- Research
- Open Access
Available connectivity analysis under free flow state in VANETs
- Chen Chen^{1}Email author,
- Lei Liu^{1},
- Xiaobo Du^{1},
- Xiaolu Wei^{1} and
- Changxing Pei^{1}
https://doi.org/10.1186/1687-1499-2012-270
© Chen et al.; licensee Springer. 2012
- Received: 27 March 2012
- Accepted: 17 July 2012
- Published: 23 August 2012
Abstract
Emerging of inter-vehicle communication gives vehicles opportunities to exchange information within limited radio ranges and self-organize in Ad Hoc manner into Vehicular Ad Hoc Networks (VANETs). However, due to strong mobility, limited market penetration rate, and lack of roadside units, connectivity is obviously a scarce resource in VANETs. Further, only depending on direct connectivity, i.e. one-hop connected links between vehicles, is far from the continuous growing communication demands in VANETs, such as inter-vehicle amusement, cooperative collision avoidance, inter-vehicle emergency notification etc. Therefore, the indirect connectivity from multi-hop forwarding and store-carry-forward strategy is also a necessary and powerful complement especially to the case where direct connections are hardly obtained. In this article, we define a new metric named available connectivity which involves both direct and indirect connectivity. By deep analyzing the statistical properties of direct and indirect connectivity in free flow state, the proposed available connectivity is obtained and quantified to increase the information dissemination opportunities for vehicles especially in a relatively slow topology changing scenario. Numerical results show that the available connectivity could provide better references for different VANETs applications and has potential relationships with many network parameters.
Keywords
- Vehicular Ad Hoc Networks
- Connectivity
- Mobility
Introduction
Vehicular Ad hoc Networks [1] (VANETs) are distributed, self-organizing communication networks built up by moving vehicles, and are thus characterized by very strong node mobility and limited degrees of freedom in the mobility patterns. The discussed IEEE 1609 Wireless Access in Vehicular Environments [2] draft is being developed for VANETs applications including mainly safety-related scenarios, such as Cooperative forward Collision Warning [3] (CCW) system, traffic signal violation warning [4], lane change warning [5] and some information applications [6]. All these applications greatly rely on the successful packets exchanging which is based on reliable links or node connectivity on a road segment. Thereupon, network connectivity is fundamental and crucial to any practical application in VANETs. Due to mobility, vehicles on a road segment can not only communicate with each other directly, they can also deliver the packets indirectly to the receiver by store-carry or multi-hop forwarding. Although connectivity can be obtained directly or indirectly, it is still a limited resource especially during sparse communication environment such as on highway or with low market penetration ratio [7], i.e. the number of equipped vehicles to the number of total vehicles. Limited connectivity directly affects the possible speed and range at which information can be disseminated over a VANET, and hence limits the up-to-datedness of the shared information that can be achieved.
The rest of the article is organized as follows. The following section provides the motivation behind our works and discusses the possible usages of available connectivity in VANETs. Section “Related studies” presents the related works. Section “Available connectivity definition and assumption” introduces our definition of available connectivity and assumptions. In Section “Statistical analysis of available connectivity”, the statistical analysis of direct and indirect connectivity is proposed. Numerical results and evaluation are presented in Section “Numerical results”. Finally, this article ended with some conclusion given in the last section.
Motivation
In this study, we define the connectivity as the number of achievable connections directly or indirectly. The definition can explore the maximum possibility for messages to be widely or quickly transmitted especially in safety-related applications contexts. The connection possibility is investigated on statistical characteristics in terms of either connected duration length or inter-vehicle distances. To reflect the high dynamism of connectivity due to different mobility in VANETs, the influence of velocity is also introduced.
The elaborate investigation to available connectivity in VANETs can make many otherwise complex problems easy. For example, to design an admission control strategy in VANETs, available connectivity can provide important reference for connectivity improvement by introducing specific vehicles. To evaluate the performance of emergent applications in VANETs, available connectivity can directly determine the successful delivery ratio of emergency notifications through given vehicles. On the other hand, to spread out the emergency notification messages quick and far enough, available connectivity could also be taken as a better choosing criterion or metric for effective forwarders. To route the packets based on available connectivity, throughput improvement can readily be obtained and backup multi-path may also be figured out by different links. In a dense network, available connectivity can also offer a threshold reference on packet generation rate to maintain sufficient connectivity but does not make the overall network congested. In summary, based on the full analysis to the proposed available connectivity, many works could be done to improve the safety, comfortableness and efficiency in VANETs.
Related studies
Although connectivity analysis is a classical problem in wireless communication networks, it has been a hot topic of interests in VANETs, especially due to the recently increasing research activities. From our viewpoint, the connectivity research in VANETs can be classified into two categories: one for investigating the lifetime properties of connected links or paths; another for investigating the inter-vehicle distances.
Researches on lifetime properties of connectivity focus on the statistical analysis of connection duration length. The authors of [8] discussed the connection duration length expression in detail considering the velocity vector involving movement directions. The numerical results in their article found high relative velocities imposing a hard task on some cooperative maneuvers including underlying routing protocols. In their work, the short connection duration was enough for emergency notification scenarios. In [12], the connection duration length between two vehicles was figured out, given their speed, direction and radio ranges. According to the obtained connected period equation, an admission control strategy was proposed to determine whether the next vehicle is allowed to inject to the current traffic flow without disrupting the ongoing connections. In [13], metrics for evaluating node connectivity in VANETs had been proposed considering the different nodal mobility effects. The connection duration length distribution was characterized by average duration of the k-hop path existing between any two nodes or vehicles. They also showed by simulation that multi-hop paths have much poorer connectivity statistics than single-hop ones in VANETs.
Researches on the connectivity statistics of inter-vehicle distances focus on disclosing the relationship between connectivity and connection distance which is defined as the length of the connected path. In [8], vehicular distance distributions had been expressed by a tuple (h,v,p), where h denotes the inter-vehicle distance, v for velocity and p as market penetration ratio, respectively. They derived the connectivity formula in terms of distance between two nodes given specific propagation model and radio range. The expression was also further extended to the connectivity of a node with degree n. In [10], the authors introduced an equivalent M/D/∞ queuing model to obtain the Laplace transform of the tail probability of the connection distance and an explicit form for the expected value of the connection distance. Their numerical results mainly showed that increasing the traffic flow and the vehicles’ radio range led to the increasing of the metrics of connectivity in term of platoon size and connectivity distance. The authors of [14] showed that, under free flow state, connectivity increases as either the vehicular density or the number of lanes growing. Especially, the distances had a major impact on connectivity when it is 3–4 times farther than the radio range. Beyond this distance, the connectivity declines slowly. In [15], a simple geometric analysis of connectivity was proposed in vehicular networks under a distance-based radio communication model. They presented the notion of connectivity robustness based on inter-vehicle distances, which can generate a locally computable function providing a sufficient reference for connectedness of the network. With the aforementioned researches, although, there were also lots of works deducing or expressing connectivity by vehicles density [16, 17], speed or equivalent speed [18], transmission range [16, 19] etc., they all can be further transformed to the equivalent expression of statistical distributions of connection duration length or inter-vehicle distances. Further, generally speaking, it is difficult to give a uniform or general Probability Density Function (PDF) or Cumulative Distribution Function (CDF) of the inter-vehicle distances or connection duration length under different application scenarios such as overtaking, platoon and collisions. However, when the traffics are in the free-flow state, some distribution function and the probability of minimum node degree could be calculated. Therefore, we focus on the connectivity statistical analysis in free flow state.
Available connectivity definition and assumption
To fully utilize the connected direct or indirect links, an elaborate statistical analysis to the available connectivity should be given first for disclosing the influential factors. The definition of available connectivity in our study is given as follows:
Definition 1: Available connectivity for a vehicle in VANET is defined as the number of achievable connected links from it within a given period.
In VANETs, due to the high mobility of vehicles, topology is time-variant and the achievable connected links for a given vehicle are also changing frequently. Therefore, accurate definition of available connectivity should reflect the instantaneous state. However, although instantaneous connectivity is precise, it is no use for us to introduce such momentary parameter or reference for future design. As a result, we intend to define available connectivity within a given period. Correspondingly, the discussed scenario in this study just considers the slow changing topology case which is accurately reasonable especially under free flow state.
Let δ be the investigation period, i.e. we calculate available connectivity of a given node every δ seconds.
where T ∈ {0, δ, 2δ, …}.
AC(T) denotes the available connectivity during [T − δ,T], and N_{ i }(T) is the number of neighbors around i at time T. θ_{ i }(j, T) is a probability of the achievable indirect connections at time T.
AC(T) represents the number of achievable links that the reference node, i.e. the investigated node for available connectivity analysis, could obtain during [T − δ,T]. High available connectivity means there are more potential nodes which can connect with the reference node. According to available connectivity, we can control speeds of vehicles and the radio range to realize high connectivity in a VANET.
Statistical analysis of available connectivity
In this section, the available connectivity will be analyzed in detail by inspecting the connection properties in direct and indirect scenarios, respectively. Thus, the final available connectivity can be obtained by combining the obtained analysis results. For explanation simplicity, we use different analyzing method for connectivity study in different scenarios as follows.
Direct connectivity
In this section, we will investigate the direct connectivity for a given reference node in VANETs. The result will be given in terms of PDF of the connection duration length in free flow state.
For a reference node, the one-hop neighbors or the nodes within the radio range of the reference node, contribute to the direct connectivity. Without loss of generality, we just investigate the contribution to direct connectivity from one specific neighbor in later discussion. Results for others are similar and will be combined when final available connectivity is calculated.
where v = (v_{ x }, v_{ y })^{ T } is the relative speed vector and p = (p_{ x }, p_{ y })^{ T } is the relative position vector. r is the communication radio range.
Collected velocity versus inter-vehicle distances cumulative frequency table and fitted normal distribution parameters
v | <8 | 8-18 | 18-28 | 28-38 | 38-48 | >48 |
---|---|---|---|---|---|---|
h | ||||||
<15 | 22 | 95 | 105 | 36 | 1 | 0 |
15-20 | 0 | 21 | 66 | 30 | 9 | 0 |
20-30 | 0 | 28 | 84 | 51 | 17 | 0 |
30-50 | 0 | 0 | 22 | 34 | 13 | 4 |
>50 | 0 | 0 | 14 | 29 | 31 | 20 |
Samples = 732 | μ = 40.15 | σ = 11.744 |
The mean velocity difference is μ = μ_{ i } − μ_{ j } and the standard deviation of velocity difference is$\sigma =\sqrt{{\sigma}_{i}^{2}+{\sigma}_{j}^{2}}\text{.}$
where v_{min} ≤ v ≤ v_{max}.
The aforementioned discussions consider that vehicles all drive in a line. However, the result is different when vehicles running on neighboring lanes.
where$Q\left(x\right)=\frac{1}{2}erfc\left(\frac{x}{\sqrt{2}}\right)$, and erfc( · ) is the complementary error function.
Indirect connectivity
In this section, we will discuss the contributions to the available connectivity from indirect communication ways, which are further classified as multi-hop forwarding in a platoon and store-carry forwarding when no any instantaneous direct connection is achievable.
Considering a section of a unidirectional highway H with length [0,L] between the sender and the receiver car, we will then give a more detailed analysis to the possible scenarios in which indirect connectivity can be obtained based on the above classifications.
Indirect connectivity by multi-hop forwarding
We will first investigate the achievable indirect connectivity based on multi-hop forwarding. Here, we discuss three cases: (1). Indirect connectivity in a platoon on a single lane when vehicles in this platoon are connected to each other; (2). Indirect connectivity for vehicles on a single lane with helps from vehicles in adjacent lanes, when multi-hop forwarding could not be obtained on this single lane; (3). Indirect connectivity for vehicles on multi-lane. In the first case, all vehicles in a lane are connected to each other forming a platoon. However, in the second case, there are communication gaps between vehicles in a single lane. Therefore, we introduce vehicles in adjacent lanes to fill the gaps and obtain the indirect connectivity. In the third case, by combining the above two cases, we obtained the general probability expression for multi-hop indirect connectivity in multi-lane scenario.
Case 1
Platoon is a typical mobility pattern of VANETs in which vehicles are forced to organize in platoons due to road congestion, legal speed limits, weather condition or other reasons resulting in the closing relative velocities between vehicles. A platoon can be defined as a set of autonomous vehicles which have to move in a convoy – i.e. following the path of the leader, through an intangible hooking [21]. Vehicles in the same direction are said to be within a platoon if and only if they can communicate with one another by one-hop or multi-hop way. Otherwise, vehicles are said to be in different platoons. To obtain the indirect connectivity in a platoon, the chained or multi-hop forwarding is an essential way by which packets are forwarded to the destination by relays. We then analyze the probability distribution of introduced connectivity by multi-hop forwarding in a platoon on a single lane.
where ${\widehat{f}}_{v}\left(v\right)=\frac{2{f}_{v}\left(v\right)}{erf\left(\frac{{\nu}_{max}-\mu}{\sigma \sqrt{2}}\right)-erf\left(\frac{{\nu}_{min}-\mu}{\sigma \sqrt{2}}\right)}$ and ${f}_{v}\left(v\right)=\frac{1}{\sqrt{2\pi}\sigma}{e}^{-\frac{{\left(v-\mu \right)}^{2}}{2{\sigma}^{2}}}\text{.}$
Based on the maximum and minimum speed limits in the highway, i.e. v_{max} and v_{min}, by replacing p_{ i }, v_{ i } in (16), We then get a truncated version of Equation (16) as (14) to avoid negative speed and zero speed.
Case 2
If the vehicles could not obtain connections on a single-lane, vehicles in adjacent lanes are needed to be introduced to provide direct links.
where $m=min\left\{n-1,\u230a\frac{l}{r}\u230b\right\}\text{.}$
Now, to explore the connection opportunities from adjacent lanes, we inspect the statistical characteristic between two platoons. We define the inter-platoon space as the distance between the last vehicle of the leading platoon and the first vehicle of the following platoon, and the intra-platoon space as the distance between the vehicles within the same platoon.
