Effective Transmission Coverage Area-Based Link Dynamics Characterization of VANET in Highway Scenario

Abstract

This paper uses the concept of effective transmission coverage area as a model for the derivation of analytic expressions in order to characterize the dynamic statistics of link lifetime, new link arrival rate, new link interarrival time, link breakage interarrival time, and so forth. Extensive simulations have been undertaken to verify the derived analytical expressions via generated mobility traces. Results demonstrate that the proposed analytical model can characterize the dynamic statistics well. Furthermore, the mathematical results of expected link life and expected new link arrival rate are provided to be used in analyzing the network connectivity. Combining with queueing theory, the upper bound connectivity of a VANET is obtained. This work thus provides a fundamental guideline on designing new mobility models, new routing protocols, and the corresponding performance analysis in VANET.

1. Introduction

VANET (Vehicular Ad Hoc Network) has emerged as a promising field of research and development, where advances in wireless and mobile ad hoc networks, global positioning systems, and sensor technologies can be collectively applied to vehicles and result in great market potential. It resembles MANET in its rapidly and dynamically changing network topologies due to the fast motion of vehicles. Vehicular communications are carried out by a set of wireless links between node pairs in multihop networks, where each node, except source and destination, behaves as a relay node to forward data packets to its next hop.

Therefore, link properties are essential to applications and services in such networks because they directly impact on many performance metrics, such as end-to-end delay, packet losses, and throughput. Moreover, link dynamics, as fundamental characteristics of network dynamics, can also be used to maximize routing performance, optimize topology control, and achieve the desired network performance. However, the characterization of link dynamics is not very easy, mainly because they are determined by a set of random factors, such as radio channels, dynamic transmission range, node mobility, and node pair distance. The mobility of vehicles in VANETs, especially, is generally constrained by predefined roads. Vehicle velocities are also restricted by speed limits, level of road congestion, and traffic control mechanisms (e.g., stop signs and traffic lights). The quality of the wireless links between mobile nodes in a VANET can still vary over time. In extreme cases, a communication link has a limited overall lifetime due to its limited transmission coverage area on road as nodes move toward and away from each other. Link dynamics may not only lead to network connectivity being time varying but significantly affect the performance of network control algorithms. In addition, different transmission coverage areas, when constrained by the predefined roads, may also lead to different link dynamics.

In this paper, we would like to systematically explore the behavior of link dynamics and its dependence on various network characteristics in VANET. In particular, considering the real scenario of VANET, we would like to focus on the concept of “effective transmission coverage area” as a function of the vehicular mobility, the intensity of vehicles on a road, and the width of the road. We use this concept to establish an analytical framework for communication links when one or both vehicles are moving on a highway with the same or opposite directions. Then we derive formal expressions for a number of link dynamic properties/measures. Their performance is evaluated by both analytical expressions and simulations. The comparisons also verify the correctness of our analysis and the success of the concept of the “effective transmission coverage area.”

The contributions of the paper are as follows. (1) The concept of “effective transmission coverage area” of node is firstly considered to analyze the link dynamics in VANET. In this concept, not only the speed limits, level of road congestion, and transmission range of node, but also the predefined road are considered, which are more able to characterize the communication environment of real highway scenario. (2) We investigate how the link properties change while varying the effective transmission coverage area. The mathematical expression is derived for link lifetime, link arrival, link breakage, and link changes. It provides a fundamental guideline on designing new mobility models and routing protocols. (3) We provide extensive simulation results to support our findings. On one hand, our results verified the derived mathematical expression in effective transmission coverage area; on the other hand, the results collectively assert that, in characterizing the link dynamics, there exists an important factor-effective transmission coverage area to be considered, which has been largely overlooked in the literature but is necessary. (4) We explore its application of the derived mathematical expression in analyzing the network connectivity. By using queueing theory and the mathematical expression of link dynamics, the upper bound connectivity of a VANET is obtained.

The organization of the rest of the paper is as follows. In Section 2, we discuss related work on the effects of mobility on link characteristics in MANET. Section 3 describes the system model and assumptions. We derive an analytical framework to study link dynamics in Section 4. In Section 5, the derived expressions are validated through simulation results. Section 6 gives the applications in connectivity analysis. Finally, Section 7 concludes this paper.

2. Related Work

Link dynamics plays an important role in topology control [1–3], routing design, data dissemination, and protocol optimization in both the traditional MANETs and the latest VANETs. Some efforts have been directed towards designing routing schemes that rely on the identification of stable links in the network. Nodes evaluate and predict stable links and then preferentially use them for routing. In [4], authors used simulations to study the probability densities of link lifetime and route lifetime for some mobility models. According to this study, the path duration seems to be a good metric for predicting the general trends in the performance of vehicular routing protocols. In SSA (Signal Strength-based Adaptive Routing) [5], signal stability and location stability are utilized to quantify the reliability of a link. With the signal stability, each node classifies its neighbors as either strongly connected or weakly connected according to the signal strength of received beacons generated periodically by its neighbors. In MORE (Movement-Prediction-based Routing) [6], the future positions of vehicles and the duration of links are predicted, by estimating the time needed for the transmission of data to decide which link is used. The RABR (Route Lifetime Assessment-Based Routing) [7–9] estimates the lifetime of a link based on the measured change rate of the signal strength averaged over the last few samples and then is used to select routes for data transmission. Also some efforts focused on the mobility trace to exact the link properties for data transmission. In Maxprop [10], the historical mobility data is used to estimate the link likelihoods. In references [11, 12], the ICTs (Intercontact-Times) of mobile nodes are estimated for data forwarding. As a random variable, ICT can reflect the link properties in some degree. The importance of link dynamics in protocol design is more and more considered; however, understanding and analyzing the link dynamics is still a big challenge nowadays.

Existing works have been focused on examining the effects of node mobility on link dynamics, such as link lifetime [13–17], link change rate [13, 18], link residual time, and link availability [13, 17, 19, 20]. The main achievements include the following. (i) Markovian model is an effective method to study relative movements and distance of a node pair. (ii) There exists a peak in the link lifetime distribution based on random mobility models. (iii) The PDF (probability density function) of link change interarrival time can be approximated by an exponential distribution with fairly high accuracy. Existing research on link properties falls into three categories: simulation-based, experimental-based, and analysis-based studies. Extensive simulation-based studies [14, 15, 21] have been performed to provide empirical distributions and statistical analysis of link lifetime and residual lifetime under different mobility models, including RWP (Random Waypoint), RW (Random Walk), Gauss-Markov, Manhattan model, and Reference Point Group Mobility model. Apart from simulation-based studies, the research literature also contains analytical work on mobility and protocol performance modeling. McDonald and Znati [22] used a RWP-like mobility model derived expressions for the probability of path availability and link availability for different initial conditions. Samar and Wicker [23] suggested a random model to analyze the link dynamics in mobile ad hoc network environment. Stochastic properties of the RW model were studied recently in [18, 24]. Su and Gerla proposed a model for link duration and evaluated it using the RW model. Based on various mobility models, Cai and Eun [25] investigated the relationship between characteristics of underlying mobility patterns and stochastic properties of contact-based metrics. Another research methodology is to study realistic mobility models extracted from real traces upon which link properties between mobile users are widely studied in [26, 27]. Bai and Krishnamachari [28] showed that the exponential model is a good fit for highway vehicle traffic. In [29], Zhu et al. studied the real taxi trace from Shanghai on the frequency and duration of transfer opportunities between taxies macroscopically and pointed out that the tail distribution of the ICT exhibits an exponential decay over a large range of timescale. Li et al. [30] carried out an extensive experiment involving tens of thousands of operational taxis in Beijing city and then found that the contact interval can be modeled by a three-segmented distribution.

The limitations of existing works on link properties are threefold. First, existing random mobility models, such as RWP model and its variants, have significant drawbacks towards the steady-state properties of moving speed and node distribution, which could lead to defective analysis and simulation on link studies. Second, the contact-based link dynamics analysis from real trace is seriously dependent on the ICT distributions; different distribution may change a lot over different region and time. Third, the prevalent assumption of the previous studies is that the transmission coverage area of each node is a constant, which is helpful in simplifying the analysis, at the cost of ignoring the effect of real network environments. For instance, in [23, 31], the authors showed that the relative movement of the transmitter-receiver pair can cause significant link variability, given that the nodes in the network are initially placed randomly and nodes move randomly at any directions. However, the results are not real in VANET environment.

There are two obvious features for VANET environment. One is that the movement of node is constrained by the predefined road and the vehicles' velocities are also restricted by speed limits; the other is that the effective transmission coverage areas for nodes are not equivalent to the transmission range of node. Up to now, there is little research work that has considered the effective coverage area to capture various link characteristics. Therefore, we aim to study the link properties considering not only the vehicular traffic variation but also the effective transmission coverage area, which include more parameters in the effective coverage area and thus we can do more systematic studies. To the best of our knowledge, such a study at this level has not been done before.

3. System Model and Assumptions

We consider a VANET in a highway scenario, where N vehicles are moving along the highway in the same or opposite directions. All the vehicles are equipped with OBU (Onboard Unit) devices for wireless communication. Each vehicle has a relay capability so that the vehicle can communicate with another one out of its transmission range through a multihop path. As a mobile node, each vehicle has the same transmission range indicated by the circle in Figure 1.

Figure 1

The effective transmission coverage area of a mobile node.

Since vehicles move on the road in just the same or opposite directions and vehicle's movement is constrained by the bound of the road, usually the transmission radius is much larger than the width of the high way. In the area outside the road, even if it is covered by a node's transmission range, there should be no nodes to communicate with the node on the road. Considering this real mobility scenario, we define the “effective transmission coverage area” of a node to be the overlapping area between the road segment and the circle as indicated by the grey area in Figure 1. For other nodes, it is similarly defined. Obviously, the transmission radius of node and the width of road are two main parameters to affect the effective transmission coverage area. A vehicle can only communicate with other vehicles that come into its effective transmission coverage area. The following list describes the assumptions made in characterizing the link dynamics:

• (1)

The terrain is flat so that we need not consider the “line of sight” effect. This would also allow us to estimate the transmitted signal when the SIR (Signal to Interference Ratio) remains high up to a certain distance R from the transmitter.

(2)

The mobile nodes have bidirectional communication links with each other when the distance is smaller than R.

(3)

A node in the network moves with a randomly chosen speed which is uniformly distributed between a m/s and b m/s. This mobility model allows for a more challenging test environment and facilitates the determination of low bounds on the link properties for a general VANET in highway scenario.

(4)

The mobility of a vehicle is independent of other vehicles on the highway. That is, the speed, direction, and location of vehicles are mutually uncorrelated.

(5)

The initial locations of vehicles in the network are modeled by a two-dimensional Poisson process with node density σ such that, for a network region D with an area A, the probability that D contains k nodes is given by

$$\begin{array}{cc}& P\left(k\text{\hspace{0.17em}\hspace{0.17em}vehicles\hspace{0.17em}\hspace{0.17em}in\hspace{0.17em}}\hspace{0.17em}D\right)=\frac{{\left(\sigma A\right)}^k{e}^{-\sigma A}}{k!}.\end{array}$$

(1)

This is frequently used to model the location distribution of nodes in VANET [27, 28] and it can model the aggregated behavior of nodes in a large network.

Assumptions (3)–(5) characterize both the individual and aggregated behavior of nodes in a large wireless network. Due to the large number of independent nodes operating in an ad hoc fashion, any correlation between nodes can be assumed to be insignificant. Although it is possible that some nodes may have similar intent and may move together, a large enough population of autonomous mobile nodes can be expected in the network so that the composite effect can be modeled by a random process.

4. Analysis of Link Dynamics

In this section, we shall analyze mathematically different performance measures of link dynamics including the expected link lifetime, the distribution of link lifetime, the expected rate of new link arrivals, the new link interarrival time distribution, the expected link breakage, and the link breakage interarrival time distribution. The mathematical expressions can be used for the modeling of link dynamics that are very important in designing good protocols in high mobility VANET environment.

4.1. Expected Link Lifetime

Figure 2 shows the transmission range of a node (say node $n_1$ ), which is a circle centered at node $n_1$ with radius R. This figure shows the trajectory of another node (say node $n_2$ ) entering the transmission range of node $n_1$ at E, traveling along $EF$ , and exiting the transmission range at F. Link lifetime is a random variable, which is defined as the time from a link appearing between a node pair to the moment of the link being broken. It is the time for node $n_2$ travelling from E to F.

Figure 2

Transmission range of node $n_1$ at O with respect to other vehicles' traveling.

To obtain the expected link lifetime, we create a Cartesian coordinate system with orthogonal unit vectors $\overrightarrow{i}$ and $\overrightarrow{j}$ along the x- and y-axes, respectively, where x-axis is parallel with the road and the origin is located at node $n_1$ . Let the velocity of node $n_1$ be ${\overrightarrow{v}}_{\mathrm{1}}=v_{\mathrm{1}}\overrightarrow{i}$ and let the velocity of node $n_2$ be ${\overrightarrow{v}}_{\mathrm{2}}=v_{\mathrm{2}}\cos \theta \overrightarrow{i}+v_{\mathrm{2}}\sin \theta \overrightarrow{j}$ , which makes an angle θ with the positive x axis. Note that θ is 0 or π in highway scenarios.

Hence, the relative velocity of node $n_2$ with respect to node $n_1$ is $\overrightarrow{v}={\overrightarrow{v}}_{\mathrm{2}}-{\overrightarrow{v}}_{\mathrm{1}}$ . The magnitude of node $n_2$ 's velocity in this coordinate system is

$$\begin{array}{cc}& v=\left|\overrightarrow{v}\right|=\sqrt{v_{\mathrm{1}}^{\mathrm{2}}+v_{\mathrm{2}}^{\mathrm{2}}+\mathrm{2}{v}_{\mathrm{1}}{v}_{\mathrm{2}}\cos \theta },\end{array}$$

(2)

where $v_{\mathrm{1}}=\left|{\overrightarrow{v}}_{\mathrm{1}}\right|$ and $v_{\mathrm{2}}=\left|{\overrightarrow{v}}_{\mathrm{2}}\right|$ . Let φ be the direction of node $n_2$ 's relative velocity to node $n_1$ . Obviously, in highway scenario, φ is also a bivalue function with value 0 or π, but it is significantly different from θ.

Let α be the angle of the point of entry E of node $n_2$ into the transmission range of $n_1$ . The value of α is measured clockwise from $O{x}^{\prime }$ which is the opposite direction of $Ox$ as shown in Figure 2. Thus point E has coordinate ( $-R\cos \alpha$ , $R\sin \alpha$ ) in the $x^{\prime }y$ coordinate system. Figure 2 shows that the direction of the relative velocity of node $n_2$ forms an angle 0 or π with the horizontal line along $EF$ .

