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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 6, JULY 2011
Exploiting Wireless Communication in Vehicle Density Estimation Sooksan Panichpapiboon, Member, IEEE, and Wasan Pattara-atikom, Member, IEEE
Abstract—Vehicle density is one of the main metrics used for assessing road traffic condition. High vehicle density indicates that traffic is congested. Currently, most vehicle density estimation approaches are designed for infrastructure-based traffic information systems. These approaches require detection devices such as inductive loop detectors or traffic surveillance cameras to be installed at various locations. Consequently, they are not appropriate for an emerging self-organizing vehicular traffic information system, where vehicles have to collect and process traffic information without relying on any fixed infrastructure. In this paper, we consider a few methods for estimating vehicle density based on the number of vehicles in the vicinity of the probe vehicle and the number of vehicles in a communication cluster. Index Terms—Traffic information systems, vehicle density estimation, vehicle-to-vehicle communication, vehicular networks.
I. I NTRODUCTION
V
EHICLE density is one of the main metrics used for assessing road traffic condition. It measures the number of vehicles per unit distance. High vehicle density usually indicates that traffic is congested, and vice versa. In addition, vehicle density can be used to estimate the average speed of vehicles on the road since the two parameters are negatively correlated. In other words, vehicles move slowly when the density is high and tend to move at a higher speed when the density is low. Vehicle density can be estimated using various methods [1]– [6]. However, one of the most common methods is deriving it from the flow rate, which is defined as the number of vehicles passing through an observation point per unit of time. In the current traffic information system, the flow rate is obtained by counting the number of vehicles that pass through a detection device, such as an inductive loop detector or a surveillance camera, over a period of time [7]–[9]. Acquiring such information, however, requires costly infrastructure. In addition, most of current traffic information systems still rely on a centralized communication model, where all the collected traffic data have to be processed by the central processing unit before being distributed back to the drivers on the street. This way of information processing and information dissemination is not suitable for an emerging self-organizing traffic information system. Manuscript received July 5, 2010; revised March 13, 2011; accepted May 6, 2011. Date of publication June 2, 2011; date of current version July 18, 2011. The review of this paper was coordinated by Prof. V. W. S. Wong. S. Panichpapiboon is with King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand (e-mail:
[email protected]). W. Pattara-atikom is with the Network Technology Laboratory, National Electronics and Computer Technology Center, Pathumthani 12120, Thailand (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2158566
In a self-organizing traffic information system, vehicles can exchange traffic information such as safety information, navigation information, and current road traffic condition among themselves without relying on any infrastructure. In this system, some (or all) vehicles may act as mobile probes that are responsible for collecting and distributing traffic information. Without any assistance from the central processing unit, the probe vehicles must be able to gather and process traffic information by themselves. The type of traffic information that will be considered in this paper is vehicle density. In particular, we will discuss how the probe vehicles can effectively use simple information to estimate vehicle density. In our perspective, the vehicle density estimation scheme for a self-organizing traffic information system should be simple and distributive. One simple way for a probe vehicle to estimate vehicle density is observing the number of neighboring vehicles in its vicinity. For example, if there are five vehicles within a distance of 100 m from the probe vehicle, then the local density measured by the probe vehicle is simply 0.05 veh/m. However, with this estimation approach, the probe vehicle only knows the local density, and it is not clear if the estimated local density is accurate enough to represent the global vehicle density, which is the ratio of the total number of vehicles on the road segment to the segment length. Whether or not the local density is sufficiently accurate to represent the global density depends on the statistical distribution of the vehicles on the road. For example, if each pair of consecutive vehicles is equally spaced, then, in this case, the local density will be exactly the same as the global density. However, this may not be true for other types of intervehicle spacing distributions. In this paper, we will evaluate the performance of a neighborbased vehicle density estimation scheme in which a probe vehicle chooses the best estimate for the vehicle density based on the number of neighbors in its vicinity and the performance of a cluster-based scheme in which the vehicle density is estimated from the number of vehicles in a communication cluster. Both the ideal scenario where intervehicle spacing distribution is assumed to be exponential and the realistic scenario where intervehicle spacing distribution is determined from real traffic data will be analyzed. The contributions of this paper can be summarized here. 1) We find expressions for the best estimate of the vehicle density under the neighbor-based scheme and under the cluster-based scheme. In fact, we show that, under the neighbor-based scheme, if the intervehicle spacing is exponentially distributed, then the local density is the best estimate for the global vehicle density.
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PANICHPAPIBOON AND PATTARA-ATIKOM: EXPLOITING WIRELESS COMMUNICATION IN VEHICLE DENSITY ESTIMATION
2) The accuracy of the neighbor-based density estimation scheme and the accuracy of the cluster-based density estimation scheme are evaluated. In particular, we quantify the errors incurred from using these statistical estimates and suggest how to improve their accuracy. 3) Finally, we characterize the interarrival time distribution and the intervehicle spacing distribution of the vehicles traveling on a highway in Saraburi, Thailand. We show that the interarrival time distribution and the intervehicle spacing distribution can be approximated by an exponential distribution. The accuracy of both the neighbor-based density estimation scheme and the cluster-based density estimation scheme is also tested on these interarrival time and intervehicle spacing data.
