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Transportation Research Part C 16 (2008) 615–634 www.elsevier.com/locate/trc

Geometric connectivity of vehicular ad hoc networks: Analytical characterization Satish Ukkusuri a,*, Lili Du b a

b

Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, USA Received 20 August 2007; received in revised form 12 December 2007; accepted 12 December 2007

Abstract Advances in wireless communications are facilitating the development of inter-vehicle communication systems that will benefit mobility and safety objectives. Recently, these systems, referred as vehicular ad hoc networks (VANETs), are gaining significant prominence from both government agencies and private organizations. VANETs are characterized by high vehicle mobility, unexpected driver behavior and variable traffic environment which bring forth challenges to maintain good connectivity. This study considers VANETs as a nominal system with disturbance. Under the nominal system, the traffic space headway is assumed to follow an approved traffic flow distributions, such as exponential distribution. Disturbance is then used to capture a set of uncertain traffic flow events caused by driver behavior and changes in traffic flow. In addition, robustness factor is incorporated to present the impact of probabilistic disturbance events that disrupt the node connectivity. Under constant disturbance conditions, the lower bound of reachable neighbors for each vehicle to maintain a high connectivity is analytically derived. Furthermore, we obtain the relationship between the number of nodes in a VANET and the reachable neighbors under which the network is asymptotically connected. Finally, in variable disturbance situations, the interaction between robustness factor and macroscopic traffic parameters are investigated based on the simulation data. The validation results demonstrate that the proposed analytical characterization can approximate VANET connectivity very well. Our results facilitate the understanding of VANET connectivity on a freeway segment under different traffic conditions. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: VANET connectivity, the number of reachable neighbors; Information propagation; Vehicular ad hoc networks; Intelligent transportation systems; Connectivity; Reachable neighbors

1. Introduction In the last 15 years, Intelligent Transportation Systems (ITS) have been researched and deployed in the US, Europe and Asia to alleviate congestion and enhance the performance of traffic networks. With the rapid advances in wireless communication technologies, vehicles in a transportation network can seamlessly *

Corresponding author. Tel.: +1 5182766033. E-mail addresses: [email protected] (S. Ukkusuri), [email protected] (L. Du).

0968-090X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2007.12.002

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S. Ukkusuri, L. Du / Transportation Research Part C 16 (2008) 615–634

communicate to obtain information about network conditions, thereby, assisting better decision making. Previous research on this topic focuses mainly on centralized Advanced Traveler Information Systems (ATIS) such as ADVANCE (Boyce et al., 2004). However, it has been recently recognized that the decentralized ATIS systems have many advantages as compared to centralized system. Primary among them are (i) they are infrastructure-light in that they do not rely on roadside sensors and traffic management centers. Instead, they exchange and collect traffic information by inter-vehicular communication; (ii) the decentralized system is robust in emergency and disaster related situations because it provides better quality of information; and (iii) information exchange in the decentralized system can be used for other applications and file sharing such as 511 (USDOT, 2007). Among different types of decentralized ATIS systems (Mahmassani, 2001), this study investigates vehicular ad hoc networks, which consists of the equipped vehicles that ‘floats’ on the traffic stream and can communicate with one another. In this, vehicles serve as data collectors and anonymously transmit traffic and road condition information from every major road within the transportation network. Such information will provide transportation agencies with the information needed to implement active strategies to relieve traffic congestion and improve safety. The importance of these systems is not lost by different transportation agencies. Research initiatives have been conducted worldwide: such as Network on Wheels (Federal Ministry of Education and Research, 2007) and FleetNet (Franz, 2007) in Europe, Advanced Safety Vehicle Program (Ministry of Land, Infrastructure and Transport, 2007) in Japan, and the Vehicle Infrastructure Initiative (VII) by the United States Department of Transportation (FHWA, 2005). Several key factors related VII motivate us to address the research questions in this paper: First, this initiative clearly demonstrated in two prominent field tests in the Detroit area and the San Francisco Bay area that integrating communication and transportation systems benefits the mobility and safety of the transportation system. Second, VII initiative is working towards reaching a collective decision in late 2009, so as to recommend pilot deployments to the US Congress. Hence, the timing is right to address the methodological issues in this area. This study addresses one of fundamental problems in VANETs: connectivity in the physical layer of the VANET architecture, which impacts the quality of information exchange significantly in VANETs (FHWA, 2005). Reliable connectivity in VANETs will determine: how far the information can be effectively propagated? And the minimum number of equipped vehicles to have good transfer of information based on vehicle density, velocity and driver behavior. For instance, an isolated node in a traffic network cannot exchange fresh traffic information with other vehicles thereby limiting its usefulness. On the other hand, if many isolated nodes exist in VANETs, the traffic advisory system cannot produce smart travel decisions for drivers since system does not have a complete traffic data set. Similar research issues have been addressed by related works in literature (such as Jin and Recker, 2006; Wang, 2007). The answer to the questions is critical to the successful design of decentralized ATIS systems incorporating parameters related to traffic density, market penetration of equipped vehicles and driver behavior. It should be noted that the connectivity described in this paper relates to the physical layer and not the media access layer (MAC) (Ni and Chandler, 1994). As such the connectivity in the physical layer is closely related to the nodal geometric distribution and the transmission range, thus we call it the geometric connectivity. Geometric connectivity in mobile ad hoc networks (MANETs) has been previously studied by researchers such as (Bettstetter, 2004; Blough et al., 2006; Desai and Manjunath, 2002; Gupta and Kumar, 1997; Piret, 1991; Santi and Blough, 2002; Spanos and Murray, 2004; Xue and Kumar, 2004). However, the following three fundamental characteristics in vehicular ad hoc networks demonstrate it is necessary for us to consider the connectivity of VANETs: First, the vehicular movement in a VANET is restricted to a predetermined traffic network, but the mobile nodes in MANETs have multiple degrees of freedom. Second, the mobility of the nodes is affected by the traffic density, which is determined by the road capacity and the underlying driver behavior, such as unexpected acceleration or deceleration (Artimy et al., 2005a; Ukkusuri et al., 2007). It should be noted that the vehicle velocity and driver behavior influence the stability of VANETs significantly. Lastly, the connectivity of the network is influenced by factors such as environmental conditions, traffic headway and vehicle mobility (Moreno et al., 2005). These distinguishing characteristics bring forth new challenges to maintain a required connectivity for VANETs. For example, the simulation results in (Blum et al., 2004) demonstrate that VANETs experience


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rapid topology change compared to general MANETs. Scho¨nhof et al. (2006) studied the connection between the two subsequent vehicles in same direction and the message transmission delay between vehicles in opposite directions, where the space headway is assumed to be an exponential distribution, and each vehicle only has one reachable neighbor in its broadcast range. Their analytical results showed the connectivity based on onehop connection is very low. Artimy et al. (2004, 2005a,b) reported that a fixed transmission range does not adapt to the frequent topology change; Dynamic transmission range is recommended; It is observed that traffic factors such as traffic density, vehicle velocity affect the connectivity of VANET significantly. In addition, Artimy et al. (2006) estimated the lower bound of the minimum transmission range in VANETs by considering the non-homogenous vehicle distribution. They found traffic jam makes higher transmission range required. Choffnes and Bustamante (2005) conducted extensive micro-simulations incorporating car-following model to obtain empirical insights of information propagation. Zhang et al. (2005) study information propagation based on the cell transmission traffic flow model where the bi-directional movement of vehicles is modeled. Jin and Recker (2006) presented an analytical model to calculate the probability of covering each vehicle along a traffic stream. Their recursive method offers a numerical computation technique to model instantaneous propagation distance. As observed by Wang (2007), the literature on inter-vehicle communication from the transportation side is relatively scant. Even though the literature cited above characterizes information propagation in VANETs from different points of view, few of them provide a rigorous technique to evaluate the network connectivity for VANETs. In addition, the relationship between the connectivity of VANETs and traffic flow features such as congestion level and driver behavior remains unclear. 1.1. Our contribution Motivated by the above points, this study analytically characterizes the geometric connectivity of VANETs under dynamic traffic flow conditions, which is assumed to be under unstable traffic density, traffic speed and relative movement between vehicles. Our modeling approach is different with previous methods used to measure network connectivity in several ways. First, instead of assuming a static network topology, VANETs are considered as a nominal system with disturbance (the idea and term, nominal system is borrowed from robust control (Zhou and Doyle, 1998)). Classical space headway distribution and dynamic traffic flow features are integrated into modeling the nominal system and disturbance respectively. Second, robustness factor is introduced as a parameter to present the impact of disturbance on the connectivity of VANETs. We further develop an explicit model by linear regression to define the robustness factors in terms of macroscopic traffic flow parameters such as average velocity, average space headway and average space headway variance. This relationship allows us to evaluate geometric connectivity under different traffic flow conditions more accurately. Third, analytical studies proposed here provide a theoretical foundation into the relationships among VANET connectivity, individual reachable neighbors, and network size.1 The rest of this paper is organized as follows: Section 2 defines the network model used in this paper. Section 3 describes our methodology and our main results. Section 4 summarizes the insights obtained and identifies future work. Throughout the paper, connectivity is synonymous with geometric connectivity as defined in Artimy et al., 2004; Santi and Blough, 2002. 2. Network model and assumptions To specify the VANETs of interest, we first describe the preliminary network platform and basic topology model. Then, we emphasize the interaction between the vehicular mobility and the space headway distribution. Last, the relationship between the transmission range and the reachable neighbors is extensively discussed. These assumptions and preliminary results provide foundation for our analytical work. The notation used in our modeling approach and the different probabilistic events are defined below: 1

Network size: the number of nodes in a network.

618


i; j; k: index for vehicles. L: the length of the freeway segment. n: the number of nodes in a VANET. r: vehicular transmission range. x: space headway. d: robustness factor. k: the traffic density of the instrumented vehicles. K c : the average number of neighbors for each vehicle, K c ¼ r k. pi : the probability that node i is connected to its forward network. Aik : the event that node i covers exactly k forward neighbors with given transmission range under nominal traffic state, pðAik Þ ¼ pik . Dk : the disturbances that interrupt T the link connecting to a node’s kth forward neighbor. P D is the probability of the event ð ki¼1 Di Þ.

2.1. Preliminary road network and communication platform In our work, VANETs is considered as a wireless network floating on the road network. Therefore, we first scope our study by specifying the assumptions for the road network and the communication platform. 2.1.1. Freeway segment For simplicity, we study a fleet of instrumented vehicles on a freeway segment without traffic control infrastructure. The freeway segment is assumed to be uni-directional with more than one lanes. Vehicles are homogenously distributed on the road. We number the space headway between consecutive vehicles from left to right, and they are denoted by xi ; i ¼ 1; . . . ; n (refer to Fig. 1). 2.1.2. Communication connections Since vehicular movement is limited to the road, we assume that the transmission range is much larger than the width of roadway. Consequently, the communication connections between vehicles mainly depend on the distance between vehicles along the road (parallel to the road line), and the distance between vehicles in the direction vertical to the road line is ignored (Vural and Ekici, 2007; Wu et al., 2004). Based on the assumptions above, VANETs on the freeway segment are modeled akin to a one dimensional network. The vehicles on a straight road are assumed to be stochastic and follows a probability distribution. VANETs at intersections are not within the scope of this study. They will be investigated in our future work. 2.1.3. DSRC communication channel We assume the nodes (vehicles) in our model broadcast information based on Dedicated Short Range Communications (DSRC) channel, which is the accepted standard for vehicle-to-vehicle and infrastructure-to-vehicle communication in the United States (based on the VII initiative). Compared to other general wireless communications such as cellular phone, FM radio and satellite, DSRC uses the directional antenna and has the relatively smaller transmission range (61 km) (Lee Armstrong, 2007; Yin et al., 2004). 2.2. Basic topology model Following the assumptions of the road network and the communication network, we discuss the basic topology model of VANETs by graph theory. At each time instant, the topology of VANETs can be represented as a graph GðV ; EÞ, where a set of vertices V represents vehicles, and a set of edges E represents communication

Kc x1

x2

x3

x4

…

xi

…

Fig. 1. K c -neighbors one dimensional vehicle ad hoc network.


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links. A communication link (eij ) exists if and only if the Euclidean distance between vehicles i and j is less than or equal to the shortest transmission range between them. (Note that in this study we ignore the influence of information routing protocols on communication connection between vehicles (Ni and Chandler, 1994).) Node i is called neighbor of node j and vice versa. The definition of communication link implies that only undirected edges are considered in this study. The degree of a node is the number nodes which are incident with this node. We further define a link as the forward-link of node i if this link is starting from a node i and ending at any node j which is in front of node i. Correspondingly, the forward degree of a node is the number of forward-links of this node, and hence is the number of forward reachable neighbors. In the study, unless otherwise mentioned, reachable neighbors indicate forward reachable neighbors. A path between two nodes is a list of consecutive links connecting the two nodes. A graph is said to be connected if at least one path exists between any two nodes in the graph. Otherwise, the graph is disconnected. A graph is k-edge-connected if and only if the removal of k 1 edges cannot disconnect the graph (Bettstetter, 2002; Chartrand and Zhang, 2005). 2.3. Unstable space headway distribution and robustness factor Before obtaining the analytical characterization of connectivity, it is important to discuss the space headway of vehicles since this is a key input in the derivation. Empirical observations in traffic science (Arasanl and Koshy, 2005; Liu et al., 2003) show that under free flow conditions, the time headway obeys an exponential distribution. In addition, as the velocity of vehicles in traffic flow is uniformly distributed, space headway obeys the same distribution with time headway. In previous work for VANETs, space headway is usually modeled as exponential distribution (Saito, 2006; Scho¨nhof et al., 2006; Tsugawa and Kato, 2003). However, in this study, space headway is considered as an unstable distribution. Our idea is supported by the following two reasons: (1) Vehicular mobility will change nodal distribution frequently. In other words, even though the space headway is initially exponential distribution, the unexpected driver behaviors transform the distribution from one exponential distribution to another or change the distribution to a non-exponential distribution and (2) the simulation results for VANETs show that the lifetime of communication links in VANETs extremely short (Artimy et al., 2004; Blum et al., 2004). This means that the connection between vehicles are interrupted frequently. These phenomena are ignored in most of the previous work. In view of the above points, this study seeks to evaluate the connectivity of VANETs robustly by introducing the disturbance and the robustness factor to the connectivity calculation procedure. Specially, VANETs is considered as a nominal system with disturbance (see Fig. 2). Under the nominal system, we consider space headway obeys some approved distribution in traffic flow. For example exponential distribution is used here, it therefore is under free flow condition. However, the actual state of traffic flow is impacted by disturbance such as unexpected acceleration, deceleration and lane changing. These uncertain events change the vehicular distribution and interrupt the connections between vehicles randomly. Robustness factor is introduced to

Disturbance: uncertain events

Nominal system: exponential distribution

Fig. 2. VANETs consists of nominal system and disturbance.

620


present the impact of disturbance on connectivity of VANETs. In addition, robustness factor here is modeled as a function of macroscopic traffic features by this study. This function will be constructed and validated by the simulation model. 2.4. Transmission range vs. reachable neighbors To calculate the geometric connectivity of a VANET, we first derive the relationship between the transmission range and the number of reachable neighbors under nominal system (free flow). Assuming space headway, xi ; i ¼ 1; . . . ; n follows independent identical exponential distribution (Saito, 2006; Scho¨nhof et al., 2006; Tsugawa and Kato, 2003), the probability that each vehicle has exactly K c neighbors is given by ! ! K Kc K c þ1 X X eðkrÞ ðkrÞ c :¼ QðrÞ: ð1Þ p xi 6 r p xi 6 r ¼ Kc! i¼1 i¼1 To find the optimal radius r which maximizes Q for a given K c we set oQ ðK 1Þ K ¼ K c kðk rÞ c eðkrÞ k ðk rÞ c eðkrÞ ¼ 0: or

ð2Þ

Solving Eq. (2), we get r ¼ Kkc . Substituting r by r in Eq. (1), the maximum value of Eq. (1) is eðK c Þ K Kc c : K c! Studying Eqs. (1) and (3), we find:

ð3Þ

Qmax ðK c Þ ¼

Qmax ðrÞ, the maximum probabilities to maintain exactly K c neighbors, is low for all K c ’s. For example, as K c ¼ 2, the probability that a vehicle will cover two neighbors when optimal broadcast radius is only 0.271. Additionally, as K c increases, QðrÞ decreases. Fig. 3 shows this observation. When r ¼ r , the probability that at least K c , K c þ 1, 2K c reachable neighbors are present for each vehicle can be calculated by the function: ! Kc X pðAt least K c reachable neighborsÞ ¼ p xi < r ð4Þ i¼1

¼1

K c 1 X i¼0

eK c K ic : i!

ð5Þ

0.4

0.35

Qmax (K ) c

0.3

0.25

0.2

0.15

0.1

0.05

0

5

10

15

20

Kc

Fig. 3. Probability that exact K c neighbors are reachable to an individual vehicle.


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0.8 at least K +1 neighbors c

At lest K neighbors

0.7

c

at least 2K neighbors c

Probability

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

20

Kc

Fig. 4. Probability that there are at least K c , K c þ 1 and 2K c reachable neighbors for an individual vehicle.

Fig. 4 shows the probability that a vehicle has at least K c or K c þ 1 neighbors in its broadcast range is nearly 0.5, which is relatively high. However, the probability that it has more than 2K c reachable neighbors is less than 0.3 when K c > 2. Thus, given r , it is reasonable to ignore the situation that more than 2K c neighbors are reachable by an individual vehicle. Based on the observations above, we further add two more assumptions to our study: (1) Each vehicle adjusts the transmission range according to the expression r ¼ Kkc ; (2) Every vehicle on an average covers the same number of reachable neighbors, say K c , therefore, we are interested in K c -edge-connection. 3. Methodology to measure the geometric connectivity of VANETs Now we present our framework to investigate the connectivity of VANETs on a freeway segment. A closed form expression to estimate the connectivity of VANETs on the freeway segment is derived first. Based on this analytical expression we further discuss several propositions about the network connectivity with a constant robustness factor. Next, based on the simulation data we build an analytical model to present the relationship between the robustness factor and the macroscopic traffic data such as average speed, average space headway and its standard variance. Lastly, integrating the function of robustness factor to our connectivity approximation we provide the closed form expression to robustly approximate VANET connectivity under various traffic flow conditions. 3.1. Connectivity calculation We begin from the methodology of connectivity calculation. The idea used to calculate geometric connectivity of a VANET on the freeway segment is based on the following proposition: Proposition 1. For a one dimensional network, the network is connected if and only if each node (except the last node) of the network connects to at least one of its forward nodes. The probability that a node connects to its forward network (see Fig. 1) is non-zero if and only if the node is connected to at least one of its forward neighbors. However, calculating this probability incorporating the defined disturbance is non-trivial. Hence, we consider its complementary event (i.e. the node does not connect to its forward network). This event is equal to the union of the events that node j covers exactly k, ðk ¼ 0; . . . 1Þ forward neighbors, but the disturbance events happen to all the neighbors simultaneously. As we already discussed in Section 2, when r ¼ r ¼ Kkc , the probability that an individual vehicle covers more

622


than 2K c forward neighbors is very low. Thus, we only need to consider the cases that node j has 0; 1; 2; . . . ; K c ; K c þ 1; K c þ 2; . . . ; 2K c forward neighbors and ignore the rest. These events are mutually exclusive, so pj , the probability that node j is disconnected to its forward network is given by !!! !!! !! 2K c 2K c 2K c 1 k k k \ \ \ \ \ \ [ [ X X : pj ¼ p Ajk Di Ajk Di p Ajk Di pjk P D ; ¼p ¼ ¼ k¼0

k¼0

i¼1

i¼1

k¼0

i¼1

k¼0

ð6Þ where Ajk represents the event that a node is reachable to exact k forward neighbor under the nominal traffic state. Di ; i ¼ 1; . . . ; k represent the event thatTa node is separated with its ith forward neighbor by the disturbance. Di ; i ¼ 1; . . . k are not independent; ð ki¼1 Di Þ denotes the event that Tk disturbance separates this node from this forward neighbor, and P denotes the probability of the event ð D i¼1 Di Þ. Here, we assume the event Tk ðAjk Þ and the event ð i¼1 Di Þ are independent. Eq. (6) shows whether a node is connected to its forward network depends on two factors. One is its transmission range, and the other is the disturbance on traffic flow. If the traffic flow is under the nominal state and no disturbance exists, then pðD0 Þ ¼ 1. Consequently, the probability that a node is disconnected to the network is reduced to the probability that the node covers zero neighbor in its transmission range. In addition, Eq. (6) is reduced to pðAj0 \ D0 Þ ¼ pðAj0 Þ, which is only related to the transmission range and the vehicular distribution under nominal traffic state. Furthermore, the event that an individual node is separated from its forward network is almost independent from node to node with assumption n 1 and r L (Bettstetter, 2004). In other words, for any given node i and node j, we suppose the probability of node i separating from its forward network is independent of the probability of node j separating from its forward network. Consequently, we obtain the probability that the vehicular ad hoc network on a freeway segment is ! 2K c n n Y X Y Pc ¼ 1 pj ¼ 1 pk P D : ð7Þ j¼1

j¼1

k¼0

p ffiffiffiffiffiffi k c To derive more comprehensive analytical result from Eq. (8), we let d ¼ max2K k¼1 f P D g. Substituting p k by Eq. (1), we obtain the lower bound for the connectivity of VANETs on freeway segments: ! 2K c n Y X eK c P D K kc Pc ¼ 1 ð8Þ k! k¼0 j¼1 ! k 2K c n X Y eK c ðdK c Þ 1 : ð9Þ P k! k¼0 j¼1 The lower bound of the connectivity is only related to the number of reachable neighbors K c and the parameter d. Referring to the definition of d, it is easy to get 0 6 d 6 1. Moreover, studying the expression of the lower bound of the connectivity, we find that to maintain the required connectivity, a smaller d means that a smaller K c is required. In other words, parameter d reflects the extent that the disturbance impacts VANET connectivity on freeway segments. We therefore define d as the robustness factor to balance the negative influence from unexpected driver behavior on connectivity. 3.2. Discussion on constant robustness factor Initially assuming robustness factor to be a constant value, we estimate the connectivity of VANETs by its lower bound in Eq. (9). Our discussion here focuses on exploring how the number of vehicles in a VANET influences its connectivity. We first perform a numerical experiment, where d ¼ 0:05, the connectivity of networks with nodes varying from 20 to 300 under different values of reachable neighbors are estimated by Eq. (9). The direct observation from the results in Fig. 5 is that P c increases with K c . This is consistent with our intuition: the higher the number of reachable neighbors for each vehicle, the higher is the network connectiv-


623

1 n=20

0.9

n=20:5:300

0.8 0.7

Pc

0.6 0.5 n=300

0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

12

14

16

Kc

Fig. 5. The connectivity of the VANETs when r ¼ r ; d ¼ 0:05.

ity. Further studying the connectivity estimated by its lower bound in Eq. (9), we develop the following proposition below. Proposition 2. The marginal effect of K c on the connectivity probability, P c decreases after reaches zero, assuming that the probability of disturbance event is constant.

K c >logn 1d ,

and finally

Proof. In the limiting case, there are n neighbors covered by a vehicle then the following derivation for Eq. (9) holds: ! !n n n n Y X X n eK c ðd K c Þk ðd K c Þk K c Pc P 1 P 1 eK c edK c P 1e k! k! j¼1 k¼0 k¼0 n

¼ ð1 eK c ðd1Þ Þ :¼ b:

ð10Þ

Letting the lower bound of P c ¼ b, and assuming K c is a continuous variable and taking the derivative of Eq. (10) with respect to K c , we get ðn1Þ b0 ðK c Þ ¼ nð1 dÞeK c ðd1Þ 1 eðK c ðd1ÞÞ > 0; ð11Þ b00 ðK c Þ ¼ nð1 dÞ2 eK c ðd1Þ ð1 eðK c ðd1ÞÞ Þðn2Þ ðneðK c ðd1ÞÞ 1Þ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} (

:A

>0

< 0; if K c > A ¼ neðK c ðd1ÞÞ 1 P 0; o:w:

ð12Þ

log n ; 1d

ð13Þ

Therefore, we obtain the following conclusions: n 1. When K c < log , b0 ðK c Þ > 0 and b00 ðK c Þ > 0, so b0 ðK c Þ increases as K c increases. 1d n 2. When K c ¼ log , b0 ðK c Þ > 0 and b00 ðK c Þ ¼ 0. b0 ðK c Þ reaches the maximum value. 1d n 3. When K c > log , b0 ðK c Þ > 0 but b00 ðK c Þ < 0 thus b0 ðK c Þ decreases as K c increases. In addition, 1d 0 limK c !1 b ¼ 0. Hence, the improvement of the lower bound of connectivity, b by the number of reachable number, K c will finally reach a constant value.

So far we proved the proposition that the improvement of the connectivity of VANETs (specifically, the lower bound of the connectivity) in terms of the number of reachable neighbors, K c will be a constant value after a certain value of K c . h

624


Since K c ¼ kr in this study, if the traffic density is constant, we can derive the similar relationship between the transmission range and the network connectivity. Santi and Blough (2002) obtained a similar result from their simulation model, although they considered two dimensional mobile ad hoc networks. Additionally, approximating the connectivity of VANETs by its lower bound, we obtain the following conclusion based on the result in Proposition 2. Corollary 1. Given that the space headway obeys exponential distribution, the lower bound of the number of pffiffi lnð1 n bÞ reachable neighbors, K c for the required connectivity probability b is d1 and the corresponding adaptive broadcast range is r ¼ K c =k. The lower bound of the number of reachable neighbors, K c depends only on the number of nodes and the probability of disturbance in the network. Rearranging Eq. (10), we get the lower bound of K c for the required connectivity pffiffiffi lnð1 n bÞ : Kc ¼ d 1

ð14Þ

The immediate problem is what if rmax < r in the light traffic situations? In order to solve this problem, let rmax ¼ r ; 0 < < 1, then K c ðrmax Þ ¼ K c ðr Þ ¼ K c , so the connectivity under these special traffic situation den grades to P c ¼ ð1 eK c ðd1Þ Þ . In the limiting situation that the traffic is very sparse and ¼ 0, the connectivity probability goes to zero at any time instant even though vehicles apply maximum broadcast range. From this point of view, VANETs are really a large scale Delay Tolerant Networks (DTN) which may not always be connected (Zheng, 2006). Recent studies show high vehicular mobility will improve the information propagation along VANETs, particularly in sparse traffic conditions (Grossglauser and Tse, 2002; Petur and Iyer, 2006). Furthermore, by asymptotic calculation in Eq. (10) we obtain additional insight into the relationship between the number of reachable neighbors and the network connectivity in limiting situation. Corollary 2. If the reachable neighbors of each vehicle is equal to Hðlog nn Þ; n P 2, the VANET with n nodes is asymptotically connected. n

n Proof. If we let K c ¼ f ðnÞ ¼ log ; n P 2, as n goes to infinite, the limit of Eq. (10): 1d 1 1 ðn1Þ nn ðn1Þ n 1 1 n ¼ lim ¼ 1: lim ð1 ef ðnÞðd1Þ Þn ¼ lim 1 n n!1 n!1 n!1 e n

ð15Þ

Thus, the network is asymptotically connected when K c ¼ Hðlog nn Þ; n P 2. h In practice, it is impossible to have infinite vehicles on a road. However, the above corollary still demonstrates a rule to choose the number of reachable neighbors for VANETs on freeway segments so that they reach a high connectivity level. In addition, using K c ¼ rk, we can get the practical transmission range for the VANETs on freeway segments for a high connectivity level. Moreover, based on the asymptotic result we explore the relationship between the number of vehicles and the number of reachable neighbors in a VANET provided the network is asymptotically connected. This result is presented next. Corollary 3. For VANETs on freeway segments, the number of reachable neighbors to obtain the required connectivity does not significantly increases with the number of vehicles in VANETs. Proof. We take the derivative of f ðnÞ respect to n: oK c n > 0; ¼ n ð1 dÞ on oK 0c n K 00c ðnÞ ¼ ¼ 2 < 0: n ð1 dÞ on K 0c ðnÞ ¼

ð16Þ ð17Þ


625

Then, 1. Since K 0c ðnÞ > 0, K c increases as n increases. This shows that as the number of vehicles in a VANET increases, each vehicle should improve its own connectivity to the network so that the connectivity of the whole network can stay at the initial level. 2. Since K 00c ðnÞ < 0, K 0c ðnÞ decreases as n increases. In addition, limn!1 K 0c ¼ 0, so the curve of K c vs. n will finally move into a flat region as n increases. In other words, K c is not sensitive to n. Therefore, if the disturbance is constant, even though large networks require individual vehicle to reach more neighbors than small networks, the difference is not significant. h The similar result is also presented by the simulation model in (Blough et al., 2006), where the values of n from 10 to 1000 are considered and network nodes are assumed randomly uniformly distributed in two dimensions. For each value of n, 10000 random nodes placements are generated to evaluate the connectivity. Their simulation results show K c is loosely dependent on n (e.g. K c ¼ 9 for n in range 50–500). So far, based on our closed form connectivity calculations, we thoroughly investigate the relationship among VANET connectivity, the number of vehicles, transmission range and the number of reachable neighbors. Most of the analytical results coincide with our intuition. 3.3. Robustness factor vs. traffic flow In the above discussion, the connectivity of VANETs is approximated by its lower bound, moreover, the robustness factor, d is assumed to be a constant value. However, from the definition of robustness factor, we can see it is closely related to many uncertain events in traffic flow, such as driver behavior, acceleration, density of vehicles and lane changing behavior. In addition, it reflects the extent that the space headway distribution deviates from the exponential distribution under various traffic flow conditions. Therefore, in order to accurately approximate connectivity of VANETs by Eq. (10), robustness factor should vary its value as traffic conditions change. In addition, it is necessary to explore an explicit relationship between the robustness factor and traffic flow features. The complexity of the problem however, makes it difficult to obtain a generalized expression of the robustness factor. In our study, we will employ data analysis techniques to characterize the causal parameters for robustness factor. Data is collected from simulation models based on MITSIMLab, a well-tested dynamic traffic simulation framework. It was developed at the MIT Intelligent Transportation Systems (ITS) Program. MITSIMLab has been widely used among transportation research community. In addition, it has an embedded car-following and lane changing models which have been well calibrated. 3.3.1. Simulation setup and experiment design The traffic simulation is setup for a one-way freeway segment in a small network. The segment is about 10 000 ft long and has three lanes. Vehicles have different speed at each time instant but their speed limit is 65 mph. Each simulation is conducted from 8:00 am 8:25 am under the constant traffic demand. We change the demand from 375 veh/h to 1125 veh/h with step size 25 veh/h to get the traffic data under different traffic flow conditions. For each demand, the simulation is run for 10 times. The data collected in the each run includes time (s), vehicle identification number, vehicle position (ft), lane identification number, vehicle speed (mph). We disregard the first 10 min of the simulation as the warm-up time and use the remaining 15 min data for our analysis. Based on the transportation data, we calculate the traffic flow parameters shown in Table 1. Through the traffic simulation data, we first demonstrate our statement that it is inaccurate to approximate connectivity of VANETs with a fixed space headway distribution. Following that, we investigate the explicit relationship between the robustness factor and traffic flow parameters. The experiment is designed as follows. 1. For each traffic demand, i, at each time instant, t ¼ 1; . . . ; T , we count the number of vehicles on the road, nit , and the traffic density, kit ¼ nit =L. For a given K c , we set the transmission range of an individual vehicle by r ¼ K c =kit . We then check if the VANET is connected at each time instant. If the VANET is connected, set cnt ¼P 1. Otherwise, set cnt ¼ 0. The connectivity of the VANET in the study time interval, T, is equal to consi ¼ t cnt =T .

626


Table 1 Defining traffic flow parameters Parameters Demand

Description Traffic demand over the segment

Definition Given parameter in simulation

AvgSpeed AvgSpeed l

Average speed of the vehicles over the segment under certain demand Average speed of the vehicles over the lane l; l ¼ 0; 1; 2 under certain demand Average standard variance of speed over the segment under certain demand Average standard variance of speed over the lane l; l ¼ 0; 1; 2 under certain demand Average space headway over the segment under certain demand Average space headway over the lane l; l ¼ 0; 1; 2 under certain demand Average standard variance of space headway over the segment under certain demand Average standard variance of space headway over lane l; l ¼ 0; 1; 2 under certain demand Avgj ðAvgt ðAvgv ðHeadwaySTDVar lv ÞÞÞ

Avgj ðAvgt ðAvgv ðspeed jtv ÞÞÞ Avgj ðAvgt ðAvgv ðspeed ljtv ÞÞÞ

SpeedSTDVar SpeedSTDVar l AvgHeadway AvgHeadway l HeadwaySTDVar HeadwaySTDVar l

Avgj ðAvgt ðSpeedSTDVarjt ÞÞ Avgj ðAvgt ðSpeedSTDVar ljt ÞÞ Avgj ðAvgt ðAvgv ðHeadway jtv ÞÞÞ Avgj ðAvgt ðAvgv ðHeadway ljtv ÞÞÞ Avgj ðAvgt ðAvgv ðHeadwaySTDVarv ÞÞÞ

j: index of run. v: index of individual vehicle. t: index of time instant. Avgindex ðX Þ: average value of X over the index.

2. For P each P traffic demand, i, under each run, j ¼ 1; . . . ; 10, we calculate the average traffic density ki ¼ j t kijt =10T . Supposing the space headway is exponential distribution, and no disturbance changes the distribution, we calculate the analytical connectivity of the VANET, conexpi (refer to Eq. (22) in Bettstetter, 2004). 3. For P each P traffic demand, i, for each run, j ¼ 1; . . . ; 10, we calculate the average traffic density ki ¼ j t kijt =10T . We still have K c ¼ ki r. Supposing the space headway is not exponential distribution, we try different d 2 ½0; 1 so that the connectivity denoted by cond i (refer to Eq. (10)) is close enough to consi . Pick the best d i which is equal to inf fd; jcond i consi j < , for any given }. 4. Repeat steps 1, 2, and 3 to obtain the robustness factor d i for different traffic demand. 5. For each demand, i, we calculate the corresponding traffic flow parameters. 6. Based on the regression model, we finally obtain the explicit relationship between the robustness factor d and the traffic flow parameters.

3.3.2. Differences between the simulation connectivity and the analytical connectivity with exponential distribution assumption Comparing VANET connectivity in the simulation, consi to the analytical connectivity, conexpi , where space headway obeys exponential distribution, we obtain the results shown in Fig. 6. It is observed that conexpi is higher than consi ; i.e. traffic flow becomes more and more congested, the gap becomes larger. The presented graphs in Fig. 6 further prove that the mobility of vehicles will change the space headway away from an exponential distribution. 3.3.3. Regression models Encouraged by the above observations, this study considers that a more accurate method to approximate connectivity of VANETs must integrate the dynamic traffic features. Based on the data we have in Table 1, we investigate the relationship between the robustness factor and traffic flow parameters. Best subset algorithm is conducted to evaluate the prediction capability of different predictor sets. The results are summarized in Table 2. The corresponding best model and their characteristics of the fitted model are also reported. In Table 2, the two models composed of the predictors in the first set is relative better than other models. The result of best subset regression based on the first predictor set are shown in Fig. 7. For these two models, Their higher R-sq and R-sq(adj) values demonstrate SSE2 is smaller. In addition, small C p values mean these 2

SSE ¼

Pn

i¼1 ðY i

Yb i Þ2 ¼

Pn

2 i¼1 ei ,

where Y i is the observed value and Y^ i is the predicted value.


627

Fig. 6. Difference between the cons and conexp under different demand.

Table 2 The best model for the possible predictors sets Predictors

Functions

R-sq (%)

R-sqðadjÞ (%)

1. demand, AvgSpeed, SpeedSTDVar, AvgHeadway, HeadwaySTDVar

d ¼ 2:86 0:102AvgSpeed 0:00164AvgHeadway þ 0:00322HeadwaySTDVar d ¼ 4:47 0:000193demand 0:139AvgSpeed 0:103SpeedSTDVar 0:00139AvgHeadway þ 0:00288HeadwaySTDVar d ¼ 0:391 þ 0:000337demand 0:00170AvgHeadwayMean þ 0:00330AvgHeadwaySTDVar

94.6

94.0

4.6

95.1

94.1

6

2. demand, RelativeSpeed, 91.5 AvgHeadway, HeadwaySTDVar 3. demand, AvgSpeed i, SpeedSTDVar i, d ¼ 0:24 þ 0:000226demand þ 0:190AvgSpeedMean 0 85.4 AvgHeadway i, HeadwaySTDVar i, 0:045AvgSpeedSTDVar 0 0:161AvgSpeedMean 1 i ¼ 0; 1; 2 0:016AvgSpeedSTDVar 1 þ 0:100AvgSpeedSTDVar 2 þ 0:00184AvgHeadwayMean 0 þ 0:00391AvgHeadwaySTDVar 0 0:00225AvgHeadwaySTDVar 1 0:00176AvgHeadwayMean 2 0:00201AvgHeadwaySTDVar 2 d ¼ 0:842 þ 0:000220demand 0:148RelativeSpeed 84.4 4. demand, RelativeSpeed, þ 0:00171AvgHeadwayMean 0 þ 0:00247AvgHeadwaySTDVar 0 AvgHeadway i, HeadwaySTDVar i, i ¼ 0; 1; 2 0:00163AvgheadwayMean 2 0:00267AvgHeadwaySTDVar 2

90.7

3.1

78.8

84.1

Cp

10

5.5

two subsets has small Mean Square Error.3 The values of C p are also close to the corresponding number of applied predictors, so the bias of the regression models are small. Clearly, just observing the R-sq and C p , it is hard to say which model is better. However, referring to hypothesis test about the coefficient of demand and SpeedSTDVar in the second model, the results (in http://www.rpi.edu/~dul/dissertation/chapter3) demonstrate the coefficient of these two predictors are zeros. By eliminating these two predictors we arrive at the final model shown as model 1: d ¼ 2:86 0:102AvgSpeed 0:00164AvgHeadway þ 0:00322HeadwaySTDVar:

ð18Þ

The corresponding fitted characteristics are reported in Table 3. In order to present the adequacy of our model, in the followed graphs, we perform a number of refined diagnostics as below: 3

MSE ¼ SSE n2.

628


Fig. 7. Result of best subset algorithm. Table 3 Regression Analysis: d versus AvgSpeed, AvgHeadway, HeadwaySTDVar Predictor

Coefficient

SE Coefficient

T

P

VIF

Constant AvgSpeed AvgHeadway HeadwaySTDVar

2.8560 0.10197 0.0016397 0.0032199

0.3669 0.01400 0.0001853 0.0003764

7.78 7.28 8.85 8.56

0.000 0.000 0.000 0.000

14.476 122.990 151.838

S = 0.0159208 PRESS = 0.00913941

R-Sq = 94.6%

R-Sq(adj) = 94.0% R-Sq(pred) = 92.76%

significance of the regression relationship; normality of error terms; non-independence of error terms; constancy of error terms; multicollinearity analysis; validation of predictive capability.

3.3.4. Testing the significance of the regression relationship using F-test First of all, we test whether the robustness factor is related to average speed, average space headway, and the space headway standard variance. The hypothesis test is below: Ho : b1 ¼ 0; b2 ¼ 0; and b3 ¼ 0: Ha : not all b0 ; b1 ; and b2 equal zero: Table 4 ANOVA table Source

DF

SS

MS

F

P

Regression Residual error Total

3 27 30

0.119369 0.006844 0.126213

0.039790 0.000253

156.98

0.000


629

Fig. 8. Normality test.

The statistics, F ¼ MSR is used for this test. The P-value of the test results shown in MiniTab output (see MSE Table 4) is 0.000. For the test risk a ¼ 0:05 (a is greater than the P-value), we conclude Ha . Therefore, Robustness factor is related to average speed, average space headway, and the space headway standard variance. 3.3.5. Normality of error terms The residual in our first-order regression model is assumed to be normal distribution. Using Ryan–Joiner test, we evaluate our assumption here. The hypothesis test is Ho : the residual meets normal distribution: Ha : the residual does not meet normal distribution: The P-value of the test results is 0.1 (see Fig. 8). For general test size a ¼ 0:05 < P-value, we accept H 0 that the residual meets normal distribution. 3.3.6. Non-independence of error terms Our data is obtained in a sequence of increasing traffic demands, so it is necessary to prepare a sequence plot of the residuals. This plot will help us to see if there is any correlation between errors terms in the sequence. Fig. 9 shows that the residuals fluctuate in a random pattern around the base line (residual = 0), thus we can confirm the error terms are independent. 3.3.7. Constancy of error terms We also need to test whether the error term has constant variance, which is one of our assumptions when we build the regression model. Plotting the residual against the predictors, we obtain Fig. 10. The residuals are evenly distributed around the base line (residual = 0) and there is no systematic tendencies to be positive or negative. Therefore, there is no indication of non-constancy of the error variance. 3.3.8. Multicollinearity analysis Multicollinearity is another important aspect that should be detected in our model since the relations between predictors might influence the adequacy of the regression model. Variance Inflation Factor (VIF) is a formal method widely used to detect the presence of multicollinearity. The maximum VIF value in Table 3 is greater than 10.4 This indicates that multicollinearity may be unduly influencing the least square estimation. However, if we remove any of the predictors, the R-sq and R-sq(adj) will decrease significantly (see 4

Usually, the maximum VIF greater than 10 or the average VIF greater than 1 indicates the presence of multicollinearity.

630


Fig. 9. Residual sequence.

Fig. 10. Scatter plot of residual vs. predictors.

Fig. 7). In addition, three predictors is not a big predictor set, and the coefficient sign of each predictor is reasonable. Therefore, we retain all the predictors in the selected regression model. 3.3.9. Validation of predictive capability Finally, we validate the prediction capability of our model by another data set. The corresponding data and estimated robustness factors from the selected regression model are presented in Table 5. The mean of the squared prediction errors (MSPR) is Pn ðd i dî Þ2 MSPR ¼ i¼1 ¼ 0:00293: ð19Þ n Table 5 Validation data set d

AvgSpeed

AvgHeadway

HeadwaySTDVar

d^

d^ d

0.081 0.16 0.167 0.18 0.177

30.55903109 29.90804692 29.76190095 29.53388976 29.08284648

756.1913077 615.1210434 525.1201003 459.6830272 406.4751723

493.5236557 426.9268335 371.6919924 330.5125446 301.7165534

0.060412604 0.169426388 0.162393892 0.161656940 0.201436243

0.000423841 8.88568E05 2.12162E05 0.000336468 0.000597130


631

MSPR based on the entire new data set is usually greater than MSE (mean square error) based on the modelbuilding data set. In our case, MSPR (= 0.00293) is slightly larger than MSE (= 0.00253). This fact implies that the MSE is a reasonable valid indicator of the predictive ability of the fitted regression model. More evidence of the internal validity of the regression model is shown in Table 3. The reasonable closeness of PRESS (prediction sum of squares = 0.009139) and SSE (sum of squared error = 0.006844) supports the validity of the regression model and the MSE as an indictor of the predictive capability. In addition, R-sq(pred) also show the good predictive capability of this regression model. These validation results support the appropriateness of the regression model selected. For a more rigorous validation, a larger data set is needed. For example, re-estimation can be conducted to obtain a new regression model based on the new data set. We then compare the new estimated regression coefficients and various characteristics of the fitted model with the corresponding values on earlier data set. If they are consistent, then they provide strong support that the selected model is applicable under broader circumstances. 3.3.10. Connectivity with dynamic traffic flow So far, we already develop a closed form expression to estimate robustness factor by average traffic speed, average space headway and standard variance of space headway. Integrating this expression (18) into Eq. (10), we can obtain the connectivity approximation: : P c ¼ ð1 eK c ðð2:860:102AvgSpeed0:00164AvgHeadwayþ0:00322HeadwaySTDVarÞ1Þ Þn ð20Þ Eq. (20) is more comprehensive to evaluate VANET connectivity since it includes the macroscopic traffic parameters: average speed, average space headway and the variance of space headway. These parameters not only characterize the traffic flow, but also reflect the vehicular mobility. In other words, Eq. (20) estimates VANET connectivity in view of both communication and transportation features. Based on the new simulation data, we compare the connectivity measured by simulation data ðconsÞ, the analytical expression with exponential distribution assumption ðconexpÞ and the closed form expression taking account of the robustness factor in Eq. (20) ðcondÞ. The results shown in Fig. 11 provide the following observations: The connectivity calculated with assumption that the space headway stably obeys exponential distribution usually overestimates the real VANET connectivity. This implies the existing of the disturbance which separates the connection between vehicles in VANET. Our analytical expression incorporating the dynamic features of traffic flow can approximate the connectivity of VANETs more accurately.

Fig. 11. Comparison of cons, conexp, and cond.

632


4. Conclusion and future work This paper discusses the connectivity of VANETs on a freeway segment. VANETs is considered as a system composed of a nominal system, in which space headway is exponential distribution, and disturbance resulting from vehicular mobility and unexpected driver behavior. The probability of maintaining a connection between any node pair is used to measure the connectivity. Robustness factor is introduced to model the effect of disturbance on VANET connectivity. This method helps us to approximate connectivity of VANETs under dynamic traffic flow. An analytical characterization of the connectivity of VANETs on freeway segments is derived. The derived expression is a function of the reachable neighbors and the robustness factor. In addition, the number of the reachable neighbors is measured by the traffic density and the transmission range. This expression provides the guidance to adaptively choose the transmission range under different traffic densities. In addition, based on the analytical expression, the connectivity of the networks with both constant and variable disturbances are discussed. This study shows that under constant disturbance, improving the individual reachable neighbors will improve the connectivity of the network but the marginal effect decreases as individual reachable neighbors increases. Moreover, the asymptotic results prove that when K c ¼ Hðlog nn Þ; n > 2, the network is asymptotically connected. Another important result is that, although a VANET with a higher number of vehicles needs more reachable neighbors to keep high connectivity as compared to a smaller network, the difference is not significant. These analytical results gives us insights into the relationships among reachable neighbors, traffic density and network connectivity. Some of them are supported by the simulation findings in (Artimy et al., 2004; Blough et al., 2006; Santi and Blough, 2002). In addition, this study explores the variable disturbance case, in which the value of robustness factor changes with the traffic flow conditions. Based on the simulation data, a linear regression model is introduced to address the relationship between the robustness factor and traffic flow parameters such as average speed, average space headway and the standard variance of space headway. Combining the robustness factor function to the connectivity calculation, we obtain a more accurate estimation for the VANET connectivity under dynamic traffic flow. This paper is a part of our ongoing efforts to address methodological issues in VANETs. One of our immediate goals is to expand the methodology presented here to the road segment with traffic control infrastructure such as traffic signals, merge and diverge freeway sections and integrating with the transit system in which vehicle locations are more predictable. In addition, validating relationship between network connectivity and node mobility will be performed with a larger data set. Another future research is to explore the opportunity for connection between vehicles over a certain period of time. The basic idea is motivated by the insight that traffic information can be transmitted from node to node, which are disconnected in space but are connected over time due to vehicle mobility. This is similar to the concept of delay and disruption tolerant broadcasting (Grossglauser and Tse, 2002; Wu et al., 2004), where a message is received by a node without a forward node, but the node can carry the message and wait for new forward opportunities due to the vehicle mobility and unexpected driver behavior. References Arasanl, V.T., Koshy, R.Z., 2005. Methodology for modeling highly heterogeneous traffic flow. J. Transp. Eng. 131 (7), 544–551. Artimy, M.M., Robertson, W., Phillips, W.J., 2004. Connectivity in inter-vehicle ad hoc network. In: CCECE 2004–CCGEI 2004, Niagara Falls, NY, USA, 2–5 May. Artimy, M.M., Phillips, W.J., Robertson, W., 2005. Connectivity with static transmission range in vehicular ad hoc networks. In: Proceedings of the 3rd Annual Communication Networks and Services Research Conference, Halifax, Canada, 16–18 May. Artimy, M.M., Robertson, W., Phillips, W.J., 2005. Assignment of dynamic transmission range based on estimation of vehicle density. In: VANET’05, Cologne, Germany, 2 September. Artimy, M.M., Robertson, W., Phillips, W.J., 2006. Minimum transmission range in vehicular ad hoc networks over uninterrupted highways. In: The 9th International IEEE Conference on Intelligent Transportation Systems, Toronto, Canada, pp. 1400– 1405. Bettstetter, Christian, 2002. On the minimum node degree and connectivity of a wireless multihop network. In: MOBIHOC’02, EPF Lausanne, Switzerland, 9–11 June.


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