Line $OM$ is perpendicular to $EF$ . The distance $d_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ that node $n_2$ travels inside node $n_1$ 's transmission range is

$$\begin{array}{cc}& d_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}=\left|\mathrm{2}R\cos \left(\alpha +\phi \right)\right|.\end{array}$$

(3)

Therefore, the time that node $n_2$ spends within the transmission range of node $n_1$ is

$$\begin{array}{cc}& t_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\left(\alpha ,v,\phi \right)=\frac{d_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}}{v}=\frac{\left|\mathrm{2}R\cos \left(\alpha +\phi \right)\right|}{v},\end{array}$$

(4)

which is the active time (and therefore lifetime) of the link between node $n_1$ and node $n_2$ . The average link lifetime of node $n_1$ as a function of its velocity $v_1$ can be calculated as the expectation of $t_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ over $\alpha ,\hspace{0.17em}\hspace{0.17em}v$ , and φ:

$$\begin{array}{cc}& T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}=E_{\alpha v\phi }\left[t_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\left(\alpha ,v,\phi \right)\right].\end{array}$$

(5)

Let the joint probability density function of α, v, and φ for nodes that enter the transmission range be $f_{\alpha v\phi }(\alpha ,v,\phi )$ . It can be expressed as

$$\begin{array}{cc}& f_{\alpha v\phi }\left(\alpha ,v,\phi \right)=f_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }\left(\alpha \hspace{0.17em}|\hspace{0.17em}v,\phi \right)·f_{v\phi }\left(v,\phi \right),\end{array}$$

(6)

where $f_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }\left(\alpha \hspace{0.17em}|\hspace{0.17em}v,\phi \right)$ is the conditional probability density function of α given the relative velocity v and φ and $f_{v\phi }\left(v,\phi \right)$ is the probability density function of v and φ. The conditional probability density function $f_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }(\alpha ,v,\phi )$ for nodes that enter the transmission range of node $n_1$ can now be derived by considering the direction of node $n_2$ 's relative velocity and the width of the highway.

Let the width of the highway be $2m$ . See Figure 2; given the direction φ of node $n_2$ 's relative velocity, the node can randomly come into the transmission range from a point on the arc of the circle $I_{\mathrm{0}}=[-\arcsin (d/R)+\phi ,\arcsin ((\mathrm{2}m-d)/R)+\phi ]$ , where d represents the distance between node $n_1$ and the roadside and the value of d should be $[0,2m]$ . For simplicity, in this paper, only the case $d=m$ is described in detail. For the other values of d, same conclusions can be obtained. For any entry point E, $OE$ makes an angle $\alpha \in I_{\mathrm{0}}$ with $O{x}^{\prime }$ . Let $OM=r$ , where M is the point of intersection of $CD$ and $EF$ . Then the value of r is $R\sin (\alpha +\phi )$ .

The direction of node $n_2$ 's relative velocity to node $n_1$ is just perpendicular to the diameter of the circle centered at node $n_1$ (i.e., origin O). Since nodes in the highway are assumed to be uniformly moved, a node entering the range with relative speed v can intersect this diameter in a range of $2m$ width with equal probability. As illustrated in Figure 3, the node's trajectory is equally likely to intersect the segment $CD$ at any point; hence the probability of location of this point of intersection is uniformly distributed on the segment.

Figure 3

Intersection of a vehicle trajectory with the transmission range of another vehicle.

Since α is the random variable representing the angle defining the point of entry of node $n_2$ in the effective transmission coverage region, for $\alpha \in I_{\mathrm{0}}$ , the conditional distribution function is

$$\begin{array}{cc}& F_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }=P\left(\alpha \hspace{0.17em}|\hspace{0.17em}v,\phi \right)={{\displaystyle \int }}_{-m}^r\frac{\mathrm{1}}{\mathrm{2}m}dr=\frac{R\sin \left(\alpha +\phi \right)+m}{\mathrm{2}m}.\end{array}$$

(7)

Hence, $f_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }(\alpha ,v,\phi )$ , the conditional probability density function, can be obtained by differentiating $F_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }$ . Its expression is

$$\begin{array}{cc}& f_{\alpha \hspace{0.17em}|\hspace{0.17em}v\phi }\left(\alpha \hspace{0.17em}|\hspace{0.17em}v,\phi \right)=\frac{R\cos \left(\alpha +\phi \right)}{\mathrm{2}m}{g}_{\mathrm{0}}\left(\alpha \right),\end{array}$$

(8)

where $g_0(\alpha )=u(\alpha -(-\arcsin (m/R)+\phi ))-u(\alpha -(\arcsin (m/R)+\phi ))$ , $u(·)$ is a step function with the equation given by

$$\begin{array}{cc}& u\left(·\right)=\left\{\begin{array}{cc}\mathrm{1}\hfill & x>\mathrm{0}\hfill \\ \mathrm{0}\hfill & x\le \mathrm{0}.\hfill \end{array}\right.\end{array}$$

(9)

$f_{v\phi }\left(v,\phi \right)$ is the probability density function of v and φ. For the mobile nodes entering the effective transmission coverage area, it can be determined by

$$\begin{array}{cc}& f_{v\phi }\left(v,\phi \right)=\frac{f\left(v_{\mathrm{2}},\theta \right)}{\left|J\left(v_{\mathrm{2}},\theta \right)\right|},\end{array}$$

(10)

where $f\left(v_{\mathrm{2}},\theta \right)$ is the probability density function of $v_2$ , θ and $J\left(v_{\mathrm{2}},\theta \right)=v_{\mathrm{2}}/\left|{\overrightarrow{v}}_{\mathrm{2}}-{\overrightarrow{v}}_{\mathrm{1}}\right|$ is the Jacobin matrix for the transform.

According to assumption (3), $v_2$ is uniformly distributed between a and b. θ is a bivalue function. Thus, their individual probability density functions are given by

$$\begin{array}{cc}& f_{v_{\mathrm{2}}}\left(v_{\mathrm{2}}\right)=\frac{\mathrm{1}}{b-a}\left\{u\left(v_{\mathrm{2}}-a\right)-u\left(v_{\mathrm{2}}-b\right)\right\},\\ f\left(\theta \right)=\frac{\mathrm{1}}{\mathrm{2}}\hspace{1em}\theta =\mathrm{0},\pi .\end{array}$$

(11)

Using $v=\left|v_{\mathrm{2}}-v_{\mathrm{1}}\right|$ and the angle φ of the relative velocity, we have

$$\begin{array}{cc}& v_{\mathrm{2}}=\sqrt{v^{\mathrm{2}}+v_{\mathrm{1}}^{\mathrm{2}}+\mathrm{2}v{v}_{\mathrm{1}}\cos \phi }.\end{array}$$

(12)

From (10) and (11), we can obtain

$$\begin{array}{cc}& f\left(v,\phi \right)=\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}vg\left(v,v_{\mathrm{1}}\right),\end{array}$$