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the average density on the road can be obtained. This estimation method is quite similar to the neighbor-based scheme, which will be discussed in this paper. However, our scheme does not require a fixed cell size. The probe vehicle can take samples of the density estimates anywhere and anytime. In addition, the accuracy of the density estimation scheme proposed in [6] has not been tested on real traffic data. Although there are many density estimation schemes proposed, most of them are not designed to be used in a selforganizing traffic information system. In addition, none of the proposed schemes that are designed for a self-organizing traffic information system has taken the distribution of intervehicle spacing into account. III. T HEORETICAL A NALYSIS
II. R ELATED W ORK There are many vehicle density estimation schemes proposed in the literature [1]–[6]. However, they can generally be divided into two categories: 1) infrastructure-based schemes and 2) infrastructure-free schemes. Infrastructure-based density estimation schemes require detection devices such as inductive loop detectors or surveillance cameras to be installed at various locations for collecting traffic data. Typical data collected are vehicle count, vehicle speed, and the instant at which each vehicle moves past the detection device [10]–[13]. The collected data are then used in estimating the vehicle density. In [1], the authors present a way to estimate the number of vehicles on a road section based on the speed and flow measurements collected at the two end points of the section. In [2], the vehicle density is estimated by using the Kalman-filtering algorithm. In [3], the author proposes a density estimation scheme that relies on the vehicle reidentification algorithm. The reidentification algorithm identifies the two instants at which a vehicle moves past two inductive loop detectors. The density is then estimated from the ratio of the number of vehicles that have moved between the two detectors and the distance between them. In [4], the authors propose a scheme for estimating the vehicle density in the section of the road where there are no detectors available. It is assumed that the traffic flow at the entry point and that at the exit point of the road section are known (e.g., there are detection devices at these two points). The vehicle density is then estimated based on the traffic flow theory. In contrast, infrastructure-free density estimation schemes do not rely on any infrastructure. These schemes are suitable for a self-organizing distributed traffic information system. In [5], vehicle density is estimated from the average fraction of vehicle stop time. In other words, the probe vehicle has to monitor the fraction of time that it is stationary during the trip period. This fraction is then used in estimating traffic density. The intuition behind this method is that the probe vehicle tends to have a large fraction of stop time if the traffic is dense. In [6], a distributed vehicle density estimation scheme is proposed. In this scheme, a road segment is divided into multiple fixedsize cells. Cell density, which is the number of vehicles in the cell, is then calculated by a designated vehicle in a cell called “group leader.” The group leader also exchanges cell density information with the leaders in the other cells so that
To get some initial insights into the effectiveness of density estimation schemes, we will first analyze an ideal scenario where intervehicle spacing is exponentially distributed with the average density ρs veh/m. The exponential intervehicle spacing corresponds to the scenario where the positions of the vehicles are randomly and uniformly distributed [14]. In other words, in this case, the vehicles are equally likely to be anywhere on the road segment. A few existing works also suggest that the intervehicle spacing distribution under the free-flow condition can be modeled with an exponential distribution [15]–[18]. Moreover, we will later show through empirical analysis that the intervehicle spacing distribution of the vehicles on a highway could be approximated with an exponential distribution. Four assumptions are used for the analysis in this section.1 1) We consider a scenario where the vehicles are moving under a free-flow condition on a highway. In this scenario, each vehicle can move freely and independently of the other vehicles. 2) We assume that each vehicle is equipped with a wireless communication device that allows it to transmit and receive data. In addition, each vehicle has a fixed transmission range, which is denoted by z (in units of meter). It is also assumed that two vehicles can directly communicate if they are within the transmission range of each other. 3) We assume that each vehicle is able to determine the relative distance between itself and a vehicle within its transmission range. This can be achieved through any of the available positioning techniques. For example, the relative distance between vehicles may be estimated from the received signal strength [19]. Alternatively, if each vehicle is equipped with an onboard global positioning system receiver, the relative distance between vehicles can straightforwardly be determined. 4) One vehicle on the road is designated as a probe vehicle, and it is responsible for gathering an information to estimate the vehicle density. In the succeeding sections, we investigate and compare the performance of the schemes where the probe vehicle uses the 1 Discussion on the practicality of these assumptions and how they can be relaxed is given in Section V.
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Fig. 1. a road.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 6, JULY 2011
Illustrative scenario where the vehicles are moving westward on
Fig. 2. Illustrative example of the vehicles on a road. Vehicle V is the probe vehicle, and vehicle U is the farthest vehicle in the range of V.
number of one-hop neighbors, the number of two-hop neighbors, and the number of vehicles in a communication cluster for density estimation.
It can be shown that, under the one-hop neighbor scheme, the best estimate for the global vehicle density is [21]
A. One-Hop-Neighbor Scheme The probe vehicle can easily obtain the number of neighbors in its one-hop vicinity. Typically, in a self-organizing vehicular ad hoc network, each vehicle learns about the existence of its neighboring vehicles through exchanges of the periodic “HELLO” packets [20]. Thus, the probe vehicle automatically knows the number of its one-hop neighbor via the process of exchanging these packets. However, in the case where the vehicles do not use a communication protocol that utilizes the HELLO packets, the probe vehicle may explicitly broadcast an inquiry packet, asking for the neighboring vehicles in its transmission range to respond. Next, we show how the probe vehicle can select the best estimate for the vehicle density based on the number of one-hop neighbors. Let us consider an illustrative scenario where the vehicles are moving westward on a road, as shown in Fig. 1. Let V be the probe vehicle and K be a random variable denoting the number of vehicles within the transmission range of the probe vehicle (i.e., in the interval of length z m). Since the intervehicle spacing is assumed to be exponentially distributed with the average density ρs , it follows that K is Poisson distributed with the following probability mass function (PMF): Pr{K = k} =
(ρs z)k −ρs z e , k!
k = 0, 1, 2, . . . .
(1)
In reality, the global average vehicle density ρs on the road is unknown; thus, it needs to be estimated by the probe vehicle. Based on maximum-likelihood estimation, if the probe vehicle V knows that there are k neighbors within the distance of z m from itself,2 and the fact that the intervehicle spacing is exponentially distributed, then the best estimate for the global vehicle density is the value of density ρ, which maximizes the following probability: p1 =
(ρz)k −ρz e . k!
(2)
2 For analytical purposes, we only consider the number of neighbors behind the probe vehicle, as shown in Fig. 1. However, one can generally consider the number of neighbors in front of the probe vehicle as well. The number of neighbors in front of the probe vehicle is also Poisson distributed with the average density ρs . Considering both the front and back neighbors of the probe vehicle is equivalent to extending the interval of interest to 2z, instead of z, as in this analysis.
