An Analytical Model of Network Connectivity in Vehicular Ad Hoc ...
An Analytical Model of Network Connectivity in Vehicular Ad Hoc Networks using Spatial Point Processes Parastoo Golmohammadi*, Payam Mokhtarian†, Farzad Safaei*, and Raad Raad* School of Electrical, Computer and Telecommunications Engineering, University of Wollongong Wollongong NSW 2522 AUSTRALIA † National Institute for Applied Statistics Research Australia, University of Wollongong, NSW 2522 AUSTRALIA Email: [email protected], {payam, farzad, raad}@uow.edu.au *
Abstract—We investigate the network connectivity of Vehicular Ad Hoc Networks (VANETs) for a typical highway at the free flow state. We propose a new approach to model the spatial distribution of vehicles in a given area using the statistical properties of spatial point patterns. We consider the VANET as a homogenous Poisson point process in which the number of vehicles per unit area has a Poisson distribution. The main advantage of such an approach is its ease of translation to a practical system. Our proposed model provides an analytical result for the vehicles’ nearest-neighbour distance distribution. We present simulation results to verify the obtained closed form solution of the network connectivity. Our model is verified by comparing it to real data obtained from the Bureau of Transport Statistics (BTS) NSW, Australia. The main advantage of our proposed model is reducing data acquisition in measuring of data, as the network connectivity can be studied dynamically using aerial images taken by aerial traffic monitoring systems. Another advantage is the ease with which arbitrary function for transmission coverage areas (e.g., antenna radiation pattern) can be included in the final closed form solution which shows capability and flexibility of the proposed model. Keywords—Nearest-neighbour distance; Network connectivity; Spatial point pattern; Transmission range; VANETs.
I. INTRODUCTION VANETs are a subset of mobile ad hoc networks (MANETs) where vehicles are equipped with wireless communication equipment to exchange messages. Vehicles can directly communicate with each other (vehicle-to-vehicle communication - V2V) or with a roadside network infrastructure (Vehicle-to-Roadside communication - V2R) [1]. The VANET applications can be categorised into safety related and user applications [2]. In 2003, the US Federal Communication Commission (FCC) established the Dedicated Short Range Communications (DSRC) Service that uses the 5.850-5.925 GHz band to provide safety and user applications [3]. In recent years, a number of studies have been published that address the VANET problem space. Vehicular communication can play an important role in improving the safety of the driver and passengers in the vehicle. Beside safety applications, many proposed commercial applications on the DSRC have triggered the fast development of VANETs. Most of these applications need a communication path between two vehicles for information dissemination. There is no guarantee that there are enough cars in the vicinity of a given transmitting vehicle to provide a complete path. To have a successful delivery of data, a connected network is required, therefore, whether inter-vehicle communication can support VANET applications or not is greatly dependent on
network connectivity. More importantly, the performance of all VANETs routing protocols is critically dependent on network connectivity. Consequently, network connectivity is a major challenge in VANETs environments that should be considered when designing an efficient routing protocol. Hence, an efficient estimation of the probability of establishing a communication path could be helpful for designing routing protocols in VANETs as routing algorithms would be able to take routes with higher connectivity probability. A number of studies have recently modelled and analysed the network connectivity (end to end connectivity) in a VANET. In [4, 5], the key focus is to derive the probability of connectivity and determine the influence of relevant factors such as traffic density and vehicular velocity on connectivity. In [6], a mobility model is developed to study the network connectivity problem. This model considers the statistical properties of random node arrivals and time-varying node speeds in different traffic scenarios. The analysis in [6] uses a geometry-assisted analytical method where the volumes of ndimensional analogue geometries are used to obtain the connectivity probability expressions. In [7], a unidirectional multilane highway is considered, in which traffic entry is assumed to follow a Poisson process. Then, the probability distribution function of the node population size and the mean cluster size on the highway are used for connectivity analysis. In [8], an analytical mobility model based on product-form queuing networks has been proposed to study the connectivity probability at both sparse and dense situations for VANETs. An analytical framework has been presented in [9] to determine the minimum number of vehicles and the critical transmission range (the transmission range required for a particular connectivity probability) required to provide the desired level of connectivity in the network. The analytical expression of connectivity probability in [9] is obtained by inter-vehicle distance distribution using vehicle density for one-way and two-way street scenarios. [10] investigates improving the connectivity problem in which a number of nodes with higher transmission range called mobile basestations are added to the network. In addition, an analytical model presented to find the optimum number of base-stations and their transmission range to have a connected network is proposed. [11] and [12] study the effect of speed distribution on the node isolation probability -the probability that a randomly chosen
node is not able to communicate with any of the other nodes in the network as well as the critical transmission range required to maintain connectivity. In [11] an analytical model for the connectivity probability of vehicles that move on a highway has been considered, in which the speed of vehicles follows a Gaussian probability distribution. Then, the distribution of inter-vehicle distance, network connectivity probability, and critical transmission range are obtained based on the average vehicle density. In [12], a common model in vehicular traffic theory is used in which the cars passing through a fixed point are separated in time based on an exponentially distributed time interval. The distribution of the number of cars in a period of time is assumed to be Poisson and also the distribution of the distances between the cars is obtained by the distribution of inter-vehicle intervals. Then queuing theory is used to study the connectivity where the VANETs environment components are assumed to be equivalent to an infinite server queuing model. Moreover, the authors in [12] use the Laplace transform of the connectivity probability distribution to obtain the explicit form of the average connectivity distance and the probability of isolation. Also, the effect of various system parameters i.e., speed distribution and traffic flow on the connectivity has been considered. As we mentioned above, all of these studies have been proposed based on classic statistical properties. They study the distribution of the distance between vehicles and employ infinite queuing theory assumptions to produce the connectivity probability. While the