 1
Connectivity models.
ReferenceSystem modelParametersScenariosMeasurement[4]Car following modelCommunication range, vehicle speed, the length of vehicle, the safe distance, and traffic lightsUrban road with intersectionsNo[5, 18, 47]The vehicle speed follows Gaussian distribution and the arrival time of vehicles obeys an exponential distributionCommunication 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 RSUsA two-way lane and an one-wayone-lane highwayNo[6]A unit disk graph radio connectivity modelVehicle movements, transmission range, and the number of neighborsHighwayYes[7, 38–40, 48]The arrival of vehicles obeys the Poisson distributionVehicle movements, transmission range, velocity, the probability of a car passing through the entrance or exit, and vehicle arrival rateHighwayNo[8]The 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 modelThe number of road lanes, vehicle density and speed, transmission range, the vehicular mobility, and the vehicle arrival rateHighwayNo[11]A simple unit disk modelThe vehicular speed, communication range, and the component speed and sizeUrbanYes[14, 15, 49]A Poisson distribution, a UD model; log-normal shadowing modelThe 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 rangeA roadNo[16]A multipriority Markov modelThe coverage of RSUs and vehicles, traffic density, the platoon ratio of VANET, and the distance of adjacent RSUsOne-way road and two-way roadNo[17]The arrival of vehicles obeys a homogeneous Poisson process; Nakagami distributionUser behavior, active vehicle ratio, traffic flow, and the channel fading factorsHighwayNo[50]The arrival of vehicles follows homogenous Poisson process; Nakagami-m distributionNakagami fading parameter, transmission power, PL exponent, and signal-to-noise ratio (SNR) threshold valueA roadNo[51]In free-flowing traffic and congestion conditions, the headway distance follows the exponential distribution and the Gaussian unitary ensemble distribution, respectively; Nakagami-m modelHeadway distance and vehicle densityHighwayYes[52, 53]Path loss modelAdverse weather conditions, traffic density, communication range, and path loss threshold valueA two-lane road; intersectionYes[19–25]A queuing theoretic model, path loss model, Nakagami, Rayleigh, Weibull, and rice fading modelPL exponent, shadow fading, fading factors, vehicle density, vehicle speed, vehicle arrival rate, transmission range, and received SNR thresholdHighwayNo[1, 37]Traffic flow model, path loss model, and small-scale fading modelDifferent traffic flows and effective communication coverageHighwayYes[27, 28]The dual-slope path loss and traffic flow modelPath loss exponent, traffic flow, and intervehicle distanceUrban; highwayNo[30, 31]An equivalent traffic model based on queuing theoryTransmission range, traffic parameters, and transmission rangeUnidirectional lane road; highwayNo[10, 32]Two mobility modelsThe mobility pattern, transmission range, bus routes, traffic lights, and background trafficStreetsNo[36, 54]The Poisson traffic model, the UD model, and log-normal shadowing modelDistance between adjacent BSs, radio coverage range of vehicle, traffic density, the number of roadside infrastructure, data sinks, the maximum number of hops in a propagation path, and intervehicle distanceThe road between two adjacent BSsNo[41]The cluster-based analytical model, the UD model, and log-normal shadowing modelPropagation distance, vehicle density, the distribution of vehicle, one-hop transmission range, and the minimum safety distance of two adjacent carsHighwayNo[55]Dynamic clustering model, the adjacent vehicle distances follow the exponential distribution, and the distribution of vehicles obeys the homogenous Poisson distributionVehicle driving, vehicle speed, and the traffic environment featuresUrbanNo[56]The arrival time of vehicles obeys exponential distribution and the vehicle speed follows the Gaussian distributionVehicle density, vehicles communication range, and the minimum safety distanceA two-lane roadNo[57]The exponential distribution model and generalized extreme value (GEV) distributionIntervehicle spacing distribution and vehicle densityHighwayNo[42]The cell transmission model and Rayleigh distributionThe vehicle flow, the message size, the communication radius, the path loss exponent, and the data rateFreewayNo[58]A new cellular automata-based mobility modelTraffic densityUrban intersectionNo[59]The triangular modelThe transmission range, vehicle density, and street widthIntersectionNo[60]BA-realtime, BA-realtime + TL, and BA-fullroadVehicle mobility model, vehicle density, vehicle speed, transmission range, and the position and number of RSUsA roadNo[61]Microscopic mobility and lane changing decision modelNetwork metric data delivery rate, vehicle density, velocity and arrival rate, deceleration or acceleration, and the safety gapHighwayNo[62]Microscopic modelsCommunication range, vehicular density, the number of highway lanes, and the speed and specific daytimeHighwayNo[63]Graph metricsVehicle rerouting capacity—No[64]The bond percolation model and Bollobas modelVehicle density and transmission range—NoIn this work, we focus on filling 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 fitted fading model for the tunnel environment. Thereafter, 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 influences of Nakagami-m fading, large-scale fading, and traffic parameters on V2V connectivity.The 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. The 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 effects of the Nakagami fading factor, the neighbor order, and the threshold value of received SNR on connectivity performance.(iii)
We study the influences of large-scale fading parameters on VANET connectivity. We demonstrate that the shadowing fading has a positive impact on V2V connectivity.The rest of this paper is organized as follows: in Section2, a channel measurement campaign in the tunnel is described. Thereafter, the PL and small-scale fading characteristics are studied in Section 3. The V2V connectivity probability considering the PL model is presented, and the influences 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 influences of various parameters on V2V connectivity probability in this section. Finally, some concluding remarks are described in Section 6.


