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In recent years, various routing metrics such as throughput, end-to-end delay, packet delivery ratio, path duration, and so forth have been used to evaluate the performance of routing protocols in VANETs. Among these routing metrics, path duration is one of the most influential metrics. Highly mobile vehicles cause frequent topology change in vehicular network environment that ultimately affects the path duration. In this paper, we have derived a mathematical model to estimate path duration using border node-based most forward progress within radius (B-MFR), a position based routing protocol. The mathematical model for estimation of path duration consists of probability of finding next-hop node in forwarding region, estimation of expected number of hops, probability distribution of velocity of nodes, and link duration between each intermediate pair of nodes. The analytical results for the path duration estimation model have been obtained using MATLAB. The model for path duration estimation has been simulated in NS2. Each of the analytical results has been verified through respective simulation results. The result analysis clearly reveals that path duration increases with the increase in transmission range and node density and decreases with the increase in the number of hops in the path and velocity of the nodes.

The intelligent transport system (ITS) has been working to make the road safer and efficient to cope up with increasing number of on-road vehicles day by day. The number of accidents on the roads is continuously increasing due to high growth in on-road vehicle population. The increasing number of accidents has become an issue of concern worldwide. It has made the roads vulnerable and threatening as every year millions of people are dying in accidents throughout the world. A modern network concept, VANETs, has become the hope for providing safer and well-organized transportation in near future [

In VANETs, routing is the process of finding optimal path for information forwarding between source and destination node. Position-based routing protocols have become one of the most investigated choices among researches due to geographic region sharing of on-road vehicles [

In VANETs, any routing path is made up of one or more links between pair of intermediate nodes. Therefore, lifetime of a link is one of the most important contributors in path duration. Path duration can be defined as the duration of time till every link of the route is active. In VANETs, the lifetime of a link is a random variable whose probability distribution depends on mobility, node density, transmission range, different traffic scenarios, and various impairments of radio communications. The links between intermediate nodes frequently break due to the high mobile nodes moving out of each other’s transmission range. Therefore, estimation of path duration between source and destination decreases the chances of path breakage. The effective use of knowledge of path duration improves the performance and efficiency of routing protocols in VANETs [

In this paper, our main contribution is in terms of probabilistic and mathematical analysis of path duration using border-node based most forward progress within radius (B-MFR) routing protocol. For this analysis, a mathematical model to estimate path duration has been developed. B-MFR is a position-based routing protocol which selects next-hop node from the nodes belonging to the border area of the transmission range. The mathematical model for path duration estimation consists of the following modules.

The rest of the paper is organized as follows. In Section

Vehicular communications in on-road traffic environments have been realized through VANETs. In this network, vehicles termed as nodes share information through wireless communication. Dedicated short range communications (DSRC) [

V2V communication is the basic and primary aim in VANETs. It is pure ad hoc communication between two vehicles [

VANETs and its communication modes.

V2R communication is the combination of ad hoc network and fixed infrastructure networks [

In spite of the fundamental importance of estimating the path duration of communication links in VANET, there have been some mathematical and experimental studies in MANETs. The estimation of path duration in MANETs is proposed using several theoretical and analytical models. Many research work and models are also proposed for the implementation and improvement of the VANET. Some of the related research works about the path duration have been carried out in the recent past decade.

The path duration is an important design parameter for the better performance and routing decision in VANETs. Authors in [

To maximize the path duration of the route, link duration of that path must be maximum. If there is any link breaks in the route, it means whole route will be expired. The link stability and route lifetime are directly proportional to each other. In [

Path selection is important to decide the path duration of the route in VANETs. The shortest path is not always the best path in terms of the path duration. To maximize the path duration, the shorter average link duration of nodes should be avoided over the longer average link duration of nodes. In [

In [

To estimate the path duration of the route in VANETs, the use of the suitable routing protocol is also a critical factor. The position-based routing protocols may be the suitable routing protocols in VANETs. The routing protocols using the position information of the node in the networks are known as the position-based routing protocols. In these protocols, the next-hop node will be selected on the basis of maximum distance covered towards the destination within the sender’s transmission range. Some position-based routing protocols such as border-node based most forward progress within radius routing (B-MFR) [

B-MFR is a modified version of MFR (most forward progress within radius) routing protocol. It avoids using the interior nodes within the transmission range for packet forwarding. Therefore, in B-MFR, a packet is sent to the next-hop node, which is positioned nearer to border or on the border of the transmission range towards the destination (see Figure

B-MFR forwarding method.