aforementioned models have provided powerful insights into VANET connectivity we believe that some are either too complicated and also result in varying degrees of accuracy. We believe that the proposed in this paper provides the following contributions: A simple and powerful analytical approach to the problem The model is shown to be accurate when compared to real data The model allows analysis extensions in a straight forward manner In classic vehicle traffic theory, the situation when the traffic density (in vehicles per kilometer per lane) is very low and the speed of vehicles (in kilometers per hour) and the traffic flow (in vehicles per hour per lane) are independent, is referred to as the free flow state [13]. In this paper, we study the network connectivity probability from a spatial perspective in the free flow state. We model the vehicular environment by a spatial point field, where vehicles are considered as points and the road is a spatial frame. Random point fields are statistical models for a sequence of events which are located in random point patterns [14]. Distribution of these fields satisfies very strong independent conditions and also statistical problems can be solved with powerful and elegant methods [15]. We consider the vehicular network as a homogenous Poisson point process in which the number of vehicles per unit area has a Poisson distribution. Following this, we study the nearest-neighbour distance distribution which results in the calculation of the network connectivity probability. Also, we study the sensitivity of the connectivity with respect to average vehicle speed for a fixed standard deviation for
various transmission ranges. The main contributions of our approach are simplicity of implementation and accuracy of the analytical model which we compare to real data. The most important advantage of this approach is reducing data acquisition in comparison with the proposed models in the literature. These models need an estimated value of vehicle density which is obtained using installed traffic monitoring systems on the road. However in our proposed model, the vehicle density can be estimated dynamically using aerial images taken by aerial traffic monitoring systems. In this model, it does not need to install any equipment to obtain data which is quite affordable and time efficient. Another advantage is that any arbitrary antenna radiation pattern for transmission coverage areas (e.g., directional or butterfly radiation pattern) can be included in the final closed form solution which shows capability and flexibility of the proposed model. II. CONNECTIVITY ANALYTICAL MODEL A. Preliminary concepts A homogeneous Poisson point process consists of n points ,…, that are randomly scattered in a set where points are uniformly and independently distributed [14]. The set is assumed to be bounded and its area is denoted by . The symbol ‘ ’ may also denote volume, if is a subset of , i.e. in the context of a three-dimensional spatial pattern. The homogenous Poisson point process has two fundamental properties. First, Poisson distribution of point counts, so that the number of points of in any bounded set follows a Poisson distribution with mean for some constant . The number is called the intensity or point density of the homogeneous Poisson point process, which describes the mean number E of points to be found in a unit area/volume and is given by E . Second, independent scattering, that is, the numbers of points of in disjoint sets form independent random variables, for arbitrary . In addition, there are four more basic properties of the homogeneous Poisson point process with intensity based on the above two fundamental properties. These properties called One-dimensional distributions, Finitedimensional distributions, Stationary and isotropic, and Void probabilities [14]. Now, given the all properties (regularity conditions), the nearest-neighbour distance distribution function for the homogenous Poisson point process is calculated. Consider a conditional nearest-neighbour distance distribution function , i.e. assume that there is a point of in , . This probability function is given by 1 Pr 0|
, ,
\ 1 .
,
(1)
This probability is the conditional probability that the distance from a point in the small sphere , to its nearestneighbour in is smaller than , under the condition that there is indeed a point of in the small sphere ( , \ , means the set that contains all those elements of , that are not in , ). This is defined for positive smaller than
, since Pr , 1 exp is positive. It is reasonable to regard the nearest-neighbour distance as the limit of the above as → ∞. Setting yields the result 1 lim → exp , . Thus the corresponding nearestneighbour distance probability density function (PDF) is /
2
exp
;
0.
(2)
B. Highway Network Connectivity Analytical Model We consider a unidirectional highway as a point process and then choose a segment of highway of length as the study area to investigate the connectivity probability. In fact, the observation window is the travelled length of highway in time , ,…, which are that is a set consists of vehicles distributed uniformly, and independently in each axis direction. Also, vehicles are moving in the segment with different speeds. The spatial distribution pattern of vehicles in the segment is a homogenous Poisson point process with parameter . The number of vehicles in period of time is considered a Poisson distribution with parameter and the area depends on the vehicle speed which has a significant influence on the segment density. Figure 1 shows three different examples of a homogenous Poisson point process and the nearest-neighbour points.
observation window is a Poisson distribution with parameter . Let be the random variable representing the and its nearestdistance between any vehicle with speed 1, … , and . neighbour with speed where , To calculate for different levels of speed, let be the number of vehicles during time (h). Thus the travelled . By these distance (∆ ) for vehicle with speed is ∆ assumptions, the average vehicle density is given by (3) E . Considering as the parameter of a homogenous Poisson point process E . In matrix notation, det in which diag 1, E . Here is a 2 2 diagonal matrix, where the first diagonal element corresponds to width transformation, which in this case is equal to one, and the second diagonal element is for the length transformation. The average vehicle density (intensity of the point process) of the observation window for the speed is (4) . E E det . However, this integral does not have a where E and closed form solution as it consists of multiplication of the truncated normal distribution. Therefore, This integral needs to be evaluated by a numerical integration method, e.g. Trapezoid integration. For this purpose, the Trapezoidal rule is used as an approximate technique for calculating the definite integral. The probability expression (2) is a function of the . This function is transmission range , that is, the area of a circle with radius . As the typical value of is greater than the road width, the probability expression (2) overestimates the corresponding probability for the highway case. For the highway, we apply the intersection area of the highway and the transmission range area in the probability expression. Suppose that vehicle is located meters from the side of road and also the road width is . Then, the road coverage by the transmission range area is shown with hachure in Figure 2.