## 2. Channel Measurement Campaign

To study the channel propagation characteristics in tunnels, we conducted the channel measurement campaign in a tunnel, as shown in Figure1. 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). The orthogonal frequency division multiplexing (OFDM) signal was emitted by the transmitter at a center frequency of 5.2 GHz. The bandwidth of the signal is 120 MHz. The receiver received the CFR Htk,fz, where tk=k·tj, tj is the measurement time grid, k denotes the time index of the measured data, k=1,2,3,⋯, and fz 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) htk,τz by taking the inverse Fourier transform of CFR Htk,fz, where τz=z·τ∆=z/B, B is the bandwidth, τ∆ denotes the delay resolution, and τ∆=1/B. Table 2 summarizes the channel measurement parameters.Figure 1
Measurement environment.Table 2
Channel measurement parameters.
ParameterValueCenter frequency5.2 GHzTransmit powerMax. 37 dBmBandwidthB120 MHzMeasurement typeSISOSignal period12.8μsNumber of carrier1537Measurement time grid1.024 msTransmit antenna height2 mReceive antenna height1.88 mTransmit and receive antennaOmnidirectionalPosition measurementGNSS, IMU, and LIDARMeasurement scenariosTunnelDuring 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. The 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. Therefore, ground-based augmentation approach is considered. Both transmitter and receiver were equipped with GNSS receivers, together with the inertial measurement unit (IMU) and LIDAR. Therefore, it is possible to combine all different 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 traffic flow 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. The vehicle carrying the transmitter is called the transmitting vehicle and the vehicle carrying the receiver is called the receiving vehicle. The 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. Through channel measurement, we can obtain CFR data and the time information of the CFR signal. The 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.Figure2 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. The 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.Figure 2
LoS scenario.Figure 3
OLoS scenario.Figure 4
NLoS scenario.