(13)

where

$$\begin{array}{cc}& g\left(v,v_{\mathrm{1}}\right)=\frac{u\left(\sqrt{v^{\mathrm{2}}+v_{\mathrm{1}}^{\mathrm{2}}+\mathrm{2}v{v}_{\mathrm{1}}\cos \phi }-a\right)-u\left(\sqrt{v^{\mathrm{2}}+v_{\mathrm{1}}^{\mathrm{2}}+\mathrm{2}v{v}_{\mathrm{1}}\cos \phi }-b\right)}{\sqrt{v^{\mathrm{2}}+v_{\mathrm{1}}^{\mathrm{2}}+\mathrm{2}v{v}_{\mathrm{1}}\cos \phi }}.\end{array}$$

(14)

Hence, the joint density of α and v can be expressed as

$$\begin{array}{cc}& f_{\alpha v\phi }\left(\alpha ,v,\phi \right)=\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{R\cos \left(\alpha +\phi \right)}{\mathrm{2}m}vg\left(v,v_{\mathrm{1}}\right)g_{\mathrm{0}}\left(\alpha \right),\end{array}$$

(15)

which is a function of $v_1$ for given values of α, v, and φ. Thus, the expected link lifetime as a function of velocity $v_1$ is

$$\begin{array}{cc}& T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\left(v_{\mathrm{1}}\right)={{\displaystyle \sum }}_{k=\mathrm{0}}^{\mathrm{1}}{{\displaystyle \int }}_{\mathrm{0}}^{\infty }{{\displaystyle \int }}_{-\pi }^{\pi }{\left.f_{\alpha v}\left(\alpha ,v,\phi \right)\right|}_{\phi =k\pi }d\alpha \hspace{0.17em}dv={{\displaystyle \int }}_{\mathrm{0}}^{b-v_{\mathrm{1}}}{{\displaystyle \int }}_{-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)}^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)}\frac{\mathrm{1}}{b-a}·\frac{R^{\mathrm{2}}{\mathrm{c}\mathrm{o}\mathrm{s}}^{\mathrm{2}}\alpha }{\mathrm{2}m}·\frac{\mathrm{1}}{v-v_{\mathrm{1}}}d\alpha \hspace{0.17em}dv+{{\displaystyle \int }}_{\mathrm{0}}^{v_{\mathrm{1}}-a}{{\displaystyle \int }}_{-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)+\pi }^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)+\pi }\frac{\mathrm{1}}{b-a}·\frac{R^{\mathrm{2}}\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}\alpha }{\mathrm{2}m}·\frac{\mathrm{1}}{v-v_{\mathrm{1}}}d\alpha \hspace{0.17em}dv+{{\displaystyle \int }}_{v_{\mathrm{1}}+a}^{v_{\mathrm{1}}+b}{{\displaystyle \int }}_{-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)+\pi }^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(m/R\right)+\pi }\frac{\mathrm{1}}{b-a}·\frac{R^{\mathrm{2}}\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}\alpha }{\mathrm{2}m}·\frac{\mathrm{1}}{v-v_{\mathrm{1}}}d\alpha \hspace{0.17em}dv=\frac{\left(R^{\mathrm{2}}\arcsin \left(m/R\right)+m\sqrt{R^{\mathrm{2}}-m^{\mathrm{2}}}\right)}{\mathrm{2}m\left(b-a\right)}\mathrm{l}\mathrm{n}\left(\frac{b}{v_{\mathrm{1}}}\right).\end{array}$$

(16)

Equation (16) describes the expected link lifetime for a node as a function of its transmission range, velocity, and the width of road, which can characterize the size of the effective coverage region when the transmission range R is fixed.

Figure 4 plots the expected link lifetime for a node with increasing velocity and different values of m, while fixing other parameters at $a=0$  m/s, $b=50$  m/s, and $R=250$  m. The velocity of the nodes in the networks is assumed to be uniformly distributed between $[0,50]$  m/s. Two trends can be observed from Figure 4. One can see that the expected link lifetime for a node decreases rapidly as its velocity is increased. As an illustration, links last more than three times longer, on average, for a node moving with a velocity of 5 m/s as compared with a node moving with a velocity of 50 m/s. This is because the magnitude of the relative velocity is between $[0,55]$  m/s when $v_1$ is 5 m/s while it is between $[0,100]$  m/s when $v_1$ is 50 m/s. When traveling the same distance, it lasts smaller average time for the latter case. Another trend is the expected link lifetime decreases with increasing m value when the velocity $v_1$ remains constant. The reason is when m is increased, the effective transmission coverage area will increase and the average distance for node to travel will decrease. Thus the expected link lifetime is decreased. Both the above two trends and observations demonstrate that effective transmission coverage area is a significant factor affecting the expected link lifetime. It is necessary to consider the effective transmission coverage area when characterizing the link statistics.

Figure 4

Expected link lifetime for different $v_1$ and $m\hspace{0.17em}\hspace{0.17em}(a=0$  m/s, $b=50$  m/s, and $R=250$  m).

4.2. Link Lifetime Distribution

Link lifetime distribution is a very important property of link. The CDF (cumulative distribution function) of link lifetime can not only demonstrate the basic link property, but also be used to analyze other properties.

For a particular vehicle moving with a velocity $v_1$ , the CDF of the link lifetime is defined as

$$\begin{array}{cc}& F_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^{v_{\mathrm{1}}}\left(t\right)=P\left\{t_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\le t\right\}.\end{array}$$

(17)

Obviously, $F_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^{v_{\mathrm{1}}}\left(t\right)=\mathrm{0}$ for $t\le c$ , where c is a nonnegative number considering the effective transmission area. Its value is a function of road width m, $v_1$ , and the relative velocity. For $t>c$ it is

$$\begin{array}{cc}& F_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^{v_{\mathrm{1}}}\left(t\right)=P\left\{\frac{\mathrm{2}R\left|\cos \left(\alpha +\phi \right)\right|}{v}\le t\right\}=\mathrm{1}-P\left\{\left|\cos \left(\alpha +\phi \right)>\frac{vt}{\mathrm{2}R}\right|\right\}.\end{array}$$

(18)

Since

$$\begin{array}{cc}& P\left\{\left|\cos \left(\alpha +\phi \right)>\frac{vt}{\mathrm{2}R}\right|\right\}={{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{2}R/t}{{\displaystyle \int }}_{\alpha =-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)}^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)}{\left.f_{\alpha v\phi \left(\alpha ,v,\phi \right)}\right|}_{\phi =\mathrm{0}}d\alpha \hspace{0.17em}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{2}R/t}{{\displaystyle \int }}_{\alpha =-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)-\pi }^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)-\pi }{\left.f_{\alpha v\phi \left(\alpha ,v,\phi \right)}\right|}_{\phi =\pi }d\alpha \hspace{0.17em}dv={{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{2}R/t}{{\displaystyle \int }}_{\alpha =-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)}^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)}\frac{\mathrm{1}}{b-a}\frac{R\cos \alpha }{\mathrm{2}m}vg\left(v,v_{\mathrm{1}}\right)g_{\mathrm{0}}\left(\alpha \right)d\alpha \hspace{0.17em}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{2}R/t}{{\displaystyle \int }}_{\alpha =-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)-\pi }^{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(vt/\mathrm{2}R\right)-\pi }\frac{\mathrm{1}}{b-a}\frac{R\cos \alpha }{\mathrm{2}m}vg\left(v,v_{\mathrm{1}}\right)g_{\mathrm{0}}\left(\alpha \right)d\alpha \hspace{0.17em}dv,\end{array}$$