In contrast, B-MFR exploits the dynamics of VANETs by choosing the node farthest from the source node to make the multihop forwarding more efficient. This routing protocol is especially important in VANETs when the node density is high and gives better performance in the dense networks like city traffic scenario. In the dense network, nodes on the border of the transmission range are easily available to forward the data. Therefore, the border-node based protocol helps to reduce the number of hops between source and destination. Moreover, B-MFR is a very useful routing concept to estimate expected distance and expected number of hops between source and destination node mathematically. These mathematical expressions can be used in estimation of path duration in VANETs.

The main drawback of the B-MFR is that the node farthest from the source cannot be sufficient selection criteria in a network with frequent topological changes. Further, B-MFR is not more suitable for sparse VANETs where smaller number of vehicles moving on the road and no or fewer road side units are present along the road. In such networks, it is important to guarantee high packet delivery rate with minimum delay and, therefore, robustness of the routing protocol is of main concern. To achieve this, we need to find the path duration by taking into consideration both the direction and speed of the vehicular nodes.

The randomness in vehicular traffic environment motivates the need for a suitable stochastic model to determine the ability of a routing protocol in terms of successful packet delivery and to analyze the network performance. Poisson distribution has been used in the model to realize dynamic traffic environment in VANETs. We followed Poisson distribution since we are interested in finding number of nodes present in specified forwarding area given the mean density of nodes in the network area. Further, the arrival of each node is independent. We assume that the nodes in the network area are deployed in a two-dimensional space according to a spatial Poisson process. The mathematical model attempts to estimate the average path duration using a position-based routing concept. Once path duration of a path is estimated well before the path breakage then the performance of a routing protocol can be enhanced significantly.

The role of routing protocol is very important in the estimation of the path duration in VANETs where the network topology frequently changes. For the proposed model, the B-MFR position-based routing protocol is used which we have explained in Section

Since we are interested in the path duration, therefore the main goal of this section is to derive a mathematical expression for path duration between two vehicles by deriving other useful mathematical expressions such as average number of hops and link duration. In this work, we use traditional traffic flow principle to describe the vehicular environment which will be more accurate for our path duration estimation. Vehicles are assumed to follow Poisson distributed arrivals for obtaining the probability distribution function (

In our proposed model, some assumptions are made which are given as follows.

Nodes are equipped with GPS receiver, digital map, and sensors.

No other fixed infrastructure for communication is present.

Transmission range for every node is same.

Average speed of the nodes in the network is constant.

Link duration for the nodes moving away from the source node.

The notations are used in our analysis. See Notations section.

Since the node closer to the border or on the border line covers maximum distance, it may reduce the number of hop counts between source and destination. It may not be possible to find even a single node at the extreme end of the sender’s transmission range. Therefore, we have considered a region around the extreme end of the transmission range towards the destination. That region is shown by shaded area in Figure

Area of shaded region

In Figure

Area of the shaded region can also be known as the area of interaction of the two circles, one with the radius

In this section, our aim is to find the probability of at least one node in the border region to improve the performance of the network. We assume that nodes are two-dimensionally Poisson distributed over the network with node density

Number of hops can be defined as the number of intermediate nodes in the route (source to destination). The main assumption is that each hop results in the same progress towards the destination, equal to the average distance covered by a node in one hop. Number of hops should be as low as possible. It will decrease the chances of link breakage and improve the path duration between nodes [