## 3. Measurement-Based Results

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


### 3.1. Path Loss and Shadow Fading

The log-distance PL model in [67] is a widely used propagation model to describe the power loss during transmission; it is defined as follows:(1)PLd=ε+10nlog10d+Xσ,where d stands for the distance between RX and TX, PLd 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. The measurement PL and theoretical PL model for different scenarios are depicted in Figures 5–7.Figure 5
The path loss model for LoS scenario in the tunnel.Figure 6
The path loss model for OLoS scenario in the tunnel.Figure 7
The path loss model for NLoS scenario in the tunnel.Generally, shadow fading in the dB scale usually follows a zero-mean Gaussian distribution [68, 69]. The shadow fading values are extracted from measurement data, and fitted with a Gaussian distribution. The extracted PL model parameters for different 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 effect in the tunnel, the obtained PL exponent n in the tunnel is less than the value in free space. This conclusion is consistent with theory in [70].Table 3
Parameters for the PL model.
ScenariosnεσLoS1.19653.7191.458OLoS1.26552.962.23NLoS1.78550.5952.385


### 3.2. Small-Scale Amplitude Fading Modelling

To mitigate the influence 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 different distributions, which are widely utilized in V2V scenario, such as Rayleigh distribution, Rice distribution, normal distribution, log-normal distribution, Nakagami-m distribution, and Weibull distribution. The Kolmogorov–Smirnov (KS) test with a confidence level95% is applied to identify the best fit distribution function [71]. The goodness of fit (GoF) value of the KS test is usually used to evaluate the best fit of a distribution. The GoF value can be calculated by ξ=suphFHh−Fh, where suph stands for the supremum operator and Fh and FHh denote the theoretical and empirical cumulative distribution functions (CDFs) of the measurement data h, respectively. The lower GoF value indicates a better fit for the model.Table4 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. This finding is consistent with the results in [72, 73].Table 4
The GoF for different distributions.
ScenarioNormalLog-normalRayleighRiceWeibullNakagami-mLoS0.02280.03510.38080.02470.06250.0184OLoS0.06760.07290.08140.06100.05480.0489NLoS0.10840.10440.08330.09020.08890.0829


## 3.1. Path Loss and Shadow Fading

The log-distance PL model in [67] is a widely used propagation model to describe the power loss during transmission; it is defined as follows:(1)PLd=ε+10nlog10d+Xσ,where d stands for the distance between RX and TX, PLd 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. The measurement PL and theoretical PL model for different scenarios are depicted in Figures 5–7.Figure 5
The path loss model for LoS scenario in the tunnel.Figure 6
The path loss model for OLoS scenario in the tunnel.Figure 7
The path loss model for NLoS scenario in the tunnel.Generally, shadow fading in the dB scale usually follows a zero-mean Gaussian distribution [68, 69]. The shadow fading values are extracted from measurement data, and fitted with a Gaussian distribution. The extracted PL model parameters for different 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 effect in the tunnel, the obtained PL exponent n in the tunnel is less than the value in free space. This conclusion is consistent with theory in [70].Table 3
Parameters for the PL model.
ScenariosnεσLoS1.19653.7191.458OLoS1.26552.962.23NLoS1.78550.5952.385


## 3.2. Small-Scale Amplitude Fading Modelling

To mitigate the influence 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 different distributions, which are widely utilized in V2V scenario, such as Rayleigh distribution, Rice distribution, normal distribution, log-normal distribution, Nakagami-m distribution, and Weibull distribution. The Kolmogorov–Smirnov (KS) test with a confidence level95% is applied to identify the best fit distribution function [71]. The goodness of fit (GoF) value of the KS test is usually used to evaluate the best fit of a distribution. The GoF value can be calculated by ξ=suphFHh−Fh, where suph stands for the supremum operator and Fh and FHh denote the theoretical and empirical cumulative distribution functions (CDFs) of the measurement data h, respectively. The lower GoF value indicates a better fit for the model.Table4 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. This finding is consistent with the results in [72, 73].Table 4
The GoF for different distributions.
ScenarioNormalLog-normalRayleighRiceWeibullNakagami-mLoS0.02280.03510.38080.02470.06250.0184OLoS0.06760.07290.08140.06100.05480.0489NLoS0.10840.10440.08330.09020.08890.0829


## 4. 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. Therefore, VANET connectivity is a meaningful topic, and it is widely studied. In this section, we firstly discuss modelling approach of V2V connectivity based on log-distance path loss model. Secondly, based on simulations, we present the influence of large-scale fading on V2V connectivity.