(19)

substituting (19) into (18), the CDF of the link lifetime for node $n_1$ moving with velocity $v_1$ can be obtained as

$$\begin{array}{cc}& F_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}^{v_{\mathrm{1}}}\left(t\right)=\mathrm{1}-\frac{R}{\mathrm{2}m\left(b-a\right)}+\left[{{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{2}R/t,b-v_{\mathrm{1}}\right)}\sqrt{\mathrm{1}-{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(\frac{vt}{\mathrm{2}R},\frac{\sqrt{R^{\mathrm{2}}-m^{\mathrm{2}}}}{R}\right)\right)}^{\mathrm{2}}}·\frac{v}{v+v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{2}R/t,v_{\mathrm{1}}-a\right)}\sqrt{\mathrm{1}-{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(\frac{vt}{\mathrm{2}R},\frac{\sqrt{R^{\mathrm{2}}-m^{\mathrm{2}}}}{R}\right)\right)}^{\mathrm{2}}}·\frac{v}{v-v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{m}\mathrm{a}\mathrm{x}\left(\mathrm{0},a+v_{\mathrm{1}}\right)}^{\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{2}R/t,b+v_{\mathrm{1}}\right)}\sqrt{\mathrm{1}-{\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(\frac{vt}{\mathrm{2}R},\frac{\sqrt{R^{\mathrm{2}}-m^{\mathrm{2}}}}{R}\right)\right)}^{\mathrm{2}}}·\frac{v}{v-v_{\mathrm{1}}}dv\right].\end{array}$$

(20)

Figure 5 plots the link lifetime CDF for different vehicular velocity and road width. The CDF increases quickly to approach probability 1 in a short time interval. For example, as shown in Figure 5(a), for $v_1=40$  m/s, the most average link lifetime is less than 15 seconds while little link lifetime lasts for more than this value. The link lifetime distribution is affected significantly by the velocity $v_1$ . With higher values of $v_1$ , the possibility that the link lifetime lasts for a small value is higher, which means the average link lifetime becomes smaller as $v_1$ increases. An interesting phenomenon is that the smallest link lifetime remains at a positive value, which is quite different from the zero values in most of the literature. This can be attributed to the use of the effective transmission coverage area and the mobility of the vehicle as presented in Section 3. The three aspects of relative direction of the vehicles, transmission radius, and road width have directly decided the smallest value of the link lifetime. Let s and $v_{\mathrm{m}\mathrm{a}\mathrm{x}}$ be the minimum value between E and F and the maximum relative velocity of other vehicles to node $n_1$ , respectively. In Figure 3, the minimum value is the length of $A_1{B}_1$ or $A_2{B}_2$ , which can be calculated as $s=\mathrm{2}\sqrt{R^{\mathrm{2}}-m^{\mathrm{2}}}$ . For velocity ${\overrightarrow{v}}_{\mathrm{1}}$ , when the direction of ${\overrightarrow{v}}_2$ is opposite and the magnitude of ${\overrightarrow{v}}_2$ is 50 m/s, $v_{\mathrm{m}\mathrm{a}\mathrm{x}}$ can be maximized; that is, $v_{\mathrm{m}\mathrm{a}\mathrm{x}}=\left|{\overrightarrow{v}}_{\mathrm{1}}\right|+\left|{\overrightarrow{v}}_{\mathrm{2}}\right|$ . Thus the smallest value of the link lifetime can be obtained by $s/v_{\mathrm{m}\mathrm{a}\mathrm{x}}$ .

Figure 5

The CDF of link lifetime ( $a=0$  m/s, $b=50$  m/s).

For example, when fixing $v_1=40$  m/s and $m=50$  m (the curve with circle in Figure 5(a)), while allowing node $n_2$ to move along roadside $A_1{B}_1$ or $A_2{B}_2$ in the opposite direction of node $n_1$ at a speed $v_2=50$  m/s, we obtain a link lifetime of 5.4433 s, which is the smallest nonzero value one can obtain in this extreme case. When fixing $v_1$ the same value and $m=10$  m, the smallest link lifetime is 7.603 s.

Similar observation can be made in Figure 5(b) for link lifetime CDF for different road widths m while fixing the speed $v_1$ at a certain value. In the case that $v_1$ is 20 m/s, $R=250$  m, and $m=250$  m, the smallest link lifetime is just zero using the above calculating method. It is consistent with the results shown in the curve with circle. The CDF of link lifetime increases quickly approaching probability 1. With the increase of road width m, the curve of the distribution shifts left, this means that the link lifetime becomes smaller when the road width m is increased in some degree.

4.3. Expected Rate of New Link Arrival

To analyze the rate of new link arrivals, we show again in Figure 6 the transmission range of node $n_1$ centered at O moving with velocity ${\overrightarrow{v}}_{\mathrm{1}}$ in the stationary x-y coordinate system. For given values of v and φ, a node with a relative velocity $\overrightarrow{v}$ would enter the transmission region within the next t seconds if it is currently located in the shaded region $D_a$ of Figure 6. This area consists of the set of points at most $vt$ meters away from the curve $\alpha \in [-\arcsin (m/R)+\phi ,\arcsin (m/R)+\phi ]$ when measured along the angle φ.

Figure 6

Analysis of the expected rate of new link arrivals.

From the geometry of Figure 6, one can show easily that the area of the shaded region $D_a$ is $A=vt·2m$ . Using assumption (5), the average number of nodes in $D_a$ is equal to $2mvt\sigma$ , where σ is the node density. Therefore, the average number of nodes in $D_a$ with velocity $\overrightarrow{v}$ is equal to ${\left.2mvt\sigma ·f(v,\phi )\right|}_{\phi =k\pi }dv$ , $k=\mathrm{0,1}$ . This is just the average number of nodes with velocity $\overrightarrow{v}$ entering the transmission range within the next t seconds. Let $\stackrel{-}{\eta }\left(v_{\mathrm{1}}\right)$ be the total number of nodes over all possible values of v and φ. Then

$$\begin{array}{cc}& \stackrel{-}{\eta }\left(v_{\mathrm{1}}\right)={{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\mathrm{2}m\sigma vtf\left(v,\phi \right)\right|}_{\phi =\mathrm{0}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\mathrm{2}m\sigma vtf\left(v,\phi \right)\right|}_{\phi =\pi }dv.\end{array}$$

(21)

Thus, the expected number of nodes entering the transmission region per second, or equivalently the rate of new link arrivals, is given by

$$\begin{array}{cc}& \dot{\stackrel{-}{\eta }}\left(v_{\mathrm{1}}\right)={{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\mathrm{2}m\sigma vf\left(v,\phi \right)\right|}_{\phi =\mathrm{0}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\mathrm{2}m\sigma vf\left(v,\phi \right)\right|}_{\phi =\pi }dv=\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\left({{\displaystyle \int }}_{v=\mathrm{0}}^{b-v_{\mathrm{1}}}\mathrm{2}m\sigma v^{\mathrm{2}}\frac{\mathrm{1}}{v+v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{v_{\mathrm{1}}-a}\mathrm{2}m\sigma v^{\mathrm{2}}\frac{\mathrm{1}}{v-v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=a+v_{\mathrm{1}}}^{b+v_{\mathrm{1}}}\mathrm{2}m\sigma v^{\mathrm{2}}\frac{\mathrm{1}}{v-v_{\mathrm{1}}}dv\right)=\frac{m\sigma }{b-a}\left({{\displaystyle \int }}_{v=\mathrm{0}}^{b-v_{\mathrm{1}}}{v}^{\mathrm{2}}\frac{\mathrm{1}}{v+v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{v_{\mathrm{1}}-a}{v}^{\mathrm{2}}\frac{\mathrm{1}}{v-v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=a+v_{\mathrm{1}}}^{b+v_{\mathrm{1}}}{v}^{\mathrm{2}}\frac{\mathrm{1}}{v-v_{\mathrm{1}}}dv\right)=\frac{m\sigma }{b-a}\left(b^{\mathrm{2}}-\mathrm{4}{v}_{\mathrm{1}}a-v_{\mathrm{1}}^{\mathrm{2}}\left(\mathrm{3}-\mathrm{2}\ln \frac{v}{b}\right)\right).\end{array}$$

(22)

An important point to observe from (22) is that the expected rate of new link arrivals for a node is directly proportional to the average density σ of nodes in the network. It is also directly proportional to the road width. These results are shown in Figure 7 as a function of the node moving velocity $v_1$ for $a=0$  m/s, $b=50$  m/s, and $R=250$  m. Assume $A_1=\pi R^2$ is the area of the transmission region of node and ρ is the number of nodes within the transmission region; then the node density is $\sigma ={\rho /A}_{\mathrm{1}}$ . For simplicity, the parameter ρ is used in the analysis. Figure 7(a) shows further the link arrival performance for a fixed road width of $m=50$  m and for different node numbers. For example, when $\rho =25$ , the expected arrival rate increases quite slowly at low velocity $v_1$ and then increases linearly at high velocity. As ρ decreases, the overall behavior is similar but at a lower arrival rate level. In the smallest case ( $\rho =1$ ) of the figure, one can see that the arrival rate just increases slowly and linearly. Figure 7(b) considers the case of $\rho =10$ and transmission radius is 250 m. Take $m=50$ meters as an example. Under this case, the expected arrival rate is proportional to the velocity $v_1$ . As a summary of Figure 7, the expected new link arrival rate always increases with the velocity of $v_1$ . It means that there are more opportunities for the mobile nodes to communicate with each other when the vehicles move faster. Although this can be seen intuitively, the more important aspect is that we obtain the mathematical expression and together with those from Figure 5, we demonstrate that there is tradeoff between opportunities and duration of the link connection. Such tradeoff should be considered in designing new network protocol based on the link dynamics model.

Figure 7

Performance of the expected new link arrival rate.

4.4. New Link Interarrival Time Distribution

The cumulative distribution function of new link interarrival time is defined as

$$\begin{array}{cc}& F_{\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}}^{v_{\mathrm{1}}}\left(t\right)=P\left(\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\hspace{0.17em}\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\le t\right).\end{array}$$

(23)

As seen in Section 4.3, a mobile node with relative velocity $\overrightarrow{v}$ currently located in $D_a$ will enter the effective transmission region within the next t s. Thus, given $\overrightarrow{v}$ , the probability that the link interarrival time is not more than t is equal to the probability that there exists at least one node in $D_a$ with $\overrightarrow{v}$ . Using assumption (5),

$$\begin{array}{cc}& P\left(\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\hspace{0.17em}\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\le t\hspace{0.17em}|\hspace{0.17em}v,\phi \right)=P\left(\mathrm{a}\mathrm{t}\hspace{0.17em}\hspace{0.17em}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}\hspace{0.17em}\hspace{0.17em}\mathrm{1}\hspace{0.17em}\hspace{0.17em}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}{D}_a\hspace{0.17em}|\hspace{0.17em}v,\phi \right)=\mathrm{1}-e^{-\mathrm{2}m\sigma tv}.\end{array}$$

(24)

Hence, the CDF of new link interarrival time can be expressed as

$$\begin{array}{cc}& F_{\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}}^{v_{\mathrm{1}}}\left(t\right)={{\displaystyle \sum }}_{k=\mathrm{0}}^{\mathrm{1}}{\left.F_{\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}}^{v_{\mathrm{1}}}\left(t\right)\right|}_{\phi =k\pi }={{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\left(\mathrm{1}-e^{-\mathrm{2}m\sigma tv}\right)f\left(v,\phi \right)\right|}_{\phi =\mathrm{0}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{\infty }{\left.\left(\mathrm{1}-e^{-\mathrm{2}m\sigma tv}\right)f\left(v,\phi \right)\right|}_{\phi =\pi }dv={{\displaystyle \int }}_{v=\mathrm{0}}^{b-v_{\mathrm{1}}}\left(\mathrm{1}-e^{-\mathrm{2}m\sigma tv}\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v+v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{v_{\mathrm{1}}-a}\left(\mathrm{1}-e^{-\mathrm{2}m\sigma tv}\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v-v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{a+v_{\mathrm{1}}}^{b+v_{\mathrm{1}}}\left(\mathrm{1}-e^{-\mathrm{2}m\sigma tv}\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v-v_{\mathrm{1}}}dv.\end{array}$$

(25)

Figure 8 illustrates the new link interarrival time distribution for a node moving with velocity $v_1$ for $a=0$  m/s, $b=50$  m/s, $R=250$  m, and $\sigma =10/A_1$ . Fixing $m=50$ meters in Figure 8(a), we observe that the new link interarrival time shifts left as the increase of $v_1$ . When $v_1=5$  m/s, $F_{\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}}^{v_{\mathrm{1}}}$   ( $t=15$ ) is about 0.8, while the probability is about 0.99 when $v_1=50$  m/s. This means the average new link interarrival time decreases with the increase of $v_1$ . Similarly, fixing $v_1=20$  m/s in Figure 8(b), it can be seen that different value of m also significantly affects the new link interarrival time. With the increase of m, the new link interarrival time decreases. This is because the bigger the value of m, the bigger the effective transmission coverage region that can be used. Consequently, more nodes may come into the transmission range and the new link interarrival time becomes smaller at some probability.

Figure 8

New link interarrival time distribution with $a=0$  m/s, $b=50$  m/s.

4.5. Expected Link Breakage Rate and Link Change Rates

Expected link breakage rate is also a very important metric to reflect the link dynamics, which is defined as the expected number of link breakages per second observed by a single node. Any change in the set of links of a node may be due to either a new link arrival or a currently active link breakage. Thus, the expected link change rate for a node is equal to the sum of expected new link arrival rate and the expected link breakage rate. The expected arrival of new link is expressed in (22).

To determine the expected link breakage rate, suppose that the network is formed at time 0. Let the total number of new link arrivals for a node between $t=0$ and $t=t_0$ be $\eta (t_0)$ and let the total number of new link breakages for the node during the same interval be $\mu (t_0)$ . Let the change of neighbor number of the node at time $t=t_0$ be $N(t_0)$ . Thus,

$$\begin{array}{cc}& \eta \left(t_{\mathrm{0}}\right)-\mu \left(t_{\mathrm{0}}\right)=N\left(t_{\mathrm{0}}\right).\end{array}$$

(26)

And $N(t_0$ ) is a stochastic process. Dividing both sides by $t_0$ and taking the limit as $t\to \infty$ ,

$$\begin{array}{cc}& \underset{t\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left(\frac{\eta \left(t_{\mathrm{0}}\right)}{t_{\mathrm{0}}}-\frac{\mu \left(t_{\mathrm{0}}\right)}{t_{\mathrm{0}}}\right)=\underset{t\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{N\left(t_{\mathrm{0}}\right)}{t_{\mathrm{0}}}.\end{array}$$

(27)

With no loss of generality, we assume $N(t)$ to be ergodic. Let $\dot{\mu }$ be the expected rate of link breakages. By periodicity, the expected average is equal to the time average in the arrival process and breakage process. That is, we have ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t_{\mathrm{0}}\to \infty }\left(\eta \left(t_{\mathrm{0}}\right)/t_{\mathrm{0}}\right)=\dot{\stackrel{-}{\eta }}$ , ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t_{\mathrm{0}}\to \infty }\left(\mu \left(t_{\mathrm{0}}\right)/t_{\mathrm{0}}\right)=\dot{\mu }$ . Since the number of neighbors of a vehicle in the highway scenario is always bounded, the change of neighbor number is ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t_{\mathrm{0}}\to \infty }\left(N\left(t_{\mathrm{0}}\right)/t_{\mathrm{0}}\right)=\mathrm{0}$ . This implies that $\dot{\mu }=\dot{\stackrel{-}{\eta }}$ ; that is, the expected rate of link breakages is equal to the expected rate of new link arrivals at equilibrium. Thus, the expected link change arrival rate $\dot{\gamma }\left(v_{\mathrm{1}}\right)=\dot{\stackrel{-}{\eta }}\left(v_{\mathrm{1}}\right)+\dot{\mu }\left(v_{\mathrm{1}}\right)=2\dot{\stackrel{-}{\eta }}\left(v_{\mathrm{1}}\right)$ , where $\dot{\stackrel{-}{\eta }}\left(v_{\mathrm{1}}\right)$ is given in (22).

The expected link change arrival rate as a function of the node velocity $v_1$ is plotted in Figure 9, where $a=0$  m/s, $b=50$  m/s, and $R=250$  m. Obviously, the expected link change arrival rate is directly proportional to the average density of nodes in the network. For same parameter values, the expected link change arrival rate is about twice as big as the new link arrival rate. The link change arrival rate increases with the increasing velocity of $v_1$ and also increases with the increasing value of ρ when fixing the value of road width. As shown in Figure 9(a), when $\rho =25$ , the expected link change rate increases quite slowly at low velocity $v_1$ and then increases linearly at high velocity. As ρ decreases, the overall behavior is similar but at a lower arrival rate level. In the smallest case ( $\rho =1$ ) of the figure, one can see that the arrival rate just increases slowly and linearly. Figure 9(b) considers the case of $\rho =10$ and transmission radius is 250 m. Take the $m=50$  m as an example. Under this case, the expected link change rate is proportional to the velocity $v_1$ .

Figure 9

Expected link change arrival rate.

4.6. Link Breakage Interarrival Time Distribution

To derive the link breakage interarrival time distribution, we proceed in a manner similar to Section 4.4. With respect to Figure 10, the shaded region $D_b$ consists of all points not more than $vt$ meters away along angle φ from the partial circle $I_{\mathrm{0}}=\left[\pi -\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(m/R)+\phi ,\pi +\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(m/R)+\phi \right]$ . It is easy to see that a node moving with an angle φ can be out of its transmission range from a point on this part of the circle. Given its relative velocity $\overrightarrow{v}$ , a node currently located in $D_b$ satisfies the condition that it will be outside the transmission region of node $n_1$ within the next t seconds, thus breaking the link between them. Note that $D_b$ also includes the possibility of nodes that are currently outside the transmission region of node $n_1$ and have yet to form a link with $n_1$ . The area of the shaded range $D_b$ is $A=2mvt$ . For given v and φ, the probability that the link breakage interarrival time is not more than t is equal to the probability that there is at least one node in $D_b$ with velocity $\overrightarrow{v}$ . Using the assumed Poisson model, we have

$$\begin{array}{cc}& P\left(\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}\hspace{0.17em}\hspace{0.17em}\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{g}\mathrm{e}\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\hspace{0.17em}\hspace{0.17em}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\le t\hspace{0.17em}|\hspace{0.17em}v,\phi \right)=P\left(\mathrm{a}\mathrm{t}\hspace{0.17em}\hspace{0.17em}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}\hspace{0.17em}\hspace{0.17em}\mathrm{1}\hspace{0.17em}\hspace{0.17em}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}{D}_b\hspace{0.17em}|\hspace{0.17em}v,\phi \right)=\mathrm{1}-e^{-\mathrm{2}mvt\sigma }.\end{array}$$

(28)

Figure 10

Analysis of the link breakage interarrival time.

Hence, the CDF of link breakage interarrival time is given by

$$\begin{array}{cc}& F_{\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}}^{v_{\mathrm{1}}}\left(t\right)={{\displaystyle \sum }}_{k=\mathrm{0}}^{\mathrm{1}}{{\displaystyle \int }}_{\mathrm{0}}^{\infty }{\left.\left(\mathrm{1}-e^{-\mathrm{2}mvt\sigma }\right)f\left(v,\phi \right)\right|}_{\phi =k\pi }dv={{\displaystyle \int }}_{v=\mathrm{0}}^{b-v_{\mathrm{1}}}\left(\mathrm{1}-e^{-\mathrm{2}mvt\sigma }\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v+v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{v=\mathrm{0}}^{v_{\mathrm{1}}-a}\left(\mathrm{1}-e^{-\mathrm{2}mvt\sigma }\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v-v_{\mathrm{1}}}dv+{{\displaystyle \int }}_{a+v_{\mathrm{1}}}^{b+v_{\mathrm{1}}}\left(\mathrm{1}-e^{-\mathrm{2}mvt\sigma }\right)\frac{\mathrm{1}}{\mathrm{2}\left(b-a\right)}\frac{v}{v-v_{\mathrm{1}}}dv.\end{array}$$

(29)

The right-hand sides of (25) and (29) are the same, implying that the distributions of the link breakage interarrival time and the new link interarrival time are the same. It is consistent with the results in Section 4.5 that the expected rate of link breakages is equal to the expected rate of new link arrivals.

5. Verification

We use SUMO [32] simulations to generate exponential mobility model of a VANET for comparisons. SUMO is open source, microscopic, space-continuous, and time discrete vehicular traffic generator that can generate the movements of the vehicle nodes. In our simulations, vehicles speed is uniformly distributed between $a=0$  m/s and 50 m/s, and they move randomly in two opposite directions. All the vehicles that moved on the highway have the capability to communicate with each other using short-range wireless interfaces such as IEEE 802.11 and DSRC (dedicated short range communication) [33]. The transmission range of the vehicles is set to be 250 m. The simulation duration is 60 minutes.

Figure 11 shows the expected link lifetime and normalized frequency of link lifetime obtained by simulations and the proposed analytical analyses, respectively, where $m=25$  m. Figure 11(a) shows that the simulation results and the analytical analyses matched very well, except that the gap is slightly bigger when the velocity is very small. This is probably due to the finite-duration simulations where there are only a limited number of nodes at very small relative velocities. These nodes may have very long link lifetimes, and their contributions are not properly reflected in our performance figures. One can also see that the difference between simulation and analytical results rapidly decreases with the decreasing expected link lifetime and increasing velocity $v_1$ . Figure 11(b) is the normalized frequency of link lifetime with $v_1=$ 20 m/s. It shows that the shape of normalized frequency has a skewness to right, and the peaked value occurs at a link lifetime of approximately 8 s. More than 90% of the link lifetime lies within $[\mathrm{6}\hspace{0.17em}\mathrm{s},\mathrm{35}\hspace{0.17em}\mathrm{s}]$ . Hence, only less than 10% of the link lifetime is longer than 35 s, which shows a good agreement with the traffic on the highway road. It is also observed that the link lifetime is larger than 6 s (since the frequency is 0 when $t<6$ ), which is in good agreement with the results about the link lifetime distribution in Section 4.2.

Figure 11

Simulation verifications. (a) Expected link lifetime and (b) normalized frequency of link lifetime.

Figure 12 gives the comparison between simulation and our earlier analysis for the link arrival rates with $\rho =10$ and the normalized interarrival frequency with $v_1=20$  m/s, where the road width is $m=50$  m. Given the intensity value of vehicles on the road, Figure 12(a) shows that the arrival rate obtained by simulation increases linearly with the increase of the velocity of $v_1$ , which is consistent with our analytical results. Figure 12(b) compares simulation with analysis for the normalized frequency of the interarrival time where $v_1$ is fixed. The good match demonstrates the correctness of the proposed analytical model as well. From this figure, we also found that the interarrival time falls off exponentially, which is consistent with the results in the literature obtained by other analysis methods [30].

Figure 12

Simulation verifications. (a) New arrival rate and (b) interarrival time frequency.

6. Analysis of Network Connectivity Based on Link Dynamics

The characterization of link dynamics gives us a formal understanding of their dynamic behavior of VANET in highway scenario. For example, expected link lifetime and expected link change rate are two important metrics to reflect the topology dynamic and network connectivity. Here, we use the knowledge of expected link lifetime $T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ and expected link arrival rate $\dot{\stackrel{-}{\eta }}$ to investigate their relationship with the average node degree for estimating the network connectivity. We use a graph, denoted $G(t)$ , to represent the VANET at time t. According to the definition of graph connectivity in [34], the connectivity of the network $G(t)$ at time t denoted by $k(G(t))$ is defined as the maximum value of k, for which a connected network $G(t)$ is k-connected. Let $d_{G(t)}^{n_i}$ denote the degree of node $n_i$ in $G(t)$ , which is the edge number of $G(t)$ incident with node $n_i$ . Let $s(G(t))$ and $E(d_{G(t)})$ be the minimum degree and the average degree of $G(t)$ , respectively, where $E(d_{G(t)})={\sum }_{i=\mathrm{1}}^{N(t)}{d}_{G(t)}^{n_i}$ , $N(t)$ is the node number in $G(t)$ at time t. From the graph theory [11], $k(G(t))$ , $s(G(t))$ and $E(d_{G(t)})$ satisfy the inequality

$$\begin{array}{cc}& k\left(G\left(t\right)\right)\le s\left(G\left(t\right)\right)\le E\left(d_{G\left(t\right)}\right).\end{array}$$

(30)

From (30), $E(d_{G(t)})$ is the upper bound of the connectivity of a VANET $G(t)$ . Thus $E(d_{G(t)})$ is a key parameter for evaluating network connectivity. Next, we study the property of VANET connectivity based on $E(d_{G(t)})$ as follows.

Each node in $G(t)$ is considered to be associated with a queueing system. As shown in Figure 1, for the queueing system of node $n_1$ , the event that any other node, denoted as $n_2$ , is moving into its effective transmission coverage area is referred to as a new arrival coming into the system. The event of node $n_2$ moving outside the effective transmission area is regarded as a departure from the system. According to Little's law of queueing system [35]: the average number of customers N in the system is equal to the average arrival rate λ of customer to the system multiplied by the average system time T per customer; that is, $N=\lambda T$ . Because the link between the node pair ( $n_1$ , $n_2$ ) immediately forms once node $n_2$ travels into node $n_1$ 's effective transmission coverage area, there is no waiting time for node $n_2$ in the queueing system of node $n_1$ . The system time that node $n_2$ spends is from the instant of its arrival into the effective transmission area of node $n_1$ to the instant of its departure from this area, which is equivalent to the expected link lifetime of the node pair ( $n_1$ , $n_2$ ); that is, $T=T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ . The average customer arrival rate $\dot{\stackrel{-}{\eta }}$ to the queueing system of node $n_1$ is equivalent to the expected new link arrival rate λ. Consequently, the average number of customers of a mobile node is equivalent to the average number of neighbors of that node; that is, $N=E(d_{G(t)})$ . Hence, with (16), (22), and (30), $E(d_{G(t)})$ is represented as

$$\begin{array}{cc}& k\left(G\left(t\right)\right)\le E\left(d_{G\left(t\right)}\right)=\dot{\stackrel{-}{\eta }}·T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}},\end{array}$$

(31)

where $T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ derived from (16) is the function of node's speed, width of the road m, and transmission range R of node. $\dot{\stackrel{-}{\eta }}$ derived from (22) is the function of node density, speed, m, and R.

Figure 13 illustrates the relationship among $E\left(d_{G\left(t\right)}\right)$ , node density, speed, and road width for a VANET $G(t)$ with a fixed transmission radius $R=250$  m. As an illustration in Figure 13(a) fixing the road width at 50 m for the network with a node density $\sigma =\mathrm{20}/A_{\mathrm{1}}$ , when nodes move at 20 m/s, the expected number of neighboring nodes $E\left(d_{G\left(t\right)}\right)$ observed by a specific node at an arbitrary time t is 4. In Figure 13(a), we found that the impact of node speed on average node degree $E\left(d_{G\left(t\right)}\right)$ is more significant than that of the node density. This is because the expected new link arrival rate $\dot{\stackrel{-}{\eta }}$ rises linearly with the increase of the node density, while the expected link lifetime $T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ decreases exponentially with the growth of node speed. According to this figure, given a fixed σ, $E\left(d_{G\left(t\right)}\right)$ is generally more than 3 times larger for nodes moving at speed of 5 m/s than those at the speed of 25 m/s. Similar results are obtained in Figure 13(b) which shows the relationship among $E\left(d_{G\left(t\right)}\right)$ , speed, and road width when fixing $\sigma =\mathrm{10}/A_{\mathrm{1}}$ . From this figure, we observe that the width of road has also significant effect on $E\left(d_{G\left(t\right)}\right)$ as well as the vehicle's speed. Thus the size of the effective transmission area can directly affect the average node degree. Given (16), (22), and (31), we found the upper bound connectivity of a VANET can be directly obtained from the values of expected link lifetime and the expected link arrival rate. Hence these two metrics establish a clear relationship between network connectivity and node mobility in VANET.

Figure 13

Upper bound of connectivity of a VANET.

7. Conclusions

Effective transmission coverage area and mobility are both significant factors affecting the performance of VANET, in which the communication links pass into and out of existence as the nodes move toward and from one another. In this paper, we mainly considered the effective coverage and vehicular mobility to explore the behavior of communication links in a systematic manner in VANET. An analytical model is proposed to address the general problem of mobility based on the effective coverage area in VANETs, which can analyze the dynamic statistics for link lifetime, new link interarrival time, link breakage interarrival time, and so forth. The formal expressions for the above link dynamic properties have been obtained. Furthermore, the analytical results are verified by extensive simulations and finally we applied two main metrics of the link dynamics metrics, expected link lifetime $T_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}$ and expected link arrival rate $\dot{\stackrel{-}{\eta }}$ , to formulate the upper bound connectivity of a VANET.

Our study focused on the effective transmission coverage area in this paper is especially helpful on mobility related research issues of VANETs. Our further work will explore a number of applications of our analytic framework, including the design of protocols for routing, neighbor discovery, and topology control and data dissemination.