ρˆ∗1 =
k z
(3)
which, in fact, coincides with the local vehicle density in the one-hop range of the probe vehicle. This has a significant implication. It suggests that, if the intervehicle spacing is exponentially distributed, then the local density is the best estimate for the global vehicle density. A graphical validation of (3) is given in [22]. B. Two-Hop-Neighbor Scheme In addition to the number of one-hop neighbors, the probe vehicle can also take the number of neighbors that its neighboring vehicles have into consideration. Let us consider the probe vehicle V and the vehicle U, where vehicle U is defined as the farthest vehicle within the transmission range of V, as shown in Fig. 2. The most useful additional information in this case is the number of neighbors behind U since none of these vehicles is in the range of V. To acquire such information, the probe vehicle V may broadcast an inquiry packet, asking all of its neighbors to respond. In the response packet, each of V’s neighbors can specify the number of neighbors behind it and its relative position, which may be obtained from an available positioning technique. Upon receiving the responses, the probe vehicle identifies the farthest vehicle and notes the number of neighbors that the farthest vehicle has. With this approach, the probe vehicle will be able to know the number of its twohop neighbors. Alternatively, each vehicle can also include the number of neighbors behind it and its current position in the periodic HELLO packet. With the number of two-hop neighbors, the best estimate for vehicle density can be computed as follows: Let kv be the number of neighbors behind vehicle V and ku be the number of neighbors behind vehicle U. Thus, from Fig. 2, it is clear that there are a total of kv + ku vehicles within the distance of length z + xu , where xu is the distance from probe vehicle V to vehicle U. Since the intervehicle spacing is exponentially distributed, it follows that the number of vehicles within the distance z + xu is Poisson distributed. If probe vehicle V knows that the number of two-hop neighbors is kv + ku and the intervehicle spacing is exponentially distributed, then the best estimate of vehicle density in this case is the value of ρ, which maximizes the following probability: p2 =
(ρ(z + xu ))(kv +ku ) −ρ(z+xu ) e . (kv + ku )!
(4)
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the intervehicle spacing is exponentially distributed with the average density ρs , the cumulative distribution function of Yi can be written as 1 − e−ρs y , y ≥ 0 (6) FYi (y) = 0, y < 0. Fig. 3. Illustrative example of vehicles moving westward on a road. The random variable Yi represents the intervehicle spacing between vehicle i and vehicle i + 1. A cluster of size n is formed if Yi ≤ z for i = 1, . . . , n − 1 and Yn > z.
Similar to the one-hop-neighbor scheme, the best estimate for the global vehicle density can be analytically solved, and in this case, it can be shown that the best estimate is [21] ρˆ∗2 =
kv + ku z + xu
The cluster will consist of n vehicles if each of the first n − 1 links is smaller than the transmission range and the nth link is larger than the transmission range. More specifically, the cluster will be of size n under these two conditions: 1) Yi ≤ z, for i = 1, . . . , n − 1; and 2) Yn > z. Let N be a random variable denoting the number of vehicles in the cluster. The PMF of the cluster size can be written as Cn = Pr{Y1 ≤ z, Y2 ≤ z, . . . , Yn−1 ≤ z, Yn > z}.
(7)
(5)
which, in fact, is the local vehicle density in the two-hop range of the probe vehicle.
Since Yi are independent and identically distributed random variables, it follows from (6) and (7) that (1 − e−ρs z )n−1 e−ρs z , n = 1, 2, 3, . . . (8) Cn = 0, otherwise.
C. Cluster-Based Scheme In general, the probe vehicle can extend the range of information collection by considering the number of neighbors in the higher tiers (e.g., the number of three-hop neighbors, fourhop neighbors, etc.). However, there is a limit on the range at which the probe vehicle can gather information. Consider a series of vehicles that are following the probe vehicle, as shown in Fig. 3. Ideally, the probe vehicle will be able to communicate and gather information from vehicle n if there is a series of communication links connecting the probe vehicle to vehicle n. This implies that the separation between any pair of consecutive vehicles (in between the probe vehicle and vehicle n) must be less than or equal to the transmission range. In addition, the probe vehicle will not be able to communicate with vehicles beyond vehicle n if the separation between n and the next vehicle is larger than the transmission range. If vehicle n is the farthest vehicle with which the probe vehicle could communicate, then we say that n is the last vehicle in the cluster. More precisely, a cluster in this paper is defined as a group of vehicles that can be reached by the probe vehicle through a series of communication links. To be consistent with the analysis in the previous sections, we will only consider the vehicles that are following the probe vehicle and not the vehicles ahead of it. The cluster size or the number of vehicles in a cluster is an important quantity that can be used in vehicle density estimation. The cluster may be of any size; however, the minimum size is 1, which means that the probe vehicle cannot reach any other vehicle but itself. Intuitively, a large cluster size suggests that the vehicle density is high, and a small cluster size suggests that the vehicles are sparsely distributed. To use the cluster size to estimate vehicle density, it is necessary to calculate the PMF of the number of vehicles in the cluster. Consider a series of vehicles shown in Fig. 3. Let Yi be a random variable denoting the space between vehicle i and vehicle i + 1, where i = 1 corresponds to the probe vehicle, and i > 1 corresponds to the vehicles following the probe vehicle. Since it is assumed that
A similar PMF of the cluster size is also derived in [16]. Next, we will use the PMF of the cluster size to estimate vehicle density. Based on the maximum-likelihood estimation, given that the probe vehicle knows that there are n vehicles in the cluster and the intervehicle spacing is exponentially distributed, then the best estimate for the vehicle density is the value of ρ that maximizes the following probability: q = (1 − e−ρz )n−1 e−ρz .