### 4.1. Single-Link Connectivity Probability

In VANETs, each vehicle can be seen as a transmission node. The distance between two vehicles denotesd. We define that these two vehicles are connected when the PL is less than the PL threshold value Pt, 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 Pcd of any two vehicles at a distance d is defined as follows:(2)Pcd=PPLd<Pt=Pε+10nlog10d+Xσ<Pt=0.5erfcε−Pt+10nlog10d2σ,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 Pt is the path loss threshold value. When the PL is larger than path loss threshold Pt, 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:(3)erfct=2π∫t∞e−x2dx.Due to the monotonicity of the functionerfc·, it indicates that the PL exponent has a negative effect on connectivity performance. The PL threshold and shadow fading have a positive impact on V2V connectivity performance.


### 4.2. Influence of Large-Scale Fading on Connectivity

Based on the measurement results, we discuss the influence 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 setε=50.595, the shadow fading standard deviation σ=2.385, path loss exponent n=1.785, PL threshold value Pt 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 8
The impacts of path loss threshold value and transmission distance on connectivity probability in NLoS scenarios.Figure9 depicts the influence 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. The 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 Pt=90 dB. It can been that connectivity probability decreases when n increases.Figure 9
The effect of PL exponentn on V2V connectivity.Figure10 illustrates the influence 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. Therefore, it is concluded that shadow fading standard deviation σ can improve connectivity performance. This conclusion is in agreement with the earlier finding, in which the greater shadow fading standard deviation σ leads to higher connectivity performance of VANETs [20, 24].Figure 10
The impact ofσ on connectivity probability.


## 4.1. Single-Link Connectivity Probability

In VANETs, each vehicle can be seen as a transmission node. The distance between two vehicles denotesd. We define that these two vehicles are connected when the PL is less than the PL threshold value Pt, 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 Pcd of any two vehicles at a distance d is defined as follows:(2)Pcd=PPLd<Pt=Pε+10nlog10d+Xσ<Pt=0.5erfcε−Pt+10nlog10d2σ,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 Pt is the path loss threshold value. When the PL is larger than path loss threshold Pt, 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:(3)erfct=2π∫t∞e−x2dx.Due to the monotonicity of the functionerfc·, it indicates that the PL exponent has a negative effect on connectivity performance. The PL threshold and shadow fading have a positive impact on V2V connectivity performance.


## 4.2. Influence of Large-Scale Fading on Connectivity

Based on the measurement results, we discuss the influence 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 setε=50.595, the shadow fading standard deviation σ=2.385, path loss exponent n=1.785, PL threshold value Pt 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 8
The impacts of path loss threshold value and transmission distance on connectivity probability in NLoS scenarios.Figure9 depicts the influence 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. The 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 Pt=90 dB. It can been that connectivity probability decreases when n increases.Figure 9
The effect of PL exponentn on V2V connectivity.Figure10 illustrates the influence 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. Therefore, it is concluded that shadow fading standard deviation σ can improve connectivity performance. This conclusion is in agreement with the earlier finding, in which the greater shadow fading standard deviation σ leads to higher connectivity performance of VANETs [20, 24].Figure 10
The impact ofσ on connectivity probability.


## 5. 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, firstly, 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 influences of Nakagami-m fading parameters on V2V connectivity.