Fig. 1. Homogenous Poisson point processes bounded by roads
Assume that , ,…, is random variable for speed 1, 2, … , and they are of vehicles where its values are ; independently and identically distributed (iid) with mean and standard deviation . Empirical studies have shown that the speeds of different vehicles in free flow state follow a Gaussian distribution [13]. We, therefore, consider the speeds of vehicles as a Gaussian distribution with average speed (mean) and standard deviation . To eliminate extreme values of speeds either negative, very high speed or close to max zero, truncating method is used so that , and where is a minimum / / possible speed in any arbitrary road and has standard Normal distribution which lies with probability 1 /2. and the random variable has a Regarding both truncated Gaussian distribution which covers 95 per cent of the Gaussian distribution with the significant level 0.05. in the The number of vehicles with speed
Fig. 2. Road coverage by transmission range area
Given Cartesian coordinates, the transmission range has a functional form as 1
.
In polar coordinates, and √1 sin cos . Therefore, the road coverage area is given by
(5)
,
2
(6)
where 4
sin 2
2
2
sin 2
,
2 .
2
Also, the nearest-neighbour distance distribution for speed based on the road coverage area is given by (7) exp . where is the derivative of respect to . In the literature [17, 18, 19] have been shown the probability that each vehicle in the network has a neighbouring vehicle within a distance can be a good approximation for the probability that a network is connected. Although, the expression (7) can be a good approximation for the network connectivity probability, this expression is not the probability of connectivity of a network. A network is connected if the distance between every two consecutive vehicles is less than the transmission . To obtain the network connectivity probability, vehicles with speed in the highway flow direction are sorted out by their locations and are indicated by 1, … , . It means that the 1 vehicle is further than vehicle. Using this assumption, the network connectivity probability, , is given by ,
(8)
is the probability of connectivity between the where vehicle and the 1 vehicle and is given by 1 ,
(9)
where is the distance between the vehicle and the 1 vehicle. Also, 1 shows that the 1 vehicle is considered as the nearest-neighbour of the vehicle even though the 1 vehicle could be closer to the vehicle. The appropriate coverage area of the conditional probability in the expression (9) shown in the Figure 3 with hachure. This is employed to calculate the probability of network connectivity
Fig. 1. Road coverage by transmission range area
, is given by
(10) 2 . Moreover, as vehicles’ locations in the highway have been considered homogeneous Poisson point process and vehicles are distributed uniformly along the highway direction, then , in which , 1, 2, … , 1 and .
and 4
This coverage area,
Consequently, the network connectivity probability, , of vehicles with random variable speed for segment of length including vehicles is 1
exp
.
(11)
Furthermore, the probability that the network in the highway , is obtained by is not connected, exp
.
(12)
where ‘ ’ stands for no connection. It is notable that the node isolation probability given based on the expression (7) is different from the probability expression (12) that the network in the highway is not connected. The reason is that in some cases, there is no isolated node in the network but the network is still disconnected. III. SIMULATION STUDY A. Model-based simulation results In this section, we present the Monte Carlo model-based simulation results of the connectivity probability for a typical highway. The idea of the Monte Carlo replication is generating a large number of synthetic data sets that are similar to the experimental data set, but each with different random normally distributed noise. Each of these new data sets is analysed, and the distributions are stored. The resulting set of distributions can then be studied, point by point, and the mean and probability contours can be calculated for inference purposes. Matlab is used as a simulation platform. To implement the simulation procedure, the following simulation set up is considered: A square with a 3 (km) side is considered as an open area A length of 10 (km) of a typical highway including 3 lanes is considered. The width of each lane is 3.5 (m) The number of vehicles is generated from a Poisson process with vehicle density rates 0, 100, … , 2000 The speeds ( ) are generated from a truncated Gaussian 15 (km/h). distribution with 60, 80, 100 and Transmission range is considered 250 (m) A total of 2000 Monte Carlo simulations are carried out to generate the output of experiments for statistical confidence To simulate the highway analytical model (11), we assumed a three lanes highway with width 3 3.5 10.5 (m) [20]. In Figure 4 the probability that these is no isolated node in the highway is plotted against the values of vehicle density rate for various vehicle average speeds with standard deviation 15 (km/h) and transmission ranges 250 (m). It is shown that this probability increases with increasing the number of vehicles. Also, the probability of the network
connectivity in the highway is less for vehicles with higher speed.
numerical evaluation, we have used real data obtained from Bureau of Transport Statistics (BTS) – Transport New South Wales Australia [21]. These data present the average weekday traffic counts by four time periods that were collected in 2008. The four time periods cover different vehicle traffic densities ranging from low to high which have been presented in Table 1 according the average weekday traffic counts obtained from Hume Highway in southbound direction in NSW [21]. TABLE I. Traffic counts of Hume Highway in southbound direction AM IP PM NT Time Period 7am – 9am 9am – 3pm 3pm – 6pm 6pm – 7am
Fig. 4. Probability of network connectivity in highway vs. number of vehicle (veh/h) for (m)
As shown in Figure 4, the simulation and analytical results are very close together and the differences are insignificant. To show the effect of vehicle density on the probability of network connectivity within highway, we plotted Figure 5 for three different values of vehicle density ( ) and the mean speed 80 (km/h). It is notable that for a given transmission range, the probability of network connectivity in the highway increases with increasing the vehicle density.
Veh/Period
2365
8675
8915
9168
Veh/h
1182.5
1445.8
2971.7
705.2
In numerical comparison process, we have employed the following specifications: A length of 15 (km) of Hume Highway in southbound direction including 3 lanes is considered (3 3.5 m) Density rate is 0, 705.2, 1182.5, 1445.8, 2971.7 The average vehicles speeds ( ) is considered 110 (km/h) according to the speed limit in Hume Highway The transmission range value is considered 250 (m) The probability of connectivity for the three models and the actual probability of connectivity obtained based on the BTS data are plotted in Figure 6.