(9)
It is possible to analytically solve for the best estimate of the vehicle density. In fact, it can be shown that the best estimate, under the cluster-based scheme, is [21] 1 −1 ∗ ln ρˆc = . (10) z n
D. Performance Comparison In this section, we compare the performance of the two neighbor-based density estimation schemes and the clusterbased density estimation scheme described in the previous sections. The performance evaluation is done via computer simulation. In the simulation of the neighbor-based scheme, 1000 vehicles are randomly generated and placed on a road segment, where the distance between any two consecutive vehicles is exponentially distributed with the average vehicle density ρs veh/m. In each simulation trial, the first vehicle is designated as a probe vehicle. Then, the number of one-hop neighbors and the number of two-hop neighbors in the range of the probe vehicle are recorded. Finally, with the number of one-hop neighbors, the best estimate for the global vehicle density in the one-hopneighbor scheme (ˆ ρ∗1 ) can be calculated from (3). Similarly, with the number of two-hop neighbors, the best estimate for the global vehicle density in the two-hop-neighbor scheme (ˆ ρ∗2 )
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can be calculated from (5). The simulation for the cluster-based scheme is similarly done. However, in this case, the cluster size, instead of the neighbor information, is collected. Finally, with the number of vehicles in the cluster, the best estimate for the global vehicle density (ˆ ρ∗c ) can be calculated from (10). Once the best estimate in the one-hop-neighbor scheme is obtained, it is compared to global density ρs , and the absolute error is computed as follows: = |ˆ ρ∗1 − ρs | .
(11)
Similarly, the absolute error of the estimate in the two-hopneighbor scheme and the absolute error of the estimate in the cluster-based scheme can be obtained by replacing ρˆ∗1 in (11) with ρˆ∗2 and ρˆ∗c , respectively. The simulation is repeated for 10 000 trials, and the mean absolute error (MAE) of the estimates in each scheme is calculated by averaging over the 10 000 absolute error samples collected from the simulation trials. In Fig. 4, we compare the MAEs of the estimates obtained from the one-hop-neighbor scheme, the two-hop-neighbor scheme, and the cluster-based scheme at various values of the global density. The transmission range is assumed to be 100 m. The values of the MAEs are shown in percentage of the global density values. For example, at ρs = 0.05 veh/m, the MAE of the estimates in the cluster-based scheme is approximately 20% of the global density value, which means that the value of the MAE is 0.01 veh/m. It can generally be observed that the accuracy of the estimates increases as the global vehicle density increases. Note that the MAEs seem to be very high when the vehicle density is low. For example, at ρs = 0.01 veh/m, the MAEs in all schemes are above 60%. This suggests that the neighbor-based schemes and the cluster-based scheme are not reliable when the vehicle density is low. The main reason is that, when the vehicles on the road are very sparse, the probe vehicle can hardly find any neighbors in its transmission range. Consequently, the probe vehicle does not have sufficient information to make an accurate estimation. In fact, with ρs = 0.01 veh/m, the probe vehicle will find, on average, only one vehicle within its 100-m transmission range. Increasing the transmission range will improve the accuracy [22]. In addition, it can be observed that the cluster-based scheme outperforms the neighborbased schemes at all vehicle densities. The main reason the cluster-based scheme has higher accuracy is that the cluster size implicitly contains the information about the number of neighbors that the probe vehicle has. Clearly, the neighbors of the probe vehicle must belong in the same cluster as the probe vehicle. Nonetheless, it is important to emphasize that the accuracy of the density estimation schemes presented so far is a result of using only one sample estimate taken by one probe vehicle. The accuracy can further be improved if more samples of the estimates are taken or more probe vehicles are deployed. E. Accuracy Improvement In the previous sections, the estimate of the density is calculated based on only one sample taken by the probe vehicle. In
Fig. 4. MAEs of the estimated vehicle density as functions of the global vehicle density. The values of the MAEs are shown in percentage of the global vehicle density. The transmission range of each vehicle is assumed to be 100 m.
this section, we investigate how the accuracy can be improved if the probe vehicle uses multiple samples of the estimates. The samples of the estimates may be taken at various locations as the probe vehicle moves. Let m be the number of sample estimates taken by the probe vehicle. To improve the accuracy, the probe vehicle can use the sample mean of the estimates to estimate the global vehicle density. The sample mean of the estimates in the one-hop-neighbor scheme can be computed as follows: m
ρ1 =
1 ∗ ρˆ m i=1 1i
(12)
where ρˆ∗1i is the ith sample density estimate taken by the probe vehicle, using the one-hop-neighbor estimation scheme. The sample mean of the estimates in the two-hop-neighbor scheme can be similarly computed. In Fig. 5, the MAEs of the estimated vehicle density in the two-hop-neighbor density estimation scheme are shown as functions of the global vehicle density. Three scenarios where the probe vehicle uses m = 1, 5, and 10 samples are considered, respectively. It can be observed that the accuracy of the neighbor-based density estimation scheme improves as the number of sample estimates increases. For example, at ρs = 0.01 veh/m, by increasing the number of samples from 1 to 5, the MAE of the estimates reduces from 48% to 25%. Moreover, with m = 5 samples, the neighbor-based density estimation scheme is able to achieve an MAE that is lower than 10% at most of the global density values. It is also worth noting that the accuracy marginally improves when the number of sample estimates is increased from five samples to ten samples. F. Effects of Intervehicle Spacing Distribution So far, the estimates in the neighbor-based scheme and the cluster-based scheme are derived based on the assumption that the intervehicle spacing distribution is exponential. It is of great interest to see how effective these density estimation
PANICHPAPIBOON AND PATTARA-ATIKOM: EXPLOITING WIRELESS COMMUNICATION IN VEHICLE DENSITY ESTIMATION
Fig. 5. MAEs of the estimated vehicle density in the two-hop-neighbor density estimation scheme as functions of the global vehicle density. Three scenarios where the probe vehicle uses m = 1, 5, and 10 samples are considered, respectively. The transmission range of each vehicle is assumed to be 200 m.
Fig. 6. MAEs of the vehicle density estimated by the two-hop-neighbor density estimation schemes. The scenarios where the intervehicle spacing distributions are lognormal and the scenario where the intervehicle spacing distribution is exponential are compared. The transmission range of each vehicle is assumed to be 200 m.
schemes would be if the intervehicle spacing distribution was not exponential. To conduct this study, we repeat the same simulations as done in the previous sections, with an exception that the intervehicle spacing is randomly generated according to a lognormal distribution.3 Note that an exponential distribution with mean 1/ρs has a variance of 1/ρ2s . Thus, to make a reasonable comparison between the lognormal distribution scenarios and the exponential distribution scenarios, we consider a lognormal distribution with mean 1/ρs and a variance of a/ρ2s , where a is a positive constant. Basically, the value of a indicates how large the variance of the lognormal distribution is in comparison with that of the exponential distribution. For example, if a = 2, the variance of the lognormal distribution will be twice that of the exponential distribution. Fig. 6 shows the MAEs of the estimates in the two-hopneighbor density estimation schemes. Three scenarios where the intervehicle spacings are lognormally distributed with mean 3 It is reported that, under a heavy-traffic condition, an intervehicle spacing distribution follows a lognormal distribution [16].
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Fig. 7. MAEs of the vehicle density estimated by the cluster-based density estimation schemes. The scenarios where the intervehicle spacing distributions are lognormal and the scenario where the intervehicle spacing distribution is exponential are compared. The transmission range of each vehicle is assumed to be 200 m.
1/ρs and variance a/ρ2s are considered. The values of a in these scenarios are given as follows: 1) a = 1; 2) a = 2; and 3) a = 1/2. The MAE of the estimates in the scenario where the intervehicle spacing is exponentially distributed with mean 1/ρs is also shown for comparison. It can be observed that the MAE of the estimates depends on the variance of the intervehicle spacing distribution. The accuracy of the estimates increases as the variance of the distribution decreases (i.e., as the value of a decreases). Moreover, in the case where the variance of the lognormal distribution is the same as that of the exponential distribution (i.e., when a = 1), the MAEs in the two scenarios are not significantly different. This suggests that the neighbor-based density estimation scheme, which uses the local density as an estimate, can still be applied, even though the intervehicle spacing distribution might not be exponential. In Fig. 7, the MAEs of the estimates in the cluster-based scheme are shown. The same scenarios as those shown in Fig. 6 are considered. As previously observed, in general, the MAE depends on the variance of the intervehicle spacing distribution. The MAE of the estimates increases as the variance increases. In contrast to the neighbor-based scheme, however, the clusterbased scheme does not perform well when the intervehicle spacing distribution is not exponential. In fact, the MAE tends to increase as the network becomes denser. The reason the cluster-based scheme poorly performs when the intervehicle spacing distribution is nonexponential is that the likelihood estimation in (10) is derived from the PMF of the cluster size under the exponential distribution assumption. Thus, when the underlying intervehicle spacing distribution changes, the estimation is expected to lose its accuracy. Nonetheless, it does not mean that the cluster-based scheme cannot be applied to nonexponential intervehicle spacing distributions. To properly apply the cluster-based density estimation method, the PMF of the cluster size under the corresponding intervehicle spacing distribution (e.g., lognormal distribution) must be derived.
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Fig. 8. PDF of the interarrival time of the vehicles traveling on the inbound Mitrapab Highway during (a) 6–8 A . M ., (b) 9–11 A . M ., (c) 12–2 P. M ., and (d) 3–5 P. M . The data were collected on February 18, 2008. The empirical pdfs are plotted with the symbols “o,” and the theoretical exponential pdfs are plotted with the dashed lines.
IV. D ENSITY E STIMATION F ROM R EAL T RAFFIC DATA In Section III, we investigate the performance of the neighbor-based density estimation scheme and the performance of the cluster-based density estimation scheme by assuming that the intervehicle spacing is exponentially distributed. In this section, we will extend the analysis to evaluate the effectiveness of these schemes when they are applied to real traffic data.
A. Interarrival Time Distribution and Intervehicle Spacing Distribution From Real Traffic Data Before being able to estimate vehicle density on the real road, it is necessary to characterize the intervehicle spacing distribution. In practice, it is difficult to directly measure the space between consecutive vehicles on the road. As a result, the intervehicle spacing distribution is usually approximated from interarrival time data. With the courtesy of the National Electronics and Computer Technology Center, we have access to real traffic data collected from various surveillance cameras. When a vehicle moves past a camera, the following data are logged into a file: 1) the estimated instantaneous speed; 2) the instant at which the vehicle enters the scene; and 3) the instant at which the vehicle leaves the scene. The logged arrival and departure times are accurate up to a unit of second. The data were collected daily from 6 A . M . to 6 P. M ., during February 2008. From the logged file, the interarrival time data, which was denoted by τ , can be acquired by observing the difference between the arrival instants of every pair of consecutive vehicles. After analyzing the empirical interarrival time data of the vehicles traveling on the highway, we hypothesize that the vehicle interarrival time follows an exponential distribution. One of the unique properties of the exponential probability density function (pdf) is that it is linear in the log-linear scale. Consequently, to test our hypothesis, we compare the empirical pdf of the interarrival time with the theoretical exponential pdf on the log-linear scale.
In Fig. 8, the empirical pdfs of the interarrival time and the theoretical exponential pdfs are shown on the log-linear scale, where the empirical pdfs are plotted with the symbols “o” and the theoretical exponential pdfs are plotted with the dashed lines. The empirical data shown in Fig. 8(a)–(d) correspond to the vehicle interarrival times on the inbound Mitrapab Highway, in Saraburi, collected on February 18, 2008, at four different observation periods, which are 6–8 A . M ., 9–11 A . M ., 12–2 P. M ., and 3–5 P. M ., respectively. The theoretical exponential pdf plotted in each subfigure is chosen such that it has the same mean as the empirical pdf plotted in the same subfigure. It can be observed from Fig. 8 that the empirical pdf in each subfigure fits well with the corresponding theoretical exponential pdf. This suggests that the interarrival time distribution could be approximated by an exponential pdf. Note also that the slopes of the pdfs shown in the four subfigures are different. This simply indicates that the mean interarrival times of the vehicles in the four periods are different. We have also analyzed the interarrival time data on different days, and in most cases, the empirical pdf also fits well with the theoretical exponential pdf. To estimate vehicle density, the interarrival time data must be translated into the intervehicle spacing data. To do so, we follow the approximated conversion method presented in [16]. Let τ be the interarrival time. If the vehicles in the flow are moving at an average speed of v m/s, then the approximated space, which is denoted by s, corresponding to the interarrival time can be written as s = v · τ.
(13)
In our analysis, the average speed v of the vehicle in the flow is obtained by averaging over the instantaneous speeds of the vehicles passing through the camera during the specified observation period. Alternatively, the approximated intervehicle spacing can be regarded as the scaled version of the interarrival time. As a result, the intervehicle spacing could also be approximated by an exponential pdf (with proper scaling). B. Performance of the Density Estimation Schemes Since the intervehicle spacing distribution could be approximated with an exponential pdf, it is possible to estimate the vehicle density by using the approaches presented in Section III. In this section, we evaluate the accuracy of the neighborbased density estimation scheme and the cluster-based density estimation scheme when they are applied to real traffic data. The experiment is conducted as follows. First, the interarrival time data during the specified observation period are converted into the intervehicle spacing data, using the conversion method described in Section IV-A. This allows us to virtually construct a map of the vehicles on the street. Next, in each experimental trial, one of the vehicles will be randomly selected and designated as a probe vehicle. The probe vehicle then calculates the best estimates of the global vehicle density according to the one-hop-neighbor, the two-hop-neighbor, and the cluster-based density estimation schemes, obtaining ρˆ∗1 , ρˆ∗2 , and ρˆ∗c respectively. The estimates are then compared with the global density,
PANICHPAPIBOON AND PATTARA-ATIKOM: EXPLOITING WIRELESS COMMUNICATION IN VEHICLE DENSITY ESTIMATION
Fig. 9. MAE of the one-hop-neighbor density estimation scheme, the MAE of the two-hop-neighbor density estimation scheme, and the MAE of the clusterbased density estimation scheme at four different observation periods. The data were collected on February 18, 2008, from the camera overlooking the inbound Mitrapab Highway.
which are denoted by ρr , and the error associated with each estimate is recorded. Global density ρr can be expressed as4 ρr =
λ v
(14)
where λ is the average flow rate obtained from the interarrival time data, and v is the average speed of the vehicles during the observation period. The experiment is repeated for 1000 trials. Then, the MAE incurred from the one-hop-neighbor density estimates, the MAE incurred from the two-hop-neighbor density estimates, and the MAE incurred from the cluster-based density estimates are calculated. In Fig. 9, the performance of the neighbor-based schemes and the performance of the cluster-based scheme are illustrated. The results presented in this figure are obtained based on the assumption that the probe vehicle takes only one sample of the estimates. Results in the scenarios where the probe vehicle takes multiple samples of the estimates will be discussed later in Fig. 10. Four observation periods (6–8 A . M ., 9–11 A . M ., 12–2 P. M ., and 3–5 P. M .) are considered. The global density (ρr ) in these periods are 0.049, 0.041, 0.040, and 0.048 veh/m, respectively. The data were collected on February 18, 2008, from the camera overlooking the inbound Mitrapab Highway. The transmission range of each vehicle is assumed to be z = 100 m. In each observation period, the MAE of the onehop-neighbor scheme (the black bar), the MAE of the twohop-neighbor scheme (the gray bar), and the MAE of the cluster-based scheme (the white bar) are compared. It can be generally observed that the MAE of the cluster-based scheme is smaller than those of the neighbor-based schemes across all the observation periods, which is in agreement with the results shown in Section III. 4 In practice, it is quite difficult to directly obtain the real global vehicle density since it will require a count of the actual number of vehicles that are simultaneously present on the entire road section at any given time. As a result, the global vehicle density is indirectly derived from the fundamental flow-density relation based on the traffic flow theory [23], which is a standard technique used in most current traffic information systems.
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Fig. 10. MAEs of the estimates in the two-hop-neighbor density estimation scheme at four different observation periods. Three scenarios where the probe vehicle uses m = 1, 5, and 10 samples in estimating the global vehicle density are considered. The data were collected on February 18, 2008, from the camera overlooking the inbound Mitrapab Highway.
As discussed earlier, the accuracy of the density estimation scheme can be improved if the probe vehicle takes multiple samples of the estimates into consideration. In Fig. 10, we compare the MAEs of the estimates in the scenarios where the probe vehicle uses multiple samples in estimating the global vehicle density. The transmission range in this case is 200 m. The considered traffic data are the same as those used in Fig. 9. In each observation period, the MAEs of the two-hopneighbor scheme when m = 1 sample (the black bar), m = 5 samples (the gray bar), and m = 10 samples (the white bar) are compared. It can be observed that the estimation accuracy significantly improves as the number of samples increases. For example, when the number of samples is increased from 1 to 5, the errors are reduced from the 20%–30% range to the 5%–10% range. However, the MAE marginally decreases when the number of samples is increased from five samples to ten samples. This is also in good agreement with the results shown in Fig. 5. V. D ISCUSSION Based on the analyses and the results presented in the previous sections, it is clear that the neighbor- and clusterbased density estimation schemes have great potential to be used in a real self-organizing traffic information system. These methods allow the vehicle density to be accurately estimated with simple exchanges of information. Nonetheless, there are some practical points that a designer needs to take into account in implementing these schemes in a real system. Here are a few issues that a designer needs to keep in mind. 1) Transmission range. In the neighbor- and the clusterbased density estimation schemes, the value of the estimated density in each scheme (i.e., ρˆ∗1 , ρˆ∗2 , and ρˆ∗c ) also depends on the transmission range. In the analyses, we have assumed that the transmission range is a constant. In reality, however, the transmission range of a communication device can vary due to environmental effects such as shadowing and fading. For example, while
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a nominal transmission range is assumed to be 100 m, in the presence of a shadowing effect, a probe vehicle may not be able to communicate with vehicles that are beyond 80 m from itself. As a result, using only a nominal value of the transmission range to compute the estimated density may cause some errors. A possible solution to circumvent this problem is to let the probe vehicle take more samples of the estimates and use the sample mean value, as discussed in Section III-E. To more accurately model the variation of the transmission range, one would need to consider the physical-layer effects such as the propagation pathloss, fading, and shadowing in details. For a treatment of a transmission range with a consideration of the physical-layer effects, one may follow an approach presented in [24]. However, it is beyond the scope of this paper. 2) Free-flow condition. In our analysis, we have assumed that the vehicles move under the free-flow condition. This implies that the vehicles can move freely and independently of the others. This is a practical assumption in the highway scenario that we are considering. However, in an urban scenario, the movements of the vehicles are typically correlated. In such a scenario, the movements of the vehicles may be more realistically modeled with a mobility model that takes the interdependence among vehicles into account. Some examples of these mobility models are the cellular automata model [25] and the carfollowing model [26]. In the cases where these mobility models are used, instead of the free-flow model, the intervehicle spacing may no longer be exponential. Nonetheless, in these cases, the density estimation schemes presented in this paper can still be applied. The key is to determine the intervehicle spacing distribution that corresponds to the mobility model of interest and then compute the estimated density accordingly. For example, to apply the cluster-based density estimation scheme to the scenario where the vehicles move according to the cellular automata model, one would need to determine the new pdf of the intervehicle spacing and the new PMF of the number of vehicles in a cluster and then use them to derive the optimal density estimate. 3) Penetration rate. In the early phase of deployment, it is likely that not all vehicles will be equipped with the communication devices. In this case, the probe vehicle cannot directly use the number of neighbors that respond to its inquiry packet to estimate the density since there may be some vehicles in its transmission range that are not able to respond (i.e., they are not equipped with the communication devices). In this case, the number of neighbors that are used in density estimation should be scaled according to the penetration rate. For example, if the current penetration rate is 50% and there are five neighbors that respond to the inquiry packet of the probe vehicle, then this means that the actual number of neighbors in the range of the probe vehicle should be around ten vehicles. However, this problem will gradually be dissolved as the number of vehicles with the communication devices increases in the network.
VI. C ONCLUSION In this paper, we have evaluated the performance of the neighbor-based vehicle density estimation scheme and the performance of the cluster-based density estimation scheme for a self-organizing traffic information system. The main findings can be summarized as follows. 1) If the intervehicle spacing is exponentially distributed, then the best estimate for the global vehicle density, under the neighbor-based scheme, is the local density. The local density is obtained from the ratio of the number of vehicles in the range of the probe vehicle and the transmission range. 2) The accuracy of the neighbor-based density estimation scheme and the accuracy of the cluster-based density estimation scheme depend on the global vehicle density. At extremely low vehicle density (sparse network), these estimation schemes are not reliable. This is due to the fact that, in the sparse network, the probe vehicle is not able to find a sufficient number of neighbors to make an accurate estimation. However, the accuracy improves as the network becomes denser. 3) The accuracy of the neighbor-based density estimation schemes can be improved by the following: 1) extending the transmission range of each vehicle; 2) acquiring additional information about the number of neighbors from vehicles in the higher tiers relative to the probe vehicle (e.g., the number of two-hop neighbors, three-hop neighbors, etc.); and 3) using a larger number of sample estimates in estimating the global density. However, out of these three methods, using a larger number of sample estimates is the most effective, in terms of improving the accuracy. Our results show that using about five samples can limit the errors to be within the 5%–10% range. 4) The cluster-based density estimation scheme can achieve higher accuracy than the neighbor-based schemes. However, it is usually more difficult to acquire the cluster size information than to acquire the information on the number of one- or two-hop neighbors. 5) Finally, we have shown that the interarrival time distribution and the intervehicle spacing distribution of the vehicles traveling on a highway in Saraburi under the free-flow traffic could be approximated with an exponential pdf. ACKNOWLEDGMENT The authors would like to thank Dr. S. Siddhichai and K. Kiratiratanapruk of the National Electronics and Computer Technology Center for providing valuable traffic data. R EFERENCES [1] D. Gazis and C. Knapp, “On-line estimation of traffic densities from timeseries of flow and speed data,” Transp. Sci., vol. 5, no. 3, pp. 283–301, Aug. 1971. [2] D. Gazis and M. Szeto, “Design of density-measuring systems for roadways,” Transp. Res. Rec., no. 495, pp. 44–52, 1974. [3] B. Coifman, “Estimating density and lane inflow on a freeway segment,” Transp. Res. Part A, vol. 37, no. 8, pp. 689–701, Oct. 2003.