### 5.1. Traffic Flow Model

We take into account the 1D VANET, the vehicles are in a free-flow state in the network, so the vehicle movements are independent of each other. For the free-flow traffic model, the number of vehicles passing through the observation point obeys the Poisson distribution. The arrival time of adjacent vehicles obeys the exponential distribution. The distance between two consecutive vehicles obeys an exponential distribution, we define 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 different 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 defined as intervehicle distance. In this paper,Xj is a random variable representing the distance between the j-th car and the j+1-th car. The probability density function (PDF) of Xj is as follows:(4)fXjx=ρe−ρx,x≥0,where ρ denotes average vehicle density.Under the free-flow 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. Therefore, in this paper, we assumed that the vehicle speed obeys a truncated Gaussian PDF, which can be calculated by [12]:(5)gv=fv∫vminvmaxfvdv,vmin≤v≤vmax,where vmin denotes the lower bound of velocity and vmax stands for the upper bound of velocity. In equation (5), fv can be defined as follows:(6)fv=1σv2πexp−v−μv22σv2,where σv stands for the standard deviation of velocity, μv represents the average velocity, vmax=μv+3σv, and vmin=μv−3σv. Then, equation (5) can be written as follows:(7)gv=2fverfvmax−μv/σv2−erfvmin−μv/σv2,where vmin≤v≤vmax, erf· represents the error function, and v denotes the vehicle speed.We define the distance between one vehicle and itsq-th neighbor as the intervehicle distance Zq. Generally, by calculating the sum of q adjacent vehicle distances Xi, we could obtain the intervehicle distance Zq. As a result, the intervehicle distance can be calculated as Zq≜∑i=1qXi. The PDF of Zq is given by [23]:(8)fZqx;q=ρqq−1!xq−1e−ρx,where ρ stands for the average vehicle density, q represents the neighbor order, and ·! stands for the factorial operator.


### 5.2. Connectivity in Nakagami-m Fading Channels

According to the measurement results in Section3, 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]. Therefore, 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.The 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 isd, the SNR at the receiver is calculated by [23](9)γd=W2βPTdnPnoise,where γ represents SNR, n denotes the PL exponent, Pnoise denotes the total additive noise power, and PT represents the transmit power; β=GTGRc2/4πfo2 denotes a constant value, where c represents the speed of light; GR and GT stand for the RX and TX gains, respectively. During the measurement, omnidirectional antennas were employed, therefore, we set GR=1 and GT=1. fo presents the carrier frequency. In equation (9), the noise power Pnoise is defined as Pnoise=KbTB, where T stands for temperature in K, B denotes the transmission bandwidth, and Kb=1.38∗10−23 J/K denotes the Boltzmann constant.It is assumed thatEW2=1, the received average SNR γ¯ with distance d can be written as [21]:(10)γ¯=βPTdnPnoise.Given the condition that the received signal magnitude follows the Nakagami distribution. The PDF of the received signal magnitudew is defined as [71]:(11)fww;m,Ω=2mmw2m−1ΩmΓmexp−mw2Ω,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]:(12)fγγ=mγ¯mγm−1Γmexp−mγγ¯,where m denotes the shape factor, γ represents the received SNR, and γ¯ denotes the average SNR.When the received SNRγ is more than the given SNR threshold Ψ, the vehicles are connected, and the transmitted message can be decoded correctly. Therefore, the probability of accurately received transmitted information at a distance d is as follows:(13)Pγd≥Ψ=∫Ψ∞fγγdγ=∫Ψ∞mγ¯mγm−1Γmexp−mγγ¯dγ=Γm,mΨ/γΓm,where Γm,mΨ/γ¯ represents the incomplete Gamma function:(14)Γm,mΨ/γ¯=∫Ψ∞mγ¯mγm−1exp−mγγ¯dγ.Ifm is a positive integer, Γm=m−1!. Γm,a is calculated by [75]:(15)Γm,a=m−1!×e−a∑k=0m−1akk!.Together with equation (15), equation (13) can be further modified and represented as follows:(16)Pγd≥Ψ=e−mΨdnPnoise/βPT∑k=0m−1mΨdnPnoise/βPTkk!,where β is a constant value, PT 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 Pnoise 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 γ is larger than the given SNR threshold value Ψ. The probability that the vehicle and its q-th neighboring vehicle are connected is given by(17)Psi=Pγx≥ΨZq=x=∫Ψ∞∫0∞fγγ·fZqx;qdxdγ=∫Ψ∞∫0∞mγ¯mγm−1Γme−mγγ¯+ρxxq−1dxdγ·ρqq−1!.After calculation, equation (17) becomes(18)Psi=∫Ψ∞∫0∞fγγfZqx;qdxdγ=ρqq−1!∑k=0m−1mΨPnoise/βPTkk!·W1,where(19)W1=∫0∞e−mΨPnoisexn/βPT+ρxxnk+q−1dx.There 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). The integral is given by(20)∫0∞xt−1e−ux+sxbdx=2π1−b/2bt−1/2u−t×G1,bb,1ubsbb1tb,…,t+b−1b,where Gr,jc,l 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(21)PS=W2∑k=0m−1mΨPnoise/βPTkk!nnk+q−1/2ρ−nk+q×G1,nn,1ρnβPTmΨPnoisenn1nk+qn,…,nk+1+q−1n,with(22)W2=ρqq−1!2π1−n/2,where β is a constant value, n denotes the PL exponent, and PT 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 Pnoise represents the noise power.Therefore, for integer values ofn 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. The 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 isfXix. As a result, the probability of two consecutive vehicles connected is calculated as [22](23)Pc=Pγx≥ΨXi=x=∫Ψ∞∫0∞fγγfXixdxdγ=ρ∫0∞e−ρxΓm,mΨ/γ¯Γmdx=ρ∑k=0m−1mΨPnoise/βPTkk!∫0∞e−ρx+mΨxnPnoise/βPTxnkdx,where n denotes the PL exponent, PT denotes the transmit power, Pnoise is the noise power, ρ denotes average vehicle density, and Ψ denotes the SNR threshold value. β is a constant value and β=GTGRc2/4πfo2.For integer values ofn, taking equation (20) into equation (23) results in [22](24)Pc=ρ2π1−n/2∑k=0m−1mΨPnoise/βPTkk!nnk+1/2ρ−nk−1×G1,nn,1ρnβPTmΨPnoisenn1nk+1n,…,nk+1n.Therefore, 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).