To determine the average number of hop counts, nodes within the transmission range

Direction of movement and speed of a node are very essential parameters for the calculation of the path duration in case of VANETs. Link duration depends on the relative velocity of the nodes as it can increase the link distance between nodes. The relative velocity between nodes is inversely proportional to the link duration. The relative velocity of the source node and next-hop node should be known to determine the expected link duration. Let

Link duration is the time for which the direct link between two nodes within the transmission range is active and it is a part of the route. It is necessary that next-hop node must be present within the transmission range of the source node to maintain the communication link between source and next-hop node. In this work, as we assumed, border node will be the next-hop node for each hop between the source and destination. Since the velocity of each node in the network is constant, it means that the links between source and next-hop node will always be maintained. As we have assumed

The path duration is one of the key parameters which could be useful to improve the performance and throughput of a highly dynamic network such as VANET. The path duration will be helpful in the process of path selection during the transmission of packet from source to destination [

In this section, extensive simulations have been performed to analyze the mathematical model for estimation of path duration presented in Section

The mathematical estimation of path duration has been simulated using network simulator (NS-2.34). MOVE (mobility model generator for vehicular networks) [

A set of five horizontal and five vertical roads crossing each other and thus making twenty five junctions is used as simulation area. The lane width used is 5 m. The velocity range 0–60 Km/h is used for node movement and transmission range varies from 100 m to 600 m. The other basic parameters used for the simulation are packet size of 512 bytes, traffic type as CBR, wireless channel, omnidirectional antenna, 802.11p as MAC wireless standard, and 300 s simulation time. The position-based routing protocol used for the simulation is B-MFR. After setting the network and traffic flow with above discussed parameters, we conducted the simulation. The average of ten different simulation runs is taken for data record where different source and different destination are selected. MATLAB is used to obtain analytical results for the mathematical formulation of the model.

The results obtained for the model have been analyzed in the following subsections. In each subsection, impact of a specific parameter on path duration has been analyzed. In each analysis, the simulation and analytical results have been discussed comparatively.

Figure

Average path duration versus transmission range.

The plots in Figure

Average path duration versus number of hops.

The node density is also a critical factor for the path duration as the possibility of finding suitable next-hop node is increased with the increased number of nodes. The results depicted in Figure

Average path duration versus node density.

VANETs are known for their mobility. The high velocity of nodes makes vehicles in and out from the transmission range of the source node, which causes most of the link or path breakage in the network. The results in Figure

Average path durations versus velocity of nodes.

In this paper, we have derived a mathematical model for estimation of path duration between source and destination nodes using position-based routing concept. This model has been verified by means of simulations and analytical results. Both the results are well approximated by the mathematical model. The mathematical model that we present is able to describe the effects of various road traffic parameters including the transmission range, number of hops, node density, and velocity of nodes on path duration. The use of the B-MFR routing protocol in this work is to find the routing path with maximum path duration. It demonstrates the selection of next-hop nodes positioned in a region around the extreme end of the transmission range towards the destination.

The mathematical analysis and simulation results reveal that high transmission range and smaller number of hops increase the path duration, but it decreases as the velocity of the nodes increases. From the above observations, it may appear that, in highly dynamic networks such as in VANET, it is very necessary to maintain the path duration between source and destination nodes. Therefore, message can be forwarded timely to reduce the large number of accidents on the road. Thus, we can say that the work in this paper helps us to improve the routing performance and decrease the number of path failures generally occurring in VANETs.

Transmission range of nodes (omnidirectional)

Region covered by transmission range

Selected region of the transmission range

Node density

Number of nodes in the shaded region

Number of nodes selected out of

Probability of successfully selecting a node

Probability of not selecting a node

Probability for selecting at least

Expected number of hops between source and destination nodes

Distance between next-hop node and destination node

Distance between source and next-hop node

Relative velocity between source node and next-hop node

Angle between

Angle between

Relative angle between source and next-hop node.

The authors declare that there is no conflict of interests regarding the publication of this paper.