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[4] L. Alvarez-Icaza, L. Munoz, X. Sun, and R. Horowitz, “Adaptive observer for traffic density estimation,” in Proc. IEEE Amer. Control Conf., Boston, MA, Jun. 2004, pp. 2705–2710. [5] M. Artimy, “Local density estimation and dynamic transmission-range assignment in vehicular ad hoc networks,” IEEE Trans. Intell. Transp. Syst., vol. 8, no. 3, pp. 400–412, Sep. 2007. [6] M. Jerbi, S. Senouci, T. Rasheed, and Y. Ghamri-Doudane, “An infrastructure-free traffic information system for vehicular networks,” in Proc. IEEE VTC, Baltimore, MD, Sep. 2007, pp. 2086–2090. [7] R. L. Anderson, “Electromagnetic loop vehicle detectors,” IEEE Trans. Veh. Technol., vol. VT-19, no. 1, pp. 23–30, Feb. 1970. [8] P. G. Michalopoulos, “Vehicle detection video through image processing: The autoscope system,” IEEE Trans. Veh. Technol., vol. 40, no. 1, pp. 21– 29, Feb. 1991. [9] C. Sun and S. G. Ritchie, Individual vehicle speed estimation using single loop inductive waveforms, pp. 1–30, Richmond, CA, Oct. 1999. [10] M. Perera and K. Harada, “An automatic system for counting and capturing the pictures of moving vehicles in real-time,” in Proc. IEEE Intell. Vehicles Symp. (IV), Columbus, OH, Jun. 2003, pp. 85–89. [11] C. Pang, W. Lam, and N. Yung, “A method for vehicle count in the presence of multiple-vehicle occlusions in traffic images,” IEEE Trans. Intell. Transp. Syst., vol. 8, no. 3, pp. 441–459, Sep. 2007. [12] E. Bas, M. Tekalp, and F. S. Salman, “Automatic vehicle counting from video for traffic flow analysis,” in Proc. IEEE Intell. Vehicles Symp. (IV), Istanbul, Turkey, Jun. 2007, pp. 392–397. [13] W. Lin, J. Dahlgren, and H. Huo, “An enhancement to speed estimation using single loop detectors,” in Proc. IEEE ITSC, Shanghai, China, Oct. 2003, pp. 417–422. [14] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. New York: McGraw-Hill, 2002. [15] M. Boban, T. Vinhoza, M. Ferreira, J. Barros, and O. K. Tonguz, “Impact of vehicles as obstacles in vehicular ad hoc networks,” IEEE J. Sel. Areas Commun., vol. 29, no. 1, pp. 15–28, Jan. 2011. [16] N. Wisitpongphan, F. Bai, P. Mudalige, V. Sadekar, and O. K. Tonguz, “Routing in sparse vehicular ad hoc wireless networks,” IEEE J. Sel. Areas Commun., vol. 25, no. 8, pp. 1538–1556, Oct. 2007. [17] L. Breiman, R. Lawrence, D. Goodwin, and B. Bailey, “The statistical properties of freeway traffic,” Transp. Res., vol. 11, no. 4, pp. 221–228, Aug. 1977. [18] R. J. Cowan, “Useful headway models,” Transp. Res., vol. 9, no. 6, pp. 371–375, Dec. 1975. [19] J. Dulmage, R. Cioffi, M. P. Fitz, and D. Cabric, “Characterization of distance error with received signal strength ranging,” in Proc. IEEE WCNC, Sydney, Australia, Apr. 2010, pp. 1–6. [20] E. M. Royer and C.-K. Toh, “A review of current routing protocols for ad hoc mobile wireless networks,” IEEE Pers. Commun., vol. 6, no. 2, pp. 46–55, Apr. 1999. [21] S. Panichpapiboon, “A study on distributing traffic information in selforganizing vehicular ad hoc networks,” Nat. Electron. Comput. Technol. Center, Pathumthani, Thailand, Aug. 2008.
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[22] S. Panichpapiboon and W. Pattara-atikom, “Evaluation of a neighborbased vehicle density estimation scheme,” in Proc. IEEE Int. Conf. ITST, Phuket, Thailand, Oct. 2008, pp. 294–298. [23] O. A. Nielsen and R. M. Jorgensen, “Estimation of speed-flow and flowdensity relations on the motorway network in the greater Copenhagen region,” IET Intell. Transp. Syst., vol. 2, no. 2, pp. 120–131, Jun. 2008. [24] S. Panichpapiboon and W. Pattara-atikom, “Connectivity requirements for self-organizing traffic information systems,” IEEE Trans. Veh. Technol., vol. 57, no. 6, pp. 3333–3340, Nov. 2008. [25] S. Maerivoet and B. De Moor, “Cellular automata models of road traffic,” Phys. Rep., vol. 419, no. 1, pp. 1–64, Nov. 2005. [26] M. Brackstone and M. McDonald, “Car-following: A historical review,” Transp. Res. Part F, vol. 2, no. 4, pp. 181–196, Dec. 1999.
Sooksan Panichpapiboon (S’05–M’07) received the B.S., M.S., and Ph.D. degrees in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, in 2000, 2002, and 2006, respectively. In April 2008, he was a Visiting Researcher with the Department of Information Engineering, University of Parma, Parma, Italy. He is currently a faculty member with the Faculty of Information Technology, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand. He has served as a technical program committee member for many international conferences. His current research interests include ad hoc wireless networks, intelligent transportation systems, radio frequency identification systems, and performance modeling. Dr. Panichpapiboon was the recipient of the Asia-Europe Meeting DUOThailand Fellowship in 2007 and the Doctoral Dissertation Award from the National Research Council of Thailand in 2011.
Wasan Pattara-atikom (M’04) received the B.E. degree in computer engineering from Khon Kaen University, Khon Kaen, Thailand, and the M.S. degree in telecommunications, the M.B.A. degree in business administration, and the Ph.D. degree in information science from the University of Pittsburgh, Pittsburgh, PA. He is currently a Senior Researcher with the Network Technology Laboratory, National Electronics and Computer Technology Center, Pathumthani, Thailand, where he leads a project on vehicular traffic prediction. His current research interests are vehicular ad hoc networks, intelligent transport systems, and machine learning.