### 5.3. 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. These parameters are directly correlated with the connectivity. In the following, we will discuss the influences of these parameters on V2V connectivity.In this section, we setβ=2.1∗10−5, average vehicle density ρ=0.02 vehicles/m, the transmit power PT=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 effect of Nakagami fading parameter m on the proposed V2V connectivity probability. Simulation results showed that as m increases, the V2V connectivity probability increases. Therefore, the Nakagami fading parameter m has a positive effect on V2V connectivity performance.Figure 11
The connectivity probability versus fading parameterm.Figure12 shows the influence of the SNR threshold Ψ on connectivity probability. Results showed that the SNR threshold has a significant impact on V2V connectivity. A larger SNR threshold Ψ results in a smaller V2V connectivity probability. This finding is consistent with the earlier conclusion in [21].Figure 12
The connectivity probability versus SNR thresholdΨ.Furthermore, the impact of average vehicle density on the V2V connectivity under Nakagami fading is illustrated in Figure13, where ρ ranges from 0.001 to 0.1 vehicles/m. The result shows that the variations in vehicle density impact the communications’ connectivity of the VANET. In free traffic state, the V2V connectivity probability increases with increasing vehicle density.Figure 13
The connectivity probability with variousρ.Figure14 illustrates the results of relation between the connectivity probability and the transmit power PT, 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. The result shows that the connectivity is significantly influenced by the neighbors’ order. The farther neighbors’ results in lower connectivity probability.Figure 14
The influence ofPT on connectivity probability.Figure 15
The connectivity probability versus neighbor order.


## 5.1. Traffic Flow Model

We take into account the 1D VANET, the vehicles are in a free-flow state in the network, so the vehicle movements are independent of each other. For the free-flow traffic model, the number of vehicles passing through the observation point obeys the Poisson distribution. The arrival time of adjacent vehicles obeys the exponential distribution. The distance between two consecutive vehicles obeys an exponential distribution, we define 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 different 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 defined as intervehicle distance. In this paper,Xj is a random variable representing the distance between the j-th car and the j+1-th car. The probability density function (PDF) of Xj is as follows:(4)fXjx=ρe−ρx,x≥0,where ρ denotes average vehicle density.Under the free-flow 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. Therefore, in this paper, we assumed that the vehicle speed obeys a truncated Gaussian PDF, which can be calculated by [12]:(5)gv=fv∫vminvmaxfvdv,vmin≤v≤vmax,where vmin denotes the lower bound of velocity and vmax stands for the upper bound of velocity. In equation (5), fv can be defined as follows:(6)fv=1σv2πexp−v−μv22σv2,where σv stands for the standard deviation of velocity, μv represents the average velocity, vmax=μv+3σv, and vmin=μv−3σv. Then, equation (5) can be written as follows:(7)gv=2fverfvmax−μv/σv2−erfvmin−μv/σv2,where vmin≤v≤vmax, erf· represents the error function, and v denotes the vehicle speed.We define the distance between one vehicle and itsq-th neighbor as the intervehicle distance Zq. Generally, by calculating the sum of q adjacent vehicle distances Xi, we could obtain the intervehicle distance Zq. As a result, the intervehicle distance can be calculated as Zq≜∑i=1qXi. The PDF of Zq is given by [23]:(8)fZqx;q=ρqq−1!xq−1e−ρx,where ρ stands for the average vehicle density, q represents the neighbor order, and ·! stands for the factorial operator.


## 5.2. Connectivity in Nakagami-m Fading Channels

According to the measurement results in Section3, 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]. Therefore, 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.The 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 isd, the SNR at the receiver is calculated by [23](9)γd=W2βPTdnPnoise,where γ represents SNR, n denotes the PL exponent, Pnoise denotes the total additive noise power, and PT represents the transmit power; β=GTGRc2/4πfo2 denotes a constant value, where c represents the speed of light; GR and GT stand for the RX and TX gains, respectively. During the measurement, omnidirectional antennas were employed, therefore, we set GR=1 and GT=1. fo presents the carrier frequency. In equation (9), the noise power Pnoise is defined as Pnoise=KbTB, where T stands for temperature in K, B denotes the transmission bandwidth, and Kb=1.38∗10−23 J/K denotes the Boltzmann constant.It is assumed thatEW2=1, the received average SNR γ¯ with distance d can be written as [21]:(10)γ¯=βPTdnPnoise.Given the condition that the received signal magnitude follows the Nakagami distribution. The PDF of the received signal magnitudew is defined as [71]:(11)fww;m,Ω=2mmw2m−1ΩmΓmexp−mw2Ω,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]:(12)fγγ=mγ¯mγm−1Γmexp−mγγ¯,where m denotes the shape factor, γ represents the received SNR, and γ¯ denotes the average SNR.When the received SNRγ is more than the given SNR threshold Ψ, the vehicles are connected, and the transmitted message can be decoded correctly. Therefore, the probability of accurately received transmitted information at a distance d is as follows:(13)Pγd≥Ψ=∫Ψ∞fγγdγ=∫Ψ∞mγ¯mγm−1Γmexp−mγγ¯dγ=Γm,mΨ/γΓm,where Γm,mΨ/γ¯ represents the incomplete Gamma function:(14)Γm,mΨ/γ¯=∫Ψ∞mγ¯mγm−1exp−mγγ¯dγ.Ifm is a positive integer, Γm=m−1!. Γm,a is calculated by [75]:(15)Γm,a=m−1!×e−a∑k=0m−1akk!.Together with equation (15), equation (13) can be further modified and represented as follows:(16)Pγd≥Ψ=e−mΨdnPnoise/βPT∑k=0m−1mΨdnPnoise/βPTkk!,where β is a constant value, PT 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 Pnoise 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 γ is larger than the given SNR threshold value Ψ. The probability that the vehicle and its q-th neighboring vehicle are connected is given by(17)Psi=Pγx≥ΨZq=x=∫Ψ∞∫0∞fγγ·fZqx;qdxdγ=∫Ψ∞∫0∞mγ¯mγm−1Γme−mγγ¯+ρxxq−1dxdγ·ρqq−1!.After calculation, equation (17) becomes(18)Psi=∫Ψ∞∫0∞fγγfZqx;qdxdγ=ρqq−1!∑k=0m−1mΨPnoise/βPTkk!·W1,where(19)W1=∫0∞e−mΨPnoisexn/βPT+ρxxnk+q−1dx.There 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). The integral is given by(20)∫0∞xt−1e−ux+sxbdx=2π1−b/2bt−1/2u−t×G1,bb,1ubsbb1tb,…,t+b−1b,where Gr,jc,l 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(21)PS=W2∑k=0m−1mΨPnoise/βPTkk!nnk+q−1/2ρ−nk+q×G1,nn,1ρnβPTmΨPnoisenn1nk+qn,…,nk+1+q−1n,with(22)W2=ρqq−1!2π1−n/2,where β is a constant value, n denotes the PL exponent, and PT 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 Pnoise represents the noise power.Therefore, for integer values ofn 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. The 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 isfXix. As a result, the probability of two consecutive vehicles connected is calculated as [22](23)Pc=Pγx≥ΨXi=x=∫Ψ∞∫0∞fγγfXixdxdγ=ρ∫0∞e−ρxΓm,mΨ/γ¯Γmdx=ρ∑k=0m−1mΨPnoise/βPTkk!∫0∞e−ρx+mΨxnPnoise/βPTxnkdx,where n denotes the PL exponent, PT denotes the transmit power, Pnoise is the noise power, ρ denotes average vehicle density, and Ψ denotes the SNR threshold value. β is a constant value and β=GTGRc2/4πfo2.For integer values ofn, taking equation (20) into equation (23) results in [22](24)Pc=ρ2π1−n/2∑k=0m−1mΨPnoise/βPTkk!nnk+1/2ρ−nk−1×G1,nn,1ρnβPTmΨPnoisenn1nk+1n,…,nk+1n.Therefore, 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).


## 5.3. 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. These parameters are directly correlated with the connectivity. In the following, we will discuss the influences of these parameters on V2V connectivity.In this section, we setβ=2.1∗10−5, average vehicle density ρ=0.02 vehicles/m, the transmit power PT=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 effect of Nakagami fading parameter m on the proposed V2V connectivity probability. Simulation results showed that as m increases, the V2V connectivity probability increases. Therefore, the Nakagami fading parameter m has a positive effect on V2V connectivity performance.Figure 11
The connectivity probability versus fading parameterm.Figure12 shows the influence of the SNR threshold Ψ on connectivity probability. Results showed that the SNR threshold has a significant impact on V2V connectivity. A larger SNR threshold Ψ results in a smaller V2V connectivity probability. This finding is consistent with the earlier conclusion in [21].Figure 12
The connectivity probability versus SNR thresholdΨ.Furthermore, the impact of average vehicle density on the V2V connectivity under Nakagami fading is illustrated in Figure13, where ρ ranges from 0.001 to 0.1 vehicles/m. The result shows that the variations in vehicle density impact the communications’ connectivity of the VANET. In free traffic state, the V2V connectivity probability increases with increasing vehicle density.Figure 13
The connectivity probability with variousρ.Figure14 illustrates the results of relation between the connectivity probability and the transmit power PT, 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. The result shows that the connectivity is significantly influenced by the neighbors’ order. The farther neighbors’ results in lower connectivity probability.Figure 14
The influence ofPT on connectivity probability.Figure 15
The connectivity probability versus neighbor order.


## 6. 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 log-normal distributions, the Nakagami-m distribution is the best fit distribution for the signal amplitude fading in the tunnel scenario. Based on this finding, 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 influences 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. The neighbor order also affect the vehicle connectivity in the VANET. The higher the neighbor order, the smaller the connectivity probability. If the PL exponent increases, the V2V connectivity probability decreases. The 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 influences of transmit power, transmission distance, and traffic 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 traffic flow, the V2V connectivity probability increases with increasing vehicle density.



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*Source: 7157721-2023-03-07.xml*