Comparison of traffic reliability index with real traffic data
- Limiao Zhang^{1, 2},
- Guanwen Zeng^{1, 2},
- Shengmin Guo^{3, 4},
- Daqing Li^{1, 2}Email author and
- Ziyou Gao^{5}
Received: 2 November 2016
Accepted: 15 August 2017
Published: 25 August 2017
Abstract
Existing studies have developed different indices based on various approaches including network connectivity, delay time and flow capacity, estimating the traffic reliability states from different angles. However, these indices mainly estimate traffic reliability from single view and rarely consider the combined effect of city traffic dynamics and underlying network structure. Based on percolation theory, Li et al. has developed a traffic reliability index to address this issue (Proc. Natl. Acad. Sci. USA 112(3):669-672, 2015) [1]. Here we compare this percolation-based index with one of the well-known index - congestion delay index (CDI). Using real traffic data of Beijing and Shenzhen (two large cities in China), we compare the two indices in the macroscopic trends and microscopic extreme values. The two indices are found to indicate the state of real-time traffic reliability in different consideration. Our results can be used for better evaluation of traffic system reliability and mitigation measures of traffic jams.
Keywords
traffic reliability reliability index percolation theory traffic data1 Introduction
Given the rapid urbanization process and the sharp growth in travel demand, people spend more time on road, which has led much economic and environmental loss. A 2015 Texas Transportation Institute report found that U.S. commuters spend about 42 hours a year stuck in traffic congestion. The total nationwide price tag: $160 billion, or $960 per commuter [2]. Behind the staggering number is the increasing concern about traffic reliability.
Traffic reliability is a critical measure to assess the performance of transportation systems, especially under unexpected events [3]. Researchers have developed different types of traffic reliability indices with different considerations. Existing traffic reliability indicators include connectivity reliability, travel time reliability, capacity reliability, travel cost reliability, traffic flow recession reliability, traffic demand satisfaction reliability, user satisfaction reliability etc.
Connectivity reliability was firstly defined by Mine and Kawai [4] in 1982, which mainly reflects the connection probability between a random pair of nodes in road network. A given road segment in the road network are classified into two states: connected or disconnected. Further works supplement the theory by extending the definition of connectivity from two nodes to k nodes [5]. However, this measure of traffic reliability neglects the limitation of real-time flow, and mainly quantifies the ability of road static capacity.
Anthony Chen et al. [6, 7] proposed the concept of capacity reliability. Capacity reliability deals with the probability of a road network to meet traffic demand under certain service level. Lindley [8] developed an index based on peak hour traffic volume of urban highways. The index is calculated by comparing volume to capacity (V/C), and roads with V/C higher than 0.77 are regarded as congested. Research at the Texas Transportation Institute [9] led to the development of the roadway congestion index (RCI) methodology to quantify the relative congestion levels in urban areas, which combines the indicator of urban area daily vehicle kilometers of travel (DVKT) per lane kilometer of roadway for both freeways and principal arterial streets.
Travel time reliability (TTR) is widely used to estimate the temporal damage caused by daily traffic congestion, which not only affects the daily travel of the public but also causes frustration among drivers [10–13]. TTR is defined as the probability of trips completed within a specified time between a given origin and destination (OD) at a certain level of service (LOS) [14]. TomTom International B.V. proposed a congestion index by comparing travel time in peak hours with travel time during non-congested periods (free flow) [15]. The difference is expressed as a percentage increase in travel time. Higher index indicates a longer delay in real-time compared with that in free-flow periods. The deformation of this index, congestion delay index (CDI) [15] that reflects the average delay time of real travel trajectories, is well-applied especially in China.
With the development of urban traffic and intelligence technology, there is a pressing need to estimate the real-time traffic performance from the system operator’s viewpoint [16, 17]. However, existing reliability studies may not be sufficient for a comprehensive network performance measurement [7]. Most of traditional reliability analyses mainly focus on the effect of single fact on the performance of the network, neglecting the combined effect of traffic dynamics and network structure. Here we use a traffic reliability index based on percolation theory [1] to measure real-time traffic reliability in a comprehensive way. We compare this traffic reliability index \(q_{c}\) with congestion delay index, and analyze the performance of these two indices in large cities of China: Beijing and Shenzhen.
By constructing a traffic dynamical network, Li et al. found that the organization of city traffic could be considered as a percolation-like transition [1]. Percolation theory [18–20] is a useful tool to study network transition, providing a possibility to overcome the limitations mentioned above. In percolation process, different clusters form as failed nodes/edges are removed (due to congestion) from original network, during which the transition can be clearly identified between a well-connected global giant cluster and isolated local clusters. Percolation theory can present a systematic viewpoint to analyze the influence of localized jam on system. The transition of traffic network phase can be quantified by the probability threshold \(q_{c}\), which can be taken as a statistical indicator of the operational limits of a network [21–23]. Therefore, the index \(q_{c}\) has three main advantages that we will discuss in the sections bellow: (a) \(q_{c}\) is the threshold distinguishing the network dynamics from connected global scale to isolated local scale; (b) \(q_{c}\) is less influenced by the trip sampling, faced with possible extreme local conditions; (c) \(q_{c}\) measures the relative traffic reliability from network operator’s viewpoint, which is scalable for comparing different cities.
In Section 2, we describe the dataset of real traffic. In Section 3, we explain the construction of dynamical traffic network and definition of index. In Section 4, we compare the two different indices in different aspects. The application in different cities is also illustrated in this part. Conclusions and discussion are presented in Section 5.
2 Data description
For the road network, intersections are represented by nodes and road segments between two intersections are represented by links. The road network of Beijing includes over 52,000 road segments (links) and 27,000 intersections (nodes). The road network of Shenzhen includes over 22,000 road segments (links) and 12,000 intersections (nodes). For each link, the velocity \(v_{ij}\) (i and j stands for the node on each end of the link respectively) is recorded according to real-time traffic. Here we consider a directed traffic network, because \(v_{ij}\) is in general different from \(v_{ji}\). The dataset covers velocity records of roads in Beijing and Shenzhen for 30 days in October 2015, including a representative holiday period in China, the National Day, from Oct. 1st to Oct. 7th. Velocity (km/h) is recorded through floating cars, with a resolution of minute.
In order to estimate traffic state of Beijing, at least the information of 31,000 floating vehicles is needed [24]. Now about 100,000 floating cars were monitored. The number of sampled floating cars varies with time. There are around 64.38% of high-level roads having more than 5 vehicles records every 5 minutes, while the percent for low-level roads is 15.21%. We use an interactive-voting based map matching algorithm to associate the measured velocity to a given road, which is introduced by reference [25]. Our GPS data includes multiple types of floating cars with different resolutions. For taxis, the resolution of GPS data is 1 min or 30 s. For private cars, the resolution of GPS data is 1 s. The vehicle position error is less than 30 m. We compute the road velocity based on Dempster-Shafer theory [26], which includes a voting process. Each road is classified into one of three categories according to a pair of thresholds (\(v_{1}\), \(v_{2}\), \(v_{1}< v_{2}\)) based on road levels. We compare the instantaneous velocities v of vehicles on a given road with its velocity thresholds. For \(v \in (0, v_{1})\), we vote for congested state. For \(v \in( v_{1}, v_{2})\), we vote for intermediate state. For \(v\in(v_{2},\infty)\), we vote for free-flow state. We regard the state with the highest votes as the real-time traffic state of this road. Then for this road, we smooth velocities within the voted category and calculate the road velocity. All sampled vehicles are pre-filtered to ensure their representativeness of road condