# Connectivity Analysis in Vehicular Networks with Slight Traffic Interference

Authors: Bin Pan; Hao Wu
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

This paper analyzes the network connectivity in vehicular networks with slight traffic interference, i.e., small-scale traffic accident. In this paper, we develop an analytical model for highway scenarios or sparse urban scenarios, where the slight traffic interference can make the vehicles slow down rather than block the traffic flow. When a traffic accident occurs, it is necessary to inform the nearby vehicles to slow down in order to reduce congestion at the accident location. Once they pass by, they can return to the normal value. Consequently, we can divide an entire road into two or three subsections, which helps us to analyze the connectivity performance. In addition, we analyze the impact of several key parameters on connectivity probability, including vehicle arrival rate, vehicle communication range, length of road, vehicle normal speed, and safe speed. All analytical results are verified through Monte Carlo simulation experiments. The simulation results are very close to analytical results, which means that the analytical results are accurate and the analytical model we propose is effective.


---

## Body

## 1. Introduction

Vehicular Ad Hoc Networks (VANETs), as special Mobile Ad Hoc Networks (MANETs), have attracted much attention in recent years. For future Intelligent Transportation System (ITS), VANETs can promise various applications in enhancing the road safety, improving traffic efficiency and providing entertainment services [1]. In VANETs, network connectivity for vehicles is one of the most significant performance [2, 3]. Many researches have focused on the network connectivity in both highway and urban scenarios. However, most previous studies considered the normal road conditions. In practice, something unexpected may happen on the road, such as traffic accident or road maintenance. We consider small-scale traffic accidents and road maintenance as slight traffic interference, which can only make the vehicles slow down, rather than block the traffic flow. When a traffic interference occurs, it is necessary to inform the vehicles which will pass by the interference location to slow down. Once they pass by the interference location, their speeds can return to the normal value. Therefore, slight traffic interference could affect the connectivity performance of vehicles driving on the road. Connectivity analysis considering the slight traffic interference is useful to design the routing protocols and vehicular network architecture in special circumstances.In this paper, we focus on the network connectivity on unidirectional road segment with a small-scale traffic accident. In particular, the road segment can be in a highway scenario or a sparse urban scenario, which means that the traffic accident can make the vehicles slow down, rather than block the traffic flow. The same as previous studies, the important metric we focus on is the connectivity probability. Here, network connectivity can be defined that if any two vehicles moving on the road segment can be connected through one-hop or multihop Vehicle-to-Vehicle (V2V) communication, the whole network can be considered connected. For calculating the steady-state connectivity probability, we derived a connectivity analytical model, taking into consideration the traffic accident. In our considered road scenario, the location of traffic accident is uniformly distributed on the road. When a traffic accident (e.g., vehicle collision) occurred, the vehicle that collided with others could inform the vehicles behind within the communication range to slow down. Therefore, the traffic accident could divide the entire road segment into two or three segments with different vehicle speeds. In sparse VANETs scenario, the vehicle arrival follows Poisson process [4]. In the analytical model, there are several key parameters that have an influence on the connectivity performance, such as vehicle arrival rate, vehicle communication range, the length of entire road, vehicle normal speed, and safe speed. We verified the effectiveness of our analytical model through simulation experiments and discussed the impact of different parameters on the connectivity performance.The main contributions of this paper are as follows:(i)
We developed an analytical model to analyze the connectivity probability of vehicles in a sparse road scenario with a traffic accident.(ii)
Based on proposed analytical model, we analyzed and discussed the impact of different key parameters on the connectivity.(iii)
All analytical results can be verified through Monte Carlo simulation method in MATLAB. The simulation results are very close to analytical results, which means that the analytical results are accurate and our proposed analytical model is effective.The remainder of this paper is organized as follows. Section2 reviews the related works about network connectivity in both highway and city scenario. Section 3 describes the system model and analyzes the network connectivity in detail. Section 4 verifies the analytical results through Monte Carlo simulation experiments and analyzes the impact of several key parameters on network connectivity. Section 5 summarizes the whole paper and proposes research prospects for future work.


## 2. Related Works

There are many studies focused on the connectivity in highway scenarios recently. In [5], Zhang et al. proposed a mathematical model to analyze both multihop uplink and downlink connectivity for infrastructure-based vehicular networks, where uplink connectivity probability indicates the user satisfaction level and downlink connectivity probability represents service coverage performance of a vehicular network. Simulation results reveal the impact of several parameters on the two performance metrics. In [6], Yousefi et al. investigated the connectivity using M/G/∞ queuing model in high speed highways. Adding some mobile nodes with larger communication range, connectivity performance can be much improved. In addition, they also studied the case with fixed Roadside Units (RSUs). In [7], based on a mearsured case study, Cheng et al. analyzed the impact of intervehicle spacing distributions on connectivity in highway scenario. The intervehicle spacing with light traffic density follows the exponential distribution, while in the scenario with high traffic density, the Generalized Extreme Value (GEV) distribution is more suitable to model the inter-vehicle spacing compared with exponential distribution. In [8], Shao et al. investigated the connectivity performance in the free-flow state and proposed a connectivity-aware routing protocol. They analyzed the connectivity probability in both V2V and Vehicle-to-Infrastructure (V2I) communication scenarios, respectively. The relationship between connectivity probability and system throughput can be derived based on a Markov analytical model. In order to improve the system throughput, Shao et al. designed a multichannel reservation scheme taking into account network connectivity and variable traffic status. In [9], Yousefi et al. proposed an analytical model to study connectivity in vehicular highway networks. They utilized a M/D/∞ queueing model to obtain the explicit form of the expected connectivity distance. The impact of different systems parameters (such as vehicle communication range, vehicle speed distribution, and traffic flow density) on the connectivity can be also obtained through the analytical model. In [10], Wu et al. investigated the connectivity properties in a mobile linear network considering high mobility of nodes and strict delay constraint. They proposed a new mobility model and a novel geometry-assisted analytical method to indicate the node distribution in steady state and to study the statistical properties of network connectivity. The conclusions can apply to delay constrained and tolerant networks.For city block scenario, researchers also paid much attention. In [11], Viriyasitavat et al. designed a comprehensive framework to study the connectivity performance in urban vehicular networks. This framework can derive some closed-form results which help to understand the relationship between network connectivity and critical system parameters (i.e., traffic density, communication range, size of a road block and traffic light mechanisms). In [12], Zhang et al. derived a closed-form polynomial expression of the connectivity probability in two-dimensional lattice networks by using a recursive decomposition approach. Then, the proposed approach can be applied to a realistic urban scenario to obtain the directed connectivity. In [13], Hoque et al. studied multihop connectivity in urban scenario on the basis of the real trajectory of taxi. A novel algorithm was proposed to analyze the connectivity property and network partitioning. Using the efficient storage technique and computation mechanism, the space-time complexity can be reduced obviously. In [14], Naboulsi et al. studied the instantaneous topology of realistic urban scenarios, i.e., Cologne and Zurich. The results showed that a poor connectivity could cause limited availability and reliability. The analysis also indicated the unrealistic connected topologies can be caused by simplistic mobility models.Under certain conditions, the vehicle density is different in different sections of one road. In [15], Khabazian et al. developed an analytical model that nodes can arrive at the highway or depart from the highway through the entry/exit points. They investigated the probability distribution of a node’s location and node population sizes. Moreover, the statistical properties for connectivity can be also derived, such as mean cluster size and the probability that nodes can form a single cluster. In [16], Zheng et al. paid attention to the connectivity problem for highway road with one entry and exit. They developed an analytical model to derive the connectivity probability considering several critical system parameters, such as vehicle arrival rate, vehicle speed, and the probability that vehicles drive through the entry and exit point. The two scenarios that no RSU deployed and one RSU deployed are both investigated.


## 3. System Model and Analysis of Connectivity

In this section, we first describe the network scenario and system model. Then we give a detailed analysis for connectivity probability of vehicles driving on unidirectional road segment with a traffic accident.


### 3.1. System Model

As shown in Figure1, we consider a multilane unidirectional road, which is denoted by [0,d], where d is the length of the entire road. The accident location z is uniformly distributed on the entire road, i.e., z∈[0,d]. As mentioned above, the road segment can be in a highway scenario or a sparse urban scenario, ensuring the accident can only make the vehicles behind within communication range Rv slow down, rather than block the traffic flow. Therefore, the traffic accident could divide the entire road segment into two or three subsegments with different vehicle speeds. Vehicles in the normal segment keep the normal speed v1, while driving in the deceleration segment slow down with the safe speed v2, i.e., v1>v2. In this paper, we neglect the acceleration and deceleration process of vehicles at the critical point of two different segment (i.e., the transition time between normal speed v1 and safe speed v2). For instance, the acceleration of regular vehicles is usually greater than 6.5 m/s2, which indicates the speed transition with a span of 10 m/s could be achieved within 1.5 s. The transition time is far shorter than the sojourn time that the vehicles stay in road segment. The vehicles arriving at the road entry follow a Poisson process with arrival rate λ. Hence, in normal segment [0,z-Rv] and [z,d], the number of vehicles per meter follows the Poisson distribution with ρ1=λ/v1, while ρ2=λ/v2 in deceleration segment [z-Rv,z].Figure 1
Network scenario.In this paper, we define the network connectivity as follows:(i)
The two vehicles are considered connected, if their intervehicle spacing is not larger than the communication range of vehicles.(ii)
The subsegment is considered connected, if any two adjacent vehicles on this subsegment are connected.(iii)
The two adjacent subsegments are considered connected, if there are at least two connected vehicles which are, respectively, located on this two subsegments.(iv)
The entire road is considered connected, if any two adjacent subsegments in this road are connected.


### 3.2. Analysis of Connectivity

According to different traffic accident locationz∈[0,d], the network connectivity can be analyzed in the four cases below (under the assumption d>3Rv):(i)
Case1: The traffic accident location z∈[0,Rv].(ii)
Case2: The traffic accident location z∈[Rv,2Rv].(iii)
Case3: The traffic accident location z∈[2Rv,d-Rv].(iv)
Case4: The traffic accident location z∈[d-Rv,d].Case 1 : (z∈[0,Rv]).
In Case1, as shown in Figure 2, the road segment [0,z] can be connected obviously, if there are vehicles driving on the road segment. Therefore, we only need to analyze the connectivity of road segment [z,d] and the connectivity between [0,z] and [z,d]. If there is no vehicle driving on the road segment [0,z], we only need to calculate the connectivity probability of road segment [z,d]. For convenience of description, we divide the entire road segment [0,d] into two sub-segments a1 and a2, denoting [0,z] and [z,d], respectively.Figure 2
The situation of Case1: z∈[0,Rv].In order to facilitate analysis, define the main notations to be used. LetPr⁡X denote the probability that the event X occurs. Let Hs denote the event that there are vehicles on segment s and Cs denote the event that the segment s is connected. Hence, the events that there is no vehicle on segment s and the segment s is disconnected can be expressed by H¯s and C¯s, respectively. Let Cs1,s2 denote the event that the segment s1 and s2 are connected; i.e., there are at least two connected vehicles, which are, respectively, driving on segment s1 and s2.On the basis of the mathematical property of Poisson distribution, the probability that there are vehicles driving on segmenta1 can be calculated by(1)PrHa1=1-ρ2·z00!·e-ρ2·z=1-e-ρ2·z.Hence, the probability that there is no vehicle driving on segment a1 can be expressed by(2)Pr⁡H¯a1=1-Pr⁡Ha1=e-ρ2·z.Similarly, the probabilities that there are vehicles driving on segment a2 and there is no vehicle on segment a2 are expressed by(3)Pr⁡Ha2=1-e-ρ1·d-zand(4)PrH¯a2=e-ρ1·d-zrespectively.Further, Case1 can be divided into two subcases as follows:(i)
Case 1.1: There are vehicles driving on segmenta1.(ii)
Case 1.2: There is no vehicle driving on segmenta1.Case 1.1. In Case 1.1, clearly, the vehicles driving on segment a1 are connected, i.e.,(5)PrCa1=1.Therefore, the vehicles on segment a2 have a great impact on network probability. Take into account the two subcases below: (i)
Case 1.1.1: There are vehicles driving on segmenta2.(ii)
Case 1.1.2: There is no vehicle driving on segmenta2.In Case 1.1.1, we will calculate the connectivity probabilityPrCa2 of the segment a2 and the connectivity probability PrCa1,a2 between segment a1 and segment a2.According to [17], the connectivity probability of one road segment with constant Poisson process can be given by(6)Prλρ,Rv,s=e-λρs∑j=0s/Rv-1jj!λρs-jRvj-1×j+λρs-jReλρs-jRv,where λρ means the vehicle density (i.e., the number of vehicles per meter follows the Poisson process with rate λρ), Rv is the communication range of vehicles, s is the length of the road segment, and ⌊x⌋ is the operation to obtain the largest integer which is not greater than x. Hence, we can calculate(7)PrCa2=Prρ1,Rv,d-z.As forPrCa1,a2, first of all, we calculate the connectivity probability that there are two connected vehicles, which are driving on segments a1 and a2, respectively. Assume that the vehicle A is located at LA (0≤LA≤z), which is uniformly distributed on segment a1, and the vehicle B is located at LB (z≤LB≤d), which is uniformly distributed on segment a2. Therefore, we can obtain the probability density function (PDF) of LA and LB below:(8)fLAx=1z,0≤x≤z(9)fLBy=1d-z,z≤y≤dDefinepAB as the probability that the vehicles A and B are connected, i.e., the probability that the distance between vehicle A and vehicle B is not greater than communication range Rv. Hence, we have(10)pAB=Pr⁡y-x≤Rv=2Rv-z2d-z,which is equal to the ratio between the dashed area and the area of rectangle with blue border, as shown in Figure 3.Figure 3
The diagram of calculatingpAB.Then, we calculate the probability that there are at least two connected vehicles, which are driving on segmentsa1 and a2, respectively. On the basis of pAB, the probability that a vehicle driving on segment a2 cannot be connected with any of vehicles on segment a1 can be expressed by(11)qa1,a2=11-Pr⁡H¯a1∑k=1∞1-pABkPr⁡na1=k=1eρ2z-1e2d-2Rv-z/2d-z·ρ2z-1,where Prns=k denote the probability that there are k vehicle nodes driving on road segment s. Therefore, we can obtain the probability that the segments a1 and a2 are connected as follows:(12)Pr⁡Ca1,a2=1-∑k=0∞qa1,a2k·Pr⁡na2=k=1-e-ρ1d-z1-qa1,a2.Thus, according to (5), (7), and (12), the connectivity probability of the entire road segment [0,d] in Case 1.1.1 can be calculated by(13)p111=Pr⁡Ca1·Pr⁡Ca2·Pr⁡Ca1,a2.In Case 1.1.2, because there is no vehicle driving on segmenta2, the connectivity probability of the entire road segment [0,d] in Case 1.1.2 can be expressed by(14)p112=Pr⁡Ca1=1.Considering the two subcases above, the connectivity probability of the entire road[0,d] in Case 1.1 can be expressed by(15)p11=Pr⁡Ha2·P111+Pr⁡H¯a2·P112.Case 1.2. In Case 1.2, we only need to calculate the connectivity probability of the segment [z,d], i.e.,(16)p12=Pr⁡Ca2.Considering the analysis mentioned above in Case1, the connectivity probability of the entire road [0,d] in Case 1 can be expressed by(17)p1=PrHa1·p11+PrH¯a1·p12.Case 2 (z∈[Rv,2Rv]).
In Case2, as shown in Figure 4, the entire road segment [0,d] can be divided into three subsegments b1, b2, and b3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively.Figure 4
The situation of Case2: z∈[Rv,2Rv].Similar to Case1, the following probabilities are easy to obtain:(18)PrHb1=1-e-ρ1z-Rv(19)PrHb2=1-e-ρ2Rv(20)Pr⁡Hb3=1-e-ρ1d-z(21)Pr⁡H¯b1=e-ρ1z-Rv(22)Pr⁡H¯b2=e-ρ2Rv(23)Pr⁡H¯b3=e-ρ1d-zAccording to whether there are vehicles driving on segmentb2, Case 2 can be divided into two subcases as follows:(i)
Case 2.1: There are vehicles driving on segmentb2.(ii)
Case 2.2: There is no vehicle driving on segmentb2.In Case 2.1, clearly, the vehicles driving on segmentb2 are connected, i.e.,(24)Pr⁡Cb2=1.Further, take in account the four subcases below:(i)
Case 2.1.1: There are vehicles driving on both segmentsb1 and b3.(ii)
Case 2.1.2: There are vehicles driving on segmentb1, but there is no vehicle driving on segment b3.(iii)
Case 2.1.3: There is no vehicle driving on segmentb1, but there are vehicles driving on segment b3.(iv)
Case 2.1.4: There is no vehicle driving on both segmentsb1 and b3.In Case 2.1.1, it is obvious that the vehicles driving on segmentb1 are connected, i.e.,(25)PrCb1=1.Thus, we only need to analyze the connectivity probability Pr{Cb3}, Pr{Cb1,b2} and Pr{Cb2,b3}. As for the connectivity probability Pr{Cb3}, it can be expressed by(26)Pr⁡Cb3=Pr⁡ρ1,Rv,d-z.Using the same derivation method asPr{Ca1,a2}, the probability Pr{Cb1,b2} that road segments b1 and b2 are connected is given by(27)Pr⁡Cb1,b2=1-e-ρ2Rv1-qb1,b2,where(28)qb1,b2=1eρ1z-Rv-1ez-Rv2/2Rv·ρ1-1.Similarly, the probabilityPrCb2,b3 that segments b2 and b3 are connected is given by(29)Pr⁡Cb2,b3=1-e-ρ1d-z1-qb2,b3,where(30)qb2,b3=1eρ2Rv-1e1-Rv/2d-z·ρ2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 2.1.1 can be expressed by(31)p211=Pr⁡Cb1·Pr⁡Cb2·Pr⁡Cb3·Pr⁡Cb1,b2·Pr⁡Cb2,b3.In Case 2.1.2, there is no vehicle driving on segmentb3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(32)p212=Pr⁡Cb1·Pr⁡Cb2·Pr⁡Cb1,b2.In Case 2.1.3, there is no vehicle driving on segmentb1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(33)p213=Pr⁡Cb2·Pr⁡Cb3·Pr⁡Cb2,b3.In Case 2.1.4, there is no vehicle driving on segmentsb1 and b3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(34)p214=Pr⁡Cb2=1.Regarding the four subcases above, the connectivity probability in Case 2.1 can be expressed by(35)p21=Pr⁡Hb1·Pr⁡Hb3·p211+Pr⁡Hb1·Pr⁡H¯b3·p212+Pr⁡H¯b1·Pr⁡Hb3·p213+Pr⁡H¯b1·Pr⁡H¯b3·p214.In Case 2.2, there is no vehicle driving on segmentb2. Only if one of the two following subcases occurs, it may be possible that the entire road segment [0,d] is connected.(i) Case 2.2.1: There are vehicles driving on segmentb1, but there is no vehicle driving on segment b3.(ii) Case 2.2.2: There is no vehicle driving on segmentb1, but there are vehicles driving on segment b3.In Case 2.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(36)p221=Pr⁡Cb1=1.In Case 2.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(37)p222=Pr⁡Cb3.Regarding the two subcases above, the connectivity probability in Case 2.2 can be expressed by(38)p22=Pr⁡Hb1·Pr⁡H¯b3·p211+Pr⁡H¯b1·Pr⁡Hb3·p222.Regarding all the analysis mentioned above in Case2, the connectivity probability of the entire road segment [0,d] in Case 2 is expressed by(39)p2=PrHb2·p21+PrH¯b2·p22.Case 3 (z∈[2Rv,d-Rv]).
In Case3, as shown in Figure 5, the entire road segment [0,d] can be divided into three subsegments c1, c2, and c3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively. The following probabilities are easy to obtain:(40)Pr⁡Hc1=1-e-ρ1z-Rv(41)Pr⁡Hc2=1-e-ρ2Rv(42)Pr⁡Hc3=1-e-ρ1d-z(43)Pr⁡H¯c1=e-ρ1z-Rv(44)Pr⁡H¯c2=e-ρ2Rv(45)Pr⁡H¯c3=e-ρ1d-zFigure 5
The situation of Case3: z∈[2Rv,d-Rv].According to whether there are vehicles driving on segmentc2, Case 3 can be divided into two subcases as follows:(i)
Case 3.1: There are vehicles driving on segmentc2.(ii)
Case 3.2: There is no vehicle driving on segmentc2.In Case 3.1, clearly, the vehicles driving on segmentc2 are connected, i.e.,(46)PrCc2=1.Further, take into account the four subcases below:(i)
Case 3.1.1: There are vehicles driving on both segmentsc1 and c3.(ii)
Case 3.1.2: There are vehicles driving on segmentc1, but there is no vehicle driving on segment c3.(iii)
Case 3.1.3: There is no vehicle driving on segmentc1, but there are vehicles driving on segment c3.(iv)
Case 3.1.4: There is no vehicle driving on both segmentsc1 and c3.In Case 3.1.1, we need to analyze the connectivity probabilitiesPrCc1, PrCc3, PrCc2,c1, and PrCc2,c3. As for connectivity probabilities PrCc1 and PrCc3, they can be expressed by(47)PrCc1=Prρ1,Rv,z-Rvand(48)PrCc3=Prρ1,Rv,d-z,respectively.Using the same derivation method asPrCa1,a2, the probability PrCc2,c1 that the segments c2 and c1 are connected can be given by(49)PrCc2,c1=1-e-ρ1z-Rv1-qc2,c1,where(50)qc2,c1=1eρ2Rv-1e1-Rv/2z-Rv·ρ2Rv-1.And the probability PrCc2,c3 that the segments c2 and c3 are connected can be given by(51)Pr⁡Cc2,c3=1-e-ρ1d-z1-qc2,c3,where(52)qc2,c3=1eρ2Rv-1e1-Rv/2d-z·ρ2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 3.1.1 can be expressed by(53)p311=PrCc1·PrCc2·PrCc3·PrCc2,c1·PrCc2,c3.In Case 3.1.2, there is no vehicle driving on segmentc3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(54)p312=PrCc1·PrCc2·PrCc2,c1.In Case 3.1.3, there is no vehicle driving on segmentc1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(55)p313=PrCc2·PrCc3·PrCc2,c3.In Case 3.1.4, there is no vehicle driving on segmentc1 and c3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(56)p314=Pr⁡Cc2=1.Regarding the four subcases above, the connectivity probability in Case 3.1 can be expressed by(57)p31=PrHc1·PrHc3·p311+PrHc1·PrH¯c3·p312+PrH¯c1·PrHc3·p313+PrH¯c1·PrH¯c3·p314.In Case 3.2, there is no vehicle driving on segmentc2. Only if one of the two following subcases occurs may it be possible that the entire road segment [0,d] is connected:(i)
Case 3.2.1: There are vehicles driving on segmentc1, but there is no vehicle driving on segment c3.(ii)
Case 3.2.2: There is no vehicle driving on segmentc1, but there are vehicles driving on segment c3.In Case 3.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(58)p321=Pr⁡Cc1.In Case 3.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(59)p322=Pr⁡Cc3.Regarding the two subcases above, the connectivity probability in Case 3.2 can be expressed by(60)p32=Pr⁡Hc1·Pr⁡H¯c3·p321+Pr⁡H¯c1·Pr⁡Hc3·p322.Regarding all the analysis mentioned above in Case3, the connectivity probability of the entire road segment [0,d] in Case 3 is expressed by(61)p3=PrHc2·p31+PrH¯c2·p32.Case 4 (z∈[d-Rv,d]).
In Case4, as shown in Figure 6, the entire road segment [0,d] can be divided into three subsegments d1, d2, and d3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively.Figure 6
The situation of Case4: z∈[d-Rv,d].Similar to Case1, the following probabilities are easy to obtain:(62)Pr⁡Hd1=1-e-ρ1z-Rv(63)Pr⁡Hd2=1-e-ρ2Rv(64)Pr⁡Hd3=1-e-ρ1d-z(65)Pr⁡H¯d1=e-ρ1z-Rv(66)Pr⁡H¯d2=e-ρ2Rv(67)Pr⁡H¯d3=e-ρ1d-zAccording to whether there are vehicles driving on segmentd2, Case 4 can be divided into two subcases as follows:(i)
Case 4.1: There are vehicles driving on segmentd2.(ii)
Case 4.2: There is no vehicle driving on segmentd2.In Case 4.1, clearly, the vehicles driving on segmentd2 are connected, i.e.,(68)Pr⁡Cd2=1.Further, take into account the four subcases below:(i)
Case 4.1.1: There are vehicles driving on both segmentsd1 and d3.(ii)
Case 4.1.2: There are vehicles driving on segmentd1, but there is no vehicle driving on segment d3.(iii)
Case 4.1.3: There is no vehicle driving on segmentd1, but there are vehicles driving on segment d3.(iv)
Case 4.1.4: There is no vehicle driving on both segmentsd1 and d3.In Case 4.1.1, it is obvious that the vehicles driving on segmentd3 are connected, i.e.,(69)PrCd3=1.Thus, we need to analyze the connectivity probabilities PrCd1, PrCd2,d1, and PrCd3,d2. As for the connectivity probability PrCd1, it can be expressed by(70)PrCd1=Prρ1,Rv,z-Rv.Using the same derivation method asPrCa1,a2, the probability PrCd2,d1 that the segments d2 and d1 are connected can be driven by(71)PrCd2,d1=1-e-ρ1z-Rv1-qd2,d1,where(72)qd2,d1=1eρ2Rv-1e1-Rv/2z-Rv·ρ2Rv-1.And the probabilityPrCd3,d2 that the segments d3 and d2 are connected can be driven by(73)Pr⁡Cd3,d2=1-e-ρ2Rv1-qd3,d2,where(74)qd3,d2=1eρ1d-z-1ed-z2ρ1/2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 4.1.1 can be expressed by(75)p411=PrCd1·PrCd2·PrCd3·PrCd2,d1·PrCd3,d2.In Case 4.1.2, there is no vehicle driving on segmentd3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(76)p412=PrCd1·PrCd2·PrCd2,d1.In Case 4.1.3, there is no vehicle driving on segmentd1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(77)p413=Pr⁡Cd2·Pr⁡Cd3·Pr⁡Cd3,d2.In Case 4.1.4, there is no vehicle driving on segmentsd1 and d3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(78)p414=PrCd2=1.Regarding the four subcases above, the connectivity probability in Case 3.1 can be expressed by(79)p41=PrHd1·PrHd3·p411+PrHd1·PrH¯d3·p412+PrH¯d1·PrHd3·p413+PrH¯d1·PrH¯d3·p414.In Case 4.2, there is no vehicle driving on segmentd2. Only if one of the two following subcases occurs may it be possible that the entire road segment [0,d] is connected:(i)
Case 4.2.1: There are vehicles driving on segmentd1, but there is no vehicle driving on segment d3.(ii)
Case 4.2.2: There is no vehicle driving on segmentd1, but there are vehicles driving on segment d3.In Case 4.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(80)p421=PrCd1.In Case 4.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(81)p422=Pr⁡Cd3=1.Regarding the two subcases above, the connectivity probability in Case 3.2 can be expressed by(82)p42=PrHd1·PrH¯d3·p411+PrH¯d1·PrHd3·p422.Regarding all the analysis mentioned above in Case4, the connectivity probability of the entire road segment [0,d] in Case 4 is expressed by(83)p4=PrHd2·p41+PrH¯d2·p42.Therefore, regarding all the analysis mentioned above, the connectivity probability of the entire road segment[0,d] can be expressed by(84)p=1d·∫0Rvp1dz+∫Rv2Rvp2dz+∫2Rvd-Rvp3dz+∫d-Rvdp4dz.


## 3.1. System Model

As shown in Figure1, we consider a multilane unidirectional road, which is denoted by [0,d], where d is the length of the entire road. The accident location z is uniformly distributed on the entire road, i.e., z∈[0,d]. As mentioned above, the road segment can be in a highway scenario or a sparse urban scenario, ensuring the accident can only make the vehicles behind within communication range Rv slow down, rather than block the traffic flow. Therefore, the traffic accident could divide the entire road segment into two or three subsegments with different vehicle speeds. Vehicles in the normal segment keep the normal speed v1, while driving in the deceleration segment slow down with the safe speed v2, i.e., v1>v2. In this paper, we neglect the acceleration and deceleration process of vehicles at the critical point of two different segment (i.e., the transition time between normal speed v1 and safe speed v2). For instance, the acceleration of regular vehicles is usually greater than 6.5 m/s2, which indicates the speed transition with a span of 10 m/s could be achieved within 1.5 s. The transition time is far shorter than the sojourn time that the vehicles stay in road segment. The vehicles arriving at the road entry follow a Poisson process with arrival rate λ. Hence, in normal segment [0,z-Rv] and [z,d], the number of vehicles per meter follows the Poisson distribution with ρ1=λ/v1, while ρ2=λ/v2 in deceleration segment [z-Rv,z].Figure 1
Network scenario.In this paper, we define the network connectivity as follows:(i)
The two vehicles are considered connected, if their intervehicle spacing is not larger than the communication range of vehicles.(ii)
The subsegment is considered connected, if any two adjacent vehicles on this subsegment are connected.(iii)
The two adjacent subsegments are considered connected, if there are at least two connected vehicles which are, respectively, located on this two subsegments.(iv)
The entire road is considered connected, if any two adjacent subsegments in this road are connected.


## 3.2. Analysis of Connectivity

According to different traffic accident locationz∈[0,d], the network connectivity can be analyzed in the four cases below (under the assumption d>3Rv):(i)
Case1: The traffic accident location z∈[0,Rv].(ii)
Case2: The traffic accident location z∈[Rv,2Rv].(iii)
Case3: The traffic accident location z∈[2Rv,d-Rv].(iv)
Case4: The traffic accident location z∈[d-Rv,d].Case 1 : (z∈[0,Rv]).
In Case1, as shown in Figure 2, the road segment [0,z] can be connected obviously, if there are vehicles driving on the road segment. Therefore, we only need to analyze the connectivity of road segment [z,d] and the connectivity between [0,z] and [z,d]. If there is no vehicle driving on the road segment [0,z], we only need to calculate the connectivity probability of road segment [z,d]. For convenience of description, we divide the entire road segment [0,d] into two sub-segments a1 and a2, denoting [0,z] and [z,d], respectively.Figure 2
The situation of Case1: z∈[0,Rv].In order to facilitate analysis, define the main notations to be used. LetPr⁡X denote the probability that the event X occurs. Let Hs denote the event that there are vehicles on segment s and Cs denote the event that the segment s is connected. Hence, the events that there is no vehicle on segment s and the segment s is disconnected can be expressed by H¯s and C¯s, respectively. Let Cs1,s2 denote the event that the segment s1 and s2 are connected; i.e., there are at least two connected vehicles, which are, respectively, driving on segment s1 and s2.On the basis of the mathematical property of Poisson distribution, the probability that there are vehicles driving on segmenta1 can be calculated by(1)PrHa1=1-ρ2·z00!·e-ρ2·z=1-e-ρ2·z.Hence, the probability that there is no vehicle driving on segment a1 can be expressed by(2)Pr⁡H¯a1=1-Pr⁡Ha1=e-ρ2·z.Similarly, the probabilities that there are vehicles driving on segment a2 and there is no vehicle on segment a2 are expressed by(3)Pr⁡Ha2=1-e-ρ1·d-zand(4)PrH¯a2=e-ρ1·d-zrespectively.Further, Case1 can be divided into two subcases as follows:(i)
Case 1.1: There are vehicles driving on segmenta1.(ii)
Case 1.2: There is no vehicle driving on segmenta1.Case 1.1. In Case 1.1, clearly, the vehicles driving on segment a1 are connected, i.e.,(5)PrCa1=1.Therefore, the vehicles on segment a2 have a great impact on network probability. Take into account the two subcases below: (i)
Case 1.1.1: There are vehicles driving on segmenta2.(ii)
Case 1.1.2: There is no vehicle driving on segmenta2.In Case 1.1.1, we will calculate the connectivity probabilityPrCa2 of the segment a2 and the connectivity probability PrCa1,a2 between segment a1 and segment a2.According to [17], the connectivity probability of one road segment with constant Poisson process can be given by(6)Prλρ,Rv,s=e-λρs∑j=0s/Rv-1jj!λρs-jRvj-1×j+λρs-jReλρs-jRv,where λρ means the vehicle density (i.e., the number of vehicles per meter follows the Poisson process with rate λρ), Rv is the communication range of vehicles, s is the length of the road segment, and ⌊x⌋ is the operation to obtain the largest integer which is not greater than x. Hence, we can calculate(7)PrCa2=Prρ1,Rv,d-z.As forPrCa1,a2, first of all, we calculate the connectivity probability that there are two connected vehicles, which are driving on segments a1 and a2, respectively. Assume that the vehicle A is located at LA (0≤LA≤z), which is uniformly distributed on segment a1, and the vehicle B is located at LB (z≤LB≤d), which is uniformly distributed on segment a2. Therefore, we can obtain the probability density function (PDF) of LA and LB below:(8)fLAx=1z,0≤x≤z(9)fLBy=1d-z,z≤y≤dDefinepAB as the probability that the vehicles A and B are connected, i.e., the probability that the distance between vehicle A and vehicle B is not greater than communication range Rv. Hence, we have(10)pAB=Pr⁡y-x≤Rv=2Rv-z2d-z,which is equal to the ratio between the dashed area and the area of rectangle with blue border, as shown in Figure 3.Figure 3
The diagram of calculatingpAB.Then, we calculate the probability that there are at least two connected vehicles, which are driving on segmentsa1 and a2, respectively. On the basis of pAB, the probability that a vehicle driving on segment a2 cannot be connected with any of vehicles on segment a1 can be expressed by(11)qa1,a2=11-Pr⁡H¯a1∑k=1∞1-pABkPr⁡na1=k=1eρ2z-1e2d-2Rv-z/2d-z·ρ2z-1,where Prns=k denote the probability that there are k vehicle nodes driving on road segment s. Therefore, we can obtain the probability that the segments a1 and a2 are connected as follows:(12)Pr⁡Ca1,a2=1-∑k=0∞qa1,a2k·Pr⁡na2=k=1-e-ρ1d-z1-qa1,a2.Thus, according to (5), (7), and (12), the connectivity probability of the entire road segment [0,d] in Case 1.1.1 can be calculated by(13)p111=Pr⁡Ca1·Pr⁡Ca2·Pr⁡Ca1,a2.In Case 1.1.2, because there is no vehicle driving on segmenta2, the connectivity probability of the entire road segment [0,d] in Case 1.1.2 can be expressed by(14)p112=Pr⁡Ca1=1.Considering the two subcases above, the connectivity probability of the entire road[0,d] in Case 1.1 can be expressed by(15)p11=Pr⁡Ha2·P111+Pr⁡H¯a2·P112.Case 1.2. In Case 1.2, we only need to calculate the connectivity probability of the segment [z,d], i.e.,(16)p12=Pr⁡Ca2.Considering the analysis mentioned above in Case1, the connectivity probability of the entire road [0,d] in Case 1 can be expressed by(17)p1=PrHa1·p11+PrH¯a1·p12.Case 2 (z∈[Rv,2Rv]).
In Case2, as shown in Figure 4, the entire road segment [0,d] can be divided into three subsegments b1, b2, and b3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively.Figure 4
The situation of Case2: z∈[Rv,2Rv].Similar to Case1, the following probabilities are easy to obtain:(18)PrHb1=1-e-ρ1z-Rv(19)PrHb2=1-e-ρ2Rv(20)Pr⁡Hb3=1-e-ρ1d-z(21)Pr⁡H¯b1=e-ρ1z-Rv(22)Pr⁡H¯b2=e-ρ2Rv(23)Pr⁡H¯b3=e-ρ1d-zAccording to whether there are vehicles driving on segmentb2, Case 2 can be divided into two subcases as follows:(i)
Case 2.1: There are vehicles driving on segmentb2.(ii)
Case 2.2: There is no vehicle driving on segmentb2.In Case 2.1, clearly, the vehicles driving on segmentb2 are connected, i.e.,(24)Pr⁡Cb2=1.Further, take in account the four subcases below:(i)
Case 2.1.1: There are vehicles driving on both segmentsb1 and b3.(ii)
Case 2.1.2: There are vehicles driving on segmentb1, but there is no vehicle driving on segment b3.(iii)
Case 2.1.3: There is no vehicle driving on segmentb1, but there are vehicles driving on segment b3.(iv)
Case 2.1.4: There is no vehicle driving on both segmentsb1 and b3.In Case 2.1.1, it is obvious that the vehicles driving on segmentb1 are connected, i.e.,(25)PrCb1=1.Thus, we only need to analyze the connectivity probability Pr{Cb3}, Pr{Cb1,b2} and Pr{Cb2,b3}. As for the connectivity probability Pr{Cb3}, it can be expressed by(26)Pr⁡Cb3=Pr⁡ρ1,Rv,d-z.Using the same derivation method asPr{Ca1,a2}, the probability Pr{Cb1,b2} that road segments b1 and b2 are connected is given by(27)Pr⁡Cb1,b2=1-e-ρ2Rv1-qb1,b2,where(28)qb1,b2=1eρ1z-Rv-1ez-Rv2/2Rv·ρ1-1.Similarly, the probabilityPrCb2,b3 that segments b2 and b3 are connected is given by(29)Pr⁡Cb2,b3=1-e-ρ1d-z1-qb2,b3,where(30)qb2,b3=1eρ2Rv-1e1-Rv/2d-z·ρ2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 2.1.1 can be expressed by(31)p211=Pr⁡Cb1·Pr⁡Cb2·Pr⁡Cb3·Pr⁡Cb1,b2·Pr⁡Cb2,b3.In Case 2.1.2, there is no vehicle driving on segmentb3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(32)p212=Pr⁡Cb1·Pr⁡Cb2·Pr⁡Cb1,b2.In Case 2.1.3, there is no vehicle driving on segmentb1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(33)p213=Pr⁡Cb2·Pr⁡Cb3·Pr⁡Cb2,b3.In Case 2.1.4, there is no vehicle driving on segmentsb1 and b3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(34)p214=Pr⁡Cb2=1.Regarding the four subcases above, the connectivity probability in Case 2.1 can be expressed by(35)p21=Pr⁡Hb1·Pr⁡Hb3·p211+Pr⁡Hb1·Pr⁡H¯b3·p212+Pr⁡H¯b1·Pr⁡Hb3·p213+Pr⁡H¯b1·Pr⁡H¯b3·p214.In Case 2.2, there is no vehicle driving on segmentb2. Only if one of the two following subcases occurs, it may be possible that the entire road segment [0,d] is connected.(i) Case 2.2.1: There are vehicles driving on segmentb1, but there is no vehicle driving on segment b3.(ii) Case 2.2.2: There is no vehicle driving on segmentb1, but there are vehicles driving on segment b3.In Case 2.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(36)p221=Pr⁡Cb1=1.In Case 2.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(37)p222=Pr⁡Cb3.Regarding the two subcases above, the connectivity probability in Case 2.2 can be expressed by(38)p22=Pr⁡Hb1·Pr⁡H¯b3·p211+Pr⁡H¯b1·Pr⁡Hb3·p222.Regarding all the analysis mentioned above in Case2, the connectivity probability of the entire road segment [0,d] in Case 2 is expressed by(39)p2=PrHb2·p21+PrH¯b2·p22.Case 3 (z∈[2Rv,d-Rv]).
In Case3, as shown in Figure 5, the entire road segment [0,d] can be divided into three subsegments c1, c2, and c3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively. The following probabilities are easy to obtain:(40)Pr⁡Hc1=1-e-ρ1z-Rv(41)Pr⁡Hc2=1-e-ρ2Rv(42)Pr⁡Hc3=1-e-ρ1d-z(43)Pr⁡H¯c1=e-ρ1z-Rv(44)Pr⁡H¯c2=e-ρ2Rv(45)Pr⁡H¯c3=e-ρ1d-zFigure 5
The situation of Case3: z∈[2Rv,d-Rv].According to whether there are vehicles driving on segmentc2, Case 3 can be divided into two subcases as follows:(i)
Case 3.1: There are vehicles driving on segmentc2.(ii)
Case 3.2: There is no vehicle driving on segmentc2.In Case 3.1, clearly, the vehicles driving on segmentc2 are connected, i.e.,(46)PrCc2=1.Further, take into account the four subcases below:(i)
Case 3.1.1: There are vehicles driving on both segmentsc1 and c3.(ii)
Case 3.1.2: There are vehicles driving on segmentc1, but there is no vehicle driving on segment c3.(iii)
Case 3.1.3: There is no vehicle driving on segmentc1, but there are vehicles driving on segment c3.(iv)
Case 3.1.4: There is no vehicle driving on both segmentsc1 and c3.In Case 3.1.1, we need to analyze the connectivity probabilitiesPrCc1, PrCc3, PrCc2,c1, and PrCc2,c3. As for connectivity probabilities PrCc1 and PrCc3, they can be expressed by(47)PrCc1=Prρ1,Rv,z-Rvand(48)PrCc3=Prρ1,Rv,d-z,respectively.Using the same derivation method asPrCa1,a2, the probability PrCc2,c1 that the segments c2 and c1 are connected can be given by(49)PrCc2,c1=1-e-ρ1z-Rv1-qc2,c1,where(50)qc2,c1=1eρ2Rv-1e1-Rv/2z-Rv·ρ2Rv-1.And the probability PrCc2,c3 that the segments c2 and c3 are connected can be given by(51)Pr⁡Cc2,c3=1-e-ρ1d-z1-qc2,c3,where(52)qc2,c3=1eρ2Rv-1e1-Rv/2d-z·ρ2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 3.1.1 can be expressed by(53)p311=PrCc1·PrCc2·PrCc3·PrCc2,c1·PrCc2,c3.In Case 3.1.2, there is no vehicle driving on segmentc3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(54)p312=PrCc1·PrCc2·PrCc2,c1.In Case 3.1.3, there is no vehicle driving on segmentc1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(55)p313=PrCc2·PrCc3·PrCc2,c3.In Case 3.1.4, there is no vehicle driving on segmentc1 and c3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(56)p314=Pr⁡Cc2=1.Regarding the four subcases above, the connectivity probability in Case 3.1 can be expressed by(57)p31=PrHc1·PrHc3·p311+PrHc1·PrH¯c3·p312+PrH¯c1·PrHc3·p313+PrH¯c1·PrH¯c3·p314.In Case 3.2, there is no vehicle driving on segmentc2. Only if one of the two following subcases occurs may it be possible that the entire road segment [0,d] is connected:(i)
Case 3.2.1: There are vehicles driving on segmentc1, but there is no vehicle driving on segment c3.(ii)
Case 3.2.2: There is no vehicle driving on segmentc1, but there are vehicles driving on segment c3.In Case 3.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(58)p321=Pr⁡Cc1.In Case 3.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(59)p322=Pr⁡Cc3.Regarding the two subcases above, the connectivity probability in Case 3.2 can be expressed by(60)p32=Pr⁡Hc1·Pr⁡H¯c3·p321+Pr⁡H¯c1·Pr⁡Hc3·p322.Regarding all the analysis mentioned above in Case3, the connectivity probability of the entire road segment [0,d] in Case 3 is expressed by(61)p3=PrHc2·p31+PrH¯c2·p32.Case 4 (z∈[d-Rv,d]).
In Case4, as shown in Figure 6, the entire road segment [0,d] can be divided into three subsegments d1, d2, and d3, denoting [0,z-Rv], [z-Rv,z], and [z,d], respectively.Figure 6
The situation of Case4: z∈[d-Rv,d].Similar to Case1, the following probabilities are easy to obtain:(62)Pr⁡Hd1=1-e-ρ1z-Rv(63)Pr⁡Hd2=1-e-ρ2Rv(64)Pr⁡Hd3=1-e-ρ1d-z(65)Pr⁡H¯d1=e-ρ1z-Rv(66)Pr⁡H¯d2=e-ρ2Rv(67)Pr⁡H¯d3=e-ρ1d-zAccording to whether there are vehicles driving on segmentd2, Case 4 can be divided into two subcases as follows:(i)
Case 4.1: There are vehicles driving on segmentd2.(ii)
Case 4.2: There is no vehicle driving on segmentd2.In Case 4.1, clearly, the vehicles driving on segmentd2 are connected, i.e.,(68)Pr⁡Cd2=1.Further, take into account the four subcases below:(i)
Case 4.1.1: There are vehicles driving on both segmentsd1 and d3.(ii)
Case 4.1.2: There are vehicles driving on segmentd1, but there is no vehicle driving on segment d3.(iii)
Case 4.1.3: There is no vehicle driving on segmentd1, but there are vehicles driving on segment d3.(iv)
Case 4.1.4: There is no vehicle driving on both segmentsd1 and d3.In Case 4.1.1, it is obvious that the vehicles driving on segmentd3 are connected, i.e.,(69)PrCd3=1.Thus, we need to analyze the connectivity probabilities PrCd1, PrCd2,d1, and PrCd3,d2. As for the connectivity probability PrCd1, it can be expressed by(70)PrCd1=Prρ1,Rv,z-Rv.Using the same derivation method asPrCa1,a2, the probability PrCd2,d1 that the segments d2 and d1 are connected can be driven by(71)PrCd2,d1=1-e-ρ1z-Rv1-qd2,d1,where(72)qd2,d1=1eρ2Rv-1e1-Rv/2z-Rv·ρ2Rv-1.And the probabilityPrCd3,d2 that the segments d3 and d2 are connected can be driven by(73)Pr⁡Cd3,d2=1-e-ρ2Rv1-qd3,d2,where(74)qd3,d2=1eρ1d-z-1ed-z2ρ1/2Rv-1.Thus, the connectivity probability of the entire road segment[0,d] in Case 4.1.1 can be expressed by(75)p411=PrCd1·PrCd2·PrCd3·PrCd2,d1·PrCd3,d2.In Case 4.1.2, there is no vehicle driving on segmentd3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(76)p412=PrCd1·PrCd2·PrCd2,d1.In Case 4.1.3, there is no vehicle driving on segmentd1. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(77)p413=Pr⁡Cd2·Pr⁡Cd3·Pr⁡Cd3,d2.In Case 4.1.4, there is no vehicle driving on segmentsd1 and d3. Therefore, the connectivity probability of the entire road segment [0,d] can be expressed by(78)p414=PrCd2=1.Regarding the four subcases above, the connectivity probability in Case 3.1 can be expressed by(79)p41=PrHd1·PrHd3·p411+PrHd1·PrH¯d3·p412+PrH¯d1·PrHd3·p413+PrH¯d1·PrH¯d3·p414.In Case 4.2, there is no vehicle driving on segmentd2. Only if one of the two following subcases occurs may it be possible that the entire road segment [0,d] is connected:(i)
Case 4.2.1: There are vehicles driving on segmentd1, but there is no vehicle driving on segment d3.(ii)
Case 4.2.2: There is no vehicle driving on segmentd1, but there are vehicles driving on segment d3.In Case 4.2.1, the connectivity probability of the entire road segment[0,d] can be calculated by(80)p421=PrCd1.In Case 4.2.2, the connectivity probability of the entire road segment[0,d] can be calculated by(81)p422=Pr⁡Cd3=1.Regarding the two subcases above, the connectivity probability in Case 3.2 can be expressed by(82)p42=PrHd1·PrH¯d3·p411+PrH¯d1·PrHd3·p422.Regarding all the analysis mentioned above in Case4, the connectivity probability of the entire road segment [0,d] in Case 4 is expressed by(83)p4=PrHd2·p41+PrH¯d2·p42.Therefore, regarding all the analysis mentioned above, the connectivity probability of the entire road segment[0,d] can be expressed by(84)p=1d·∫0Rvp1dz+∫Rv2Rvp2dz+∫2Rvd-Rvp3dz+∫d-Rvdp4dz.


## 4. Simulation Results and Discussions

In this section, we verify the accuracy and effectiveness of our proposed analytical model through simulation experiments. In addition, we analyze and discuss the impact of different key parameters on connectivity performance, including vehicle arrival rateλ, vehicle communication range Rv, the length of entire road d, vehicle normal speed v1, and safe speed v2. To perform the simulation experiments, we adopt Monte Carlo simulation method in MATLAB. For each simulation result, it is an average of 105 trials. The simulation results are very close to analytical results, which means that the analytical results are accurate and our proposed analytical model is effective.Figure7 shows the impact of accident location on connectivity probability. From the curve in traffic accident condition, we can see that the connectivity probability increases first and then decreases with different accident location, which shows that the vehicles in the middle part of the road have a greater influence on the connectivity. When the traffic accident occurs at location 0, the traffic flow in [0,d] can be same as that in normal condition. Hence, the connectivity probability with accident location 0 is equal to that with normal condition. Note that the traffic accident, which makes the vehicles slow down rather than block the traffic flow, can improve the connectivity to a certain degree.Figure 7
The impact of accident location on connectivity probability (λ=0.2 veh/s, v1=20 m/s, v2=10 m/s, Rv=300m, and d=2000 m).Figures8–11 all show the impact of vehicle arrival rate on connectivity probability. Obviously, the connectivity probability increases with the increase of vehicle arrival rate. It is because that the vehicle density can increase with the increase of arrival rate, which improves the road connectivity. What is more, from Figures 8–11, we can see that the deviation between analytical result and simulation result decreases with the increase of vehicle arrival rate. The reason is that the vehicles have a relatively large arrival randomness and inter-vehicle spacing with a small arrival rate.Figure 8
The impact of vehicle arrival rate and communication range on connectivity probability (v1=20 m/s, v2=10 m/s, and d=2000 m).Figure 9
The impact of vehicle arrival rate and length of road on connectivity probability (Rv=300 m, v1=20 m/s, and v2=10 m/s).Figure 10
The impact of vehicle arrival rate and vehicle normal speed on connectivity probability (Rv=300 m, v2=10 m/s, d=2000 m).Figure 11
The impact of vehicle arrival rate and vehicle safe speed on connectivity probability (Rv=300 m, v1=20 m/s, and d=2000 m).In addition, Figure8 shows the impact of vehicle communication range on connectivity probability. The connectivity probability increases with the increase of vehicle communication range, which is because that the two vehicles can obtain a greater opportunity to be connected with a larger communication range. Note that the connectivity probability has a larger improvement with the communication range from 200 m to 300 m, comparing with other communication range gaps. It can offer guidance to the design of signal transmitting power.Figure9 shows the impact of the length of entire road on connectivity probability. It can be seen that the connectivity probability decreases with the increase of the length of road. The reason is that, with constant probability that two adjacent vehicles are connected, the larger the length of road is, the larger the number of vehicles is, the smaller the connectivity probability of all vehicles is. Consequently, the connectivity probability decreases.Figure10 shows the impact of vehicle normal speed on connectivity probability. It can be seen that the connectivity probability decreases with the increase of vehicle normal speed. The reason is that the vehicle density in normal segment decreases with the increase of normal speed. Moreover, the normal segment occupies a relatively large proportion of the entire road. Therefore, the connectivity probability of the entire road decreases relatively obviously.Figure11 shows the impact of vehicle safe speed on connectivity probability. It can be seen that the connectivity probability decreases slightly with the increase of vehicle safe speed. The reason is the similar to the impact of normal speed on connectivity probability. However, the deceleration segment occupies a small proportion of the entire road, resulting in a slow decline in probability.


## 5. Conclusions

In this paper, we analyzed the connectivity in vehicular networks with slight traffic interference, i.e., small-scale traffic accident. An analytical model was proposed to calculate the connectivity probability of the entire road segment. The road segment can be in a highway scenario or a sparse urban scenario, which means that the slight traffic interference can only make the vehicles slow down, rather than block the traffic flow. When a traffic accident occurs, it is necessary to inform the vehicles within communication range which will pass by the accident location to slow down. Once passing by the accident location, their speeds can return to the normal value. Moreover, on the basis of the analytical model, we analyzed the impact of several key parameters on connectivity probability, including vehicle arrival rate, vehicle communication range, the length of road, vehicle normal speed, and safe speed. All analytical results are verified through simulation experiments. The simulation results are very close to analytical results, which means that the analytical results are accurate and our proposed analytical model is effective.In the future, we will consider the acceleration and deceleration process in a more practical scenario, which is much more complicated. Note that we assume the vehicle arrival follows the Poisson process; namely, intervehicle spacing follows the exponential distribution. How about the network connectivity with other intervehicle spacing distribution? In addition, in order to facilitate the analysis, the length of deceleration segment is set to the value of vehicle communication range. The different cases with different values of the length of deceleration segment are also our future work.



---

*Source: 9340239-2018-12-05.xml*