Fig. 6. Comparison between actual connectivity probability and the three different connectivity study approaches
Fig. 5. Probability of network connectivity in highway vs. transmission range (km) for (km/h)
B. Emeprical evaluation results In the VANET’s context, there are few approaches to investigate network connectivity probability. Here we have compared two typical models [11] and [12] in the literature with our proposed model in terms of prediction and performance with numerical evaluation empirically. These analytical models are referred as Model I [11], Model II [12] and Model PPP (our proposed analytical model), where PPP stands for Poisson Point Process. To impediment a reliable
As shown in Figure 6, model I and model II have a slight overprediction for calculation of network connectivity probability in the time period of 6:00 pm to 7:00 am (NT) when the traffic density is low, while model PPP is quite similar to the actual probability for spare vehicle traffic densities. For time period of 7:00 am to 9:00 am (AM), both model I and model II have overprediction and model PPP has slightly underprediction. For time period of 9:00 am to 3:00 pm (IP) when the traffic density is medium, both model I and model II have significant overprediction, while model PPP has slightly underprediction. In high traffic density, for time period 3:00 pm to 6:00 pm (PM) all three models perform well and quite similar to the actual probability. Totally, the trend of
network connectivity probability of model PPP is much closer to the actual probability trend than model I and II for different vehicle traffic densities. We have also plotted the probability that the network is disconnected vs. the number of vehicles per hours in Figure 7 for model I, model II, and model PPP and compare them with the actual probability.
scenarios such as road congestion, and inhomogeneous traffic densities including sparse and dense scenarios. REFERENCES [1]
[2]
[3] [4]
[5]
[6]
[7]
[8] Fig. 7. Comparison between actual probability of disconnected network and the three different connectivity study approaches
The comparison shows that all of these three models perform well in time periods AM, IP, and PM. However, model I and II have a significant underprediction in time period NT when the traffic density is low, while model PPP has an insignificant difference with the actual probability of disconnected network. Generally, the trend of the probability that the network is not connected in model PPP is very close to the trend of actual probability for whole range of traffic densities. IV. CONCLUSION This paper presents a new analytical approach for studying connectivity in VANETs. We have derived the analytical expressions for a typical highway for vehicular networks using the spatial point pattern statistical concepts for the free flow traffic model. In the proposed method, we have used the nearest-neighbor distance distribution function of vehicles to obtain a closed form for connectivity probability in vehicular networks. The model-based simulation results provide evidence for the accuracy of our model. The numerical evaluation results of the BTS data have clearly shown that our proposed model performs more accurately than the other proposed models in the literature in term of network connectivity probability estimation. As long as we are able to obtain the distribution function of the nearest-neighbor distances and that of the transmission range of vehicles, the connectivity analysis performed in this paper can be extended to more general scenarios. Our proposed model is able to study the connectivity in two directions traffic and forward/backward data communications where all speeds in both directions should be considered by relative speeds using parallelogram rule called vector summation. Our future work is extension of the proposed model for different traffic
[9]
[10]
[11]
[12]
[13] [14] [15] [16] [17]
[18] [19]
[20] [21]
A. Borkerche, H. Oliveira, E. Nakamura, and A. Loureiro, Vehicular ad hoc networks: a new challenge for localization-based system. Computer Communications, 31, pp. 2838-2849, 2008. Y. Toor, P. Muhlethaler, and A. Laouiti, Vehicular ad hoc networks: applications and related technical issues. IEEE Communication surveys and tutorials, 10, No. 3, pp. 74-88, 2008. The US Federal Communication Commission (FCC) - DSRC website. http://wireless.fcc.gov/services/its/dsrc. M. M. Artimy, W. Robertson, and W. J. Phillips, Connectivity with static transmission range in vehicular ad hoc networks. Proceeding of 3rd Annual Conference on Communication Networks and Services Research, pp. 237-242, 2005. M. M. Artimy, W. Robertson, and W. J. Phillips, Assignment of dynamic transmission range based on estimation of vehicular density. Proceeding of ACM VANET, Germany, 2005. J. Wu, Connectivity of mobile linear networks with dynamic node population and delay constraint. IEEE Journal of Selected Areas in Communications (JSAC), 27, No. 7, pp. 1215-1218, 2009. M. Khabazian, and M. K. Ali, A performance modelling of connectivity in vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 57, pp. 2440-2450, 2008. G. H. Mohimani, F. Ashtiani, A. Javanmard, and M. Hamdi, Mobility modelling, spatial traffic distribution, and probability of connectivity for spare and dense vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 58, No. 4, pp. 1998-2007, 2008. S. Panichpapiboon, and W. Pattara-Atikom, Connectivity requirements for a self-organizing traffic information systems. IEEE Transaction on Vehicular Technology, 57, No. 6, pp. 3333-3340, 2008. S. Yousefi, E Altman, R. El-Azouzi, and M. Fathy, Improving connectivity in vehicular ad hoc networks: An analytical study. Computer Communications, 31, No. 9, pp. 1653-1659, 2008. V. K. M. Ajeer, P. C. Neelakantan, and A. V. Babu, Network connectivity of one-dimensional vehicular ad hoc network. IEEE ICCSP, pp. 241-245, 2011. S. Yousefi, E. Altman, R. El-Azouzi, and M. Fathy, Analytical model for connectivity in vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 57, No. 6, pp. 3341-3356, 2008. R. P. Roess, E. S. Prassas, and W. R. McShane, Traffic Engineering (4th edition), Prentice Hall, New Jersey, 2010. N. Cressie, Statistics for Spatial Data, John Wiley, New York, 1993. J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan, Statistical Analysis and Modeling of Spatial Point Patterns. Wiley, England, 2008. K. E. Atkinson, An Introduction to Numerical Analysis (2nd edition). Wiley, New York, 1989. C. Bettstetter, On the minimum node degree and connectivity of a wireless multihop network. Proceedings of the 3rd ACM International Symposium (MobiHoc'02), pp. 80-91, New York, USA, 2002. C. Bettstetter, On the connectivity of ad hoc networks. The Computer Journal, 47, No. 4, pp. 432-447, 2004. C. Bettstetter, Topology properties of ad hoc networks with random waypoint mobility. Proceeding of MOBIHOC, Annapolis, MD, USA, 2003. The ACT Government Australia - Territory and Municipal Services Website: http://www.tams.act.gov.au. The NSW Government Australia – Bureau of Transport Statistics Website: http://www.bts.nsw.gov.au.
Abstract—We investigate the network connectivity of Vehicular Ad Hoc Networks (VANETs) for a typical highway at the free flow state. We propose a new approach to model the spatial distribution of vehicles in a given area using the statistical properties of spatial point patterns. We consider the VANET as a homogenous Poisson point process in which the number of vehicles per unit area has a Poisson distribution. The main advantage of such an approach is its ease of translation to a practical system. Our proposed model provides an analytical result for the vehicles’ nearest-neighbour distance distribution. We present simulation results to verify the obtained closed form solution of the network connectivity. Our model is verified by comparing it to real data obtained from the Bureau of Transport Statistics (BTS) NSW, Australia. The main advantage of our proposed model is reducing data acquisition in measuring of data, as the network connectivity can be studied dynamically using aerial images taken by aerial traffic monitoring systems. Another advantage is the ease with which arbitrary function for transmission coverage areas (e.g., antenna radiation pattern) can be included in the final closed form solution which shows capability and flexibility of the proposed model. Keywords—Nearest-neighbour distance; Network connectivity; Spatial point pattern; Transmission range; VANETs.
I. INTRODUCTION VANETs are a subset of mobile ad hoc networks (MANETs) where vehicles are equipped with wireless communication equipment to exchange messages. Vehicles can directly communicate with each other (vehicle-to-vehicle communication - V2V) or with a roadside network infrastructure (Vehicle-to-Roadside communication - V2R) [1]. The VANET applications can be categorised into safety related and user applications [2]. In 2003, the US Federal Communication Commission (FCC) established the Dedicated Short Range Communications (DSRC) Service that uses the 5.850-5.925 GHz band to provide safety and user applications [3]. In recent years, a number of studies have been published that address the VANET problem space. Vehicular communication can play an important role in improving the safety of the driver and passengers in the vehicle. Beside safety applications, many proposed commercial applications on the DSRC have triggered the fast development of VANETs. Most of these applications need a communication path between two vehicles for information dissemination. There is no guarantee that there are enough cars in the vicinity of a given transmitting vehicle to provide a complete path. To have a successful delivery of data, a connected network is required, therefore, whether inter-vehicle communication can support VANET applications or not is greatly dependent on
network connectivity. More importantly, the performance of all VANETs routing protocols is critically dependent on network connectivity. Consequently, network connectivity is a major challenge in VANETs environments that should be considered when designing an efficient routing protocol. Hence, an efficient estimation of the probability of establishing a communication path could be helpful for designing routing protocols in VANETs as routing algorithms would be able to take routes with higher connectivity probability. A number of studies have recently modelled and analysed the network connectivity (end to end connectivity) in a VANET. In [4, 5], the key focus is to derive the probability of connectivity and determine the influence of relevant factors such as traffic density and vehicular velocity on connectivity. In [6], a mobility model is developed to study the network connectivity problem. This model considers the statistical properties of random node arrivals and time-varying node speeds in different traffic scenarios. The analysis in [6] uses a geometry-assisted analytical method where the volumes of ndimensional analogue geometries are used to obtain the connectivity probability expressions. In [7], a unidirectional multilane highway is considered, in which traffic entry is assumed to follow a Poisson process. Then, the probability distribution function of the node population size and the mean cluster size on the highway are used for connectivity analysis. In [8], an analytical mobility model based on product-form queuing networks has been proposed to study the connectivity probability at both sparse and dense situations for VANETs. An analytical framework has been presented in [9] to determine the minimum number of vehicles and the critical transmission range (the transmission range required for a particular connectivity probability) required to provide the desired level of connectivity in the network. The analytical expression of connectivity probability in [9] is obtained by inter-vehicle distance distribution using vehicle density for one-way and two-way street scenarios. [10] investigates improving the connectivity problem in which a number of nodes with higher transmission range called mobile basestations are added to the network. In addition, an analytical model presented to find the optimum number of base-stations and their transmission range to have a connected network is proposed. [11] and [12] study the effect of speed distribution on the node isolation probability -the probability that a randomly chosen
node is not able to communicate with any of the other nodes in the network as well as the critical transmission range required to maintain connectivity. In [11] an analytical model for the connectivity probability of vehicles that move on a highway has been considered, in which the speed of vehicles follows a Gaussian probability distribution. Then, the distribution of inter-vehicle distance, network connectivity probability, and critical transmission range are obtained based on the average vehicle density. In [12], a common model in vehicular traffic theory is used in which the cars passing through a fixed point are separated in time based on an exponentially distributed time interval. The distribution of the number of cars in a period of time is assumed to be Poisson and also the distribution of the distances between the cars is obtained by the distribution of inter-vehicle intervals. Then queuing theory is used to study the connectivity where the VANETs environment components are assumed to be equivalent to an infinite server queuing model. Moreover, the authors in [12] use the Laplace transform of the connectivity probability distribution to obtain the explicit form of the average connectivity distance and the probability of isolation. Also, the effect of various system parameters i.e., speed distribution and traffic flow on the connectivity has been considered. As we mentioned above, all of these studies have been proposed based on classic statistical properties. They study the distribution of the distance between vehicles and employ infinite queuing theory assumptions to produce the connectivity probability. While the aforementioned models have provided powerful insights into VANET connectivity we believe that some are either too complicated and also result in varying degrees of accuracy. We believe that the proposed in this paper provides the following contributions: A simple and powerful analytical approach to the problem The model is shown to be accurate when compared to real data The model allows analysis extensions in a straight forward manner In classic vehicle traffic theory, the situation when the traffic density (in vehicles per kilometer per lane) is very low and the speed of vehicles (in kilometers per hour) and the traffic flow (in vehicles per hour per lane) are independent, is referred to as the free flow state [13]. In this paper, we study the network connectivity probability from a spatial perspective in the free flow state. We model the vehicular environment by a spatial point field, where vehicles are considered as points and the road is a spatial frame. Random point fields are statistical models for a sequence of events which are located in random point patterns [14]. Distribution of these fields satisfies very strong independent conditions and also statistical problems can be solved with powerful and elegant methods [15]. We consider the vehicular network as a homogenous Poisson point process in which the number of vehicles per unit area has a Poisson distribution. Following this, we study the nearest-neighbour distance distribution which results in the calculation of the network connectivity probability. Also, we study the sensitivity of the connectivity with respect to average vehicle speed for a fixed standard deviation for
various transmission ranges. The main contributions of our approach are simplicity of implementation and accuracy of the analytical model which we compare to real data. The most important advantage of this approach is reducing data acquisition in comparison with the proposed models in the literature. These models need an estimated value of vehicle density which is obtained using installed traffic monitoring systems on the road. However in our proposed model, the vehicle density can be estimated dynamically using aerial images taken by aerial traffic monitoring systems. In this model, it does not need to install any equipment to obtain data which is quite affordable and time efficient. Another advantage is that any arbitrary antenna radiation pattern for transmission coverage areas (e.g., directional or butterfly radiation pattern) can be included in the final closed form solution which shows capability and flexibility of the proposed model. II. CONNECTIVITY ANALYTICAL MODEL A. Preliminary concepts A homogeneous Poisson point process consists of n points ,…, that are randomly scattered in a set where points are uniformly and independently distributed [14]. The set is assumed to be bounded and its area is denoted by . The symbol ‘ ’ may also denote volume, if is a subset of , i.e. in the context of a three-dimensional spatial pattern. The homogenous Poisson point process has two fundamental properties. First, Poisson distribution of point counts, so that the number of points of in any bounded set follows a Poisson distribution with mean for some constant . The number is called the intensity or point density of the homogeneous Poisson point process, which describes the mean number E of points to be found in a unit area/volume and is given by E . Second, independent scattering, that is, the numbers of points of in disjoint sets form independent random variables, for arbitrary . In addition, there are four more basic properties of the homogeneous Poisson point process with intensity based on the above two fundamental properties. These properties called One-dimensional distributions, Finitedimensional distributions, Stationary and isotropic, and Void probabilities [14]. Now, given the all properties (regularity conditions), the nearest-neighbour distance distribution function for the homogenous Poisson point process is calculated. Consider a conditional nearest-neighbour distance distribution function , i.e. assume that there is a point of in , . This probability function is given by 1 Pr 0|
, ,
\ 1 .
,
(1)
This probability is the conditional probability that the distance from a point in the small sphere , to its nearestneighbour in is smaller than , under the condition that there is indeed a point of in the small sphere ( , \ , means the set that contains all those elements of , that are not in , ). This is defined for positive smaller than
, since Pr , 1 exp is positive. It is reasonable to regard the nearest-neighbour distance as the limit of the above as → ∞. Setting yields the result 1 lim → exp , . Thus the corresponding nearestneighbour distance probability density function (PDF) is /
2
exp
;
0.
(2)
B. Highway Network Connectivity Analytical Model We consider a unidirectional highway as a point process and then choose a segment of highway of length as the study area to investigate the connectivity probability. In fact, the observation window is the travelled length of highway in time , ,…, which are that is a set consists of vehicles distributed uniformly, and independently in each axis direction. Also, vehicles are moving in the segment with different speeds. The spatial distribution pattern of vehicles in the segment is a homogenous Poisson point process with parameter . The number of vehicles in period of time is considered a Poisson distribution with parameter and the area depends on the vehicle speed which has a significant influence on the segment density. Figure 1 shows three different examples of a homogenous Poisson point process and the nearest-neighbour points.
observation window is a Poisson distribution with parameter . Let be the random variable representing the and its nearestdistance between any vehicle with speed 1, … , and . neighbour with speed where , To calculate for different levels of speed, let be the number of vehicles during time (h). Thus the travelled . By these distance (∆ ) for vehicle with speed is ∆ assumptions, the average vehicle density is given by (3) E . Considering as the parameter of a homogenous Poisson point process E . In matrix notation, det in which diag 1, E . Here is a 2 2 diagonal matrix, where the first diagonal element corresponds to width transformation, which in this case is equal to one, and the second diagonal element is for the length transformation. The average vehicle density (intensity of the point process) of the observation window for the speed is (4) . E E det . However, this integral does not have a where E and closed form solution as it consists of multiplication of the truncated normal distribution. Therefore, This integral needs to be evaluated by a numerical integration method, e.g. Trapezoid integration. For this purpose, the Trapezoidal rule is used as an approximate technique for calculating the definite integral. The probability expression (2) is a function of the . This function is transmission range , that is, the area of a circle with radius . As the typical value of is greater than the road width, the probability expression (2) overestimates the corresponding probability for the highway case. For the highway, we apply the intersection area of the highway and the transmission range area in the probability expression. Suppose that vehicle is located meters from the side of road and also the road width is . Then, the road coverage by the transmission range area is shown with hachure in Figure 2.
Fig. 1. Homogenous Poisson point processes bounded by roads
Assume that , ,…, is random variable for speed 1, 2, … , and they are of vehicles where its values are ; independently and identically distributed (iid) with mean and standard deviation . Empirical studies have shown that the speeds of different vehicles in free flow state follow a Gaussian distribution [13]. We, therefore, consider the speeds of vehicles as a Gaussian distribution with average speed (mean) and standard deviation . To eliminate extreme values of speeds either negative, very high speed or close to max zero, truncating method is used so that , and where is a minimum / / possible speed in any arbitrary road and has standard Normal distribution which lies with probability 1 /2. and the random variable has a Regarding both truncated Gaussian distribution which covers 95 per cent of the Gaussian distribution with the significant level 0.05. in the The number of vehicles with speed
Fig. 2. Road coverage by transmission range area
Given Cartesian coordinates, the transmission range has a functional form as 1
.
In polar coordinates, and √1 sin cos . Therefore, the road coverage area is given by
(5)
,
2
(6)
where 4
sin 2
2
2
sin 2
,
2 .
2
Also, the nearest-neighbour distance distribution for speed based on the road coverage area is given by (7) exp . where is the derivative of respect to . In the literature [17, 18, 19] have been shown the probability that each vehicle in the network has a neighbouring vehicle within a distance can be a good approximation for the probability that a network is connected. Although, the expression (7) can be a good approximation for the network connectivity probability, this expression is not the probability of connectivity of a network. A network is connected if the distance between every two consecutive vehicles is less than the transmission . To obtain the network connectivity probability, vehicles with speed in the highway flow direction are sorted out by their locations and are indicated by 1, … , . It means that the 1 vehicle is further than vehicle. Using this assumption, the network connectivity probability, , is given by ,
(8)
is the probability of connectivity between the where vehicle and the 1 vehicle and is given by 1 ,
(9)
where is the distance between the vehicle and the 1 vehicle. Also, 1 shows that the 1 vehicle is considered as the nearest-neighbour of the vehicle even though the 1 vehicle could be closer to the vehicle. The appropriate coverage area of the conditional probability in the expression (9) shown in the Figure 3 with hachure. This is employed to calculate the probability of network connectivity
Fig. 1. Road coverage by transmission range area
, is given by
(10) 2 . Moreover, as vehicles’ locations in the highway have been considered homogeneous Poisson point process and vehicles are distributed uniformly along the highway direction, then , in which , 1, 2, … , 1 and .
and 4
This coverage area,
Consequently, the network connectivity probability, , of vehicles with random variable speed for segment of length including vehicles is 1
exp
.
(11)
Furthermore, the probability that the network in the highway , is obtained by is not connected, exp
.
(12)
where ‘ ’ stands for no connection. It is notable that the node isolation probability given based on the expression (7) is different from the probability expression (12) that the network in the highway is not connected. The reason is that in some cases, there is no isolated node in the network but the network is still disconnected. III. SIMULATION STUDY A. Model-based simulation results In this section, we present the Monte Carlo model-based simulation results of the connectivity probability for a typical highway. The idea of the Monte Carlo replication is generating a large number of synthetic data sets that are similar to the experimental data set, but each with different random normally distributed noise. Each of these new data sets is analysed, and the distributions are stored. The resulting set of distributions can then be studied, point by point, and the mean and probability contours can be calculated for inference purposes. Matlab is used as a simulation platform. To implement the simulation procedure, the following simulation set up is considered: A square with a 3 (km) side is considered as an open area A length of 10 (km) of a typical highway including 3 lanes is considered. The width of each lane is 3.5 (m) The number of vehicles is generated from a Poisson process with vehicle density rates 0, 100, … , 2000 The speeds ( ) are generated from a truncated Gaussian 15 (km/h). distribution with 60, 80, 100 and Transmission range is considered 250 (m) A total of 2000 Monte Carlo simulations are carried out to generate the output of experiments for statistical confidence To simulate the highway analytical model (11), we assumed a three lanes highway with width 3 3.5 10.5 (m) [20]. In Figure 4 the probability that these is no isolated node in the highway is plotted against the values of vehicle density rate for various vehicle average speeds with standard deviation 15 (km/h) and transmission ranges 250 (m). It is shown that this probability increases with increasing the number of vehicles. Also, the probability of the network
connectivity in the highway is less for vehicles with higher speed.
numerical evaluation, we have used real data obtained from Bureau of Transport Statistics (BTS) – Transport New South Wales Australia [21]. These data present the average weekday traffic counts by four time periods that were collected in 2008. The four time periods cover different vehicle traffic densities ranging from low to high which have been presented in Table 1 according the average weekday traffic counts obtained from Hume Highway in southbound direction in NSW [21]. TABLE I. Traffic counts of Hume Highway in southbound direction AM IP PM NT Time Period 7am – 9am 9am – 3pm 3pm – 6pm 6pm – 7am
Fig. 4. Probability of network connectivity in highway vs. number of vehicle (veh/h) for (m)
As shown in Figure 4, the simulation and analytical results are very close together and the differences are insignificant. To show the effect of vehicle density on the probability of network connectivity within highway, we plotted Figure 5 for three different values of vehicle density ( ) and the mean speed 80 (km/h). It is notable that for a given transmission range, the probability of network connectivity in the highway increases with increasing the vehicle density.
Veh/Period
2365
8675
8915
9168
Veh/h
1182.5
1445.8
2971.7
705.2
In numerical comparison process, we have employed the following specifications: A length of 15 (km) of Hume Highway in southbound direction including 3 lanes is considered (3 3.5 m) Density rate is 0, 705.2, 1182.5, 1445.8, 2971.7 The average vehicles speeds ( ) is considered 110 (km/h) according to the speed limit in Hume Highway The transmission range value is considered 250 (m) The probability of connectivity for the three models and the actual probability of connectivity obtained based on the BTS data are plotted in Figure 6.
Fig. 6. Comparison between actual connectivity probability and the three different connectivity study approaches
Fig. 5. Probability of network connectivity in highway vs. transmission range (km) for (km/h)
B. Emeprical evaluation results In the VANET’s context, there are few approaches to investigate network connectivity probability. Here we have compared two typical models [11] and [12] in the literature with our proposed model in terms of prediction and performance with numerical evaluation empirically. These analytical models are referred as Model I [11], Model II [12] and Model PPP (our proposed analytical model), where PPP stands for Poisson Point Process. To impediment a reliable
As shown in Figure 6, model I and model II have a slight overprediction for calculation of network connectivity probability in the time period of 6:00 pm to 7:00 am (NT) when the traffic density is low, while model PPP is quite similar to the actual probability for spare vehicle traffic densities. For time period of 7:00 am to 9:00 am (AM), both model I and model II have overprediction and model PPP has slightly underprediction. For time period of 9:00 am to 3:00 pm (IP) when the traffic density is medium, both model I and model II have significant overprediction, while model PPP has slightly underprediction. In high traffic density, for time period 3:00 pm to 6:00 pm (PM) all three models perform well and quite similar to the actual probability. Totally, the trend of
network connectivity probability of model PPP is much closer to the actual probability trend than model I and II for different vehicle traffic densities. We have also plotted the probability that the network is disconnected vs. the number of vehicles per hours in Figure 7 for model I, model II, and model PPP and compare them with the actual probability.
scenarios such as road congestion, and inhomogeneous traffic densities including sparse and dense scenarios. REFERENCES [1]
[2]
[3] [4]
[5]
[6]
[7]
[8] Fig. 7. Comparison between actual probability of disconnected network and the three different connectivity study approaches
The comparison shows that all of these three models perform well in time periods AM, IP, and PM. However, model I and II have a significant underprediction in time period NT when the traffic density is low, while model PPP has an insignificant difference with the actual probability of disconnected network. Generally, the trend of the probability that the network is not connected in model PPP is very close to the trend of actual probability for whole range of traffic densities. IV. CONCLUSION This paper presents a new analytical approach for studying connectivity in VANETs. We have derived the analytical expressions for a typical highway for vehicular networks using the spatial point pattern statistical concepts for the free flow traffic model. In the proposed method, we have used the nearest-neighbor distance distribution function of vehicles to obtain a closed form for connectivity probability in vehicular networks. The model-based simulation results provide evidence for the accuracy of our model. The numerical evaluation results of the BTS data have clearly shown that our proposed model performs more accurately than the other proposed models in the literature in term of network connectivity probability estimation. As long as we are able to obtain the distribution function of the nearest-neighbor distances and that of the transmission range of vehicles, the connectivity analysis performed in this paper can be extended to more general scenarios. Our proposed model is able to study the connectivity in two directions traffic and forward/backward data communications where all speeds in both directions should be considered by relative speeds using parallelogram rule called vector summation. Our future work is extension of the proposed model for different traffic
[9]
[10]
[11]
[12]
[13] [14] [15] [16] [17]
[18] [19]
[20] [21]
A. Borkerche, H. Oliveira, E. Nakamura, and A. Loureiro, Vehicular ad hoc networks: a new challenge for localization-based system. Computer Communications, 31, pp. 2838-2849, 2008. Y. Toor, P. Muhlethaler, and A. Laouiti, Vehicular ad hoc networks: applications and related technical issues. IEEE Communication surveys and tutorials, 10, No. 3, pp. 74-88, 2008. The US Federal Communication Commission (FCC) - DSRC website. http://wireless.fcc.gov/services/its/dsrc. M. M. Artimy, W. Robertson, and W. J. Phillips, Connectivity with static transmission range in vehicular ad hoc networks. Proceeding of 3rd Annual Conference on Communication Networks and Services Research, pp. 237-242, 2005. M. M. Artimy, W. Robertson, and W. J. Phillips, Assignment of dynamic transmission range based on estimation of vehicular density. Proceeding of ACM VANET, Germany, 2005. J. Wu, Connectivity of mobile linear networks with dynamic node population and delay constraint. IEEE Journal of Selected Areas in Communications (JSAC), 27, No. 7, pp. 1215-1218, 2009. M. Khabazian, and M. K. Ali, A performance modelling of connectivity in vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 57, pp. 2440-2450, 2008. G. H. Mohimani, F. Ashtiani, A. Javanmard, and M. Hamdi, Mobility modelling, spatial traffic distribution, and probability of connectivity for spare and dense vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 58, No. 4, pp. 1998-2007, 2008. S. Panichpapiboon, and W. Pattara-Atikom, Connectivity requirements for a self-organizing traffic information systems. IEEE Transaction on Vehicular Technology, 57, No. 6, pp. 3333-3340, 2008. S. Yousefi, E Altman, R. El-Azouzi, and M. Fathy, Improving connectivity in vehicular ad hoc networks: An analytical study. Computer Communications, 31, No. 9, pp. 1653-1659, 2008. V. K. M. Ajeer, P. C. Neelakantan, and A. V. Babu, Network connectivity of one-dimensional vehicular ad hoc network. IEEE ICCSP, pp. 241-245, 2011. S. Yousefi, E. Altman, R. El-Azouzi, and M. Fathy, Analytical model for connectivity in vehicular ad hoc networks. IEEE Transaction on Vehicular Technology, 57, No. 6, pp. 3341-3356, 2008. R. P. Roess, E. S. Prassas, and W. R. McShane, Traffic Engineering (4th edition), Prentice Hall, New Jersey, 2010. N. Cressie, Statistics for Spatial Data, John Wiley, New York, 1993. J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan, Statistical Analysis and Modeling of Spatial Point Patterns. Wiley, England, 2008. K. E. Atkinson, An Introduction to Numerical Analysis (2nd edition). Wiley, New York, 1989. C. Bettstetter, On the minimum node degree and connectivity of a wireless multihop network. Proceedings of the 3rd ACM International Symposium (MobiHoc'02), pp. 80-91, New York, USA, 2002. C. Bettstetter, On the connectivity of ad hoc networks. The Computer Journal, 47, No. 4, pp. 432-447, 2004. C. Bettstetter, Topology properties of ad hoc networks with random waypoint mobility. Proceeding of MOBIHOC, Annapolis, MD, USA, 2003. The ACT Government Australia - Territory and Municipal Services Website: http://www.tams.act.gov.au. The NSW Government Australia – Bureau of Transport Statistics Website: http://www.bts.nsw.gov.au.