With Equations (24), (25) and (26), (23) can be derived.
Case 3
Indirect connectivity by store-carry-forward
In this subsection, we will analyze the indirect connectivity from store-carry-forward strategy where no any ongoing single-hop/multi-hop link is achievable between the sender and the intended receiver. We then discuss the store-carry-forward scenarios under the same-direction and opposite-direction cases respectively.
Same-direction store-carry-forward
and $m\left(v,z\right)=\frac{t-z}{t}{\widehat{f}}_{v}\left(v\right)\text{.}$
With Equations (30), (31), (32), (33) and (34), (28) can be derived.
Next, we investigate the statistical properties of the time τ_{ 1 } needed for the first transversal hop from C to A, where the distance between C and A is D 1 > r.
We first introduce a stochastic variable x defined by the PDF f(x) = ϑe^{ ϑx }θ( − x) expressing vehicle C’s coordinate. The θ(x) is a function with result 1 when x is larger than zero and zero for others.
If x < –r, a finite intervalτ_{ 1 }is needed for C moving into the radio range of A.
Then, we investigate the CDF of intervalτ_{ 2 } by which C could carry and forward the messages to B.
Opposite-direction store-carry-forward
We still assume that the distance between the sender and the receiver is L. The sender and the receiver have a constant speed v_{ 1 } whereas the store-carry forwarders drive in the opposite direction with speedv_{ 2 }. The radio ranges for all vehicles are r and the forwarders are further assumed to form a connected platoon denoted by Z as shown in Figure 18(a). Such hypothesis is reasonable because it equals to the case that there is only single available forwarder within radio range of the sender. In other words, regarding the little contribution to the final connection duration, we neglect the information propagation time among vehicles in Z.
- 1.
If there is any vehicle in Z falling on the radio range of the receiver, with the above assumptions, we could take the needed duration ast_{1} = 0.
- 2.
If there is no any vehicle in Z falling on the radio range of the receiver, we should analyze CDF of the needed duration t _{ 2 }to further explore the connectivity characteristic.
where ${N}_{p}=\u230a\frac{{e}^{-\lambda \frac{r}{{v}_{2}}}}{2}\u230b\text{.}$
where ${P}_{d}=1-{\displaystyle \underset{0}{\overset{\frac{r}{{v}_{2}}}{\int}}f\left(\tau \right)d\tau}={e}^{-\lambda \frac{r}{{v}_{2}}}\text{.}$
Based on the assumption that the sender is paralleling to the center of Z, we have ${N}_{p}=\u230a\frac{{C}_{\mathit{platoon}}}{2}\u230b$. Therefore, the sum of the arrival intervals between the N_{ p } vehicles is$T={\displaystyle \sum _{i=1}^{{N}_{p}}{T}_{i}}$. With N_{ p } fold convolution to the PDF of T_{ i }, Equation (44) can be derived.
where g(x) = x^{ n } and g^{ n }(x) denote the n times differential to g(x).
With aforementioned analysis, we could find that, in VANETs environment, the connectivity may be obtained by many different ways instead of only direct links. Based on the definition of available connectivity by Equation (2), the total achievable number of connections for a given topology at a determined moment could be calculated. This important statistic can be used to improve the network performance especially for safety-related applications where the vehicles with larger available connectivity may be the better candidates for emergent messages forwarding to make alerts spreading quickly and widely.
Numerical results
Distribution parameters for vehicles’ speed
μ[km/h] | σ[km/h] | A[h/km] |
---|---|---|
70 | 21 | 0.016105 |
90 | 27 | 0.012526 |
110 | 33 | 0.010249 |
130 | 39 | 0.008672 |
150 | 45 | 0.007516 |
Distribution parameters for toy vehicles’ speed
μ[km/h] | σ[km/h] | A[h/km] |
---|---|---|
32.2 | 6.4 | 0.032407 |
17.6 | 2.8 | 0.058327 |
Conclusion
In this article, a useful metric, i.e. available connectivity has been introduced in VANETs. The available connectivity not only comes from direct connections from neighbors, but also from the indirection links by multi-hop and store-carry forwarding. The elaborate investigation to the statistical properties of the available connectivity has been given. Numerical results show that our proposed available connectivity has many potential relationships with network parameters and may provide important references for future protocols design in VANETs.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (61201133; 61172055;60832005; U0835004; 61072067), the Postdoctoral Science Foundation of China (20100481323), the Program for New Century Ex-cellent Talents (NCET-11-0691), the “111 Project” of China (B08038),and the Foundation of Guangxi Key Lab of Wireless Wideband Commu-nication & Signal Processing (11105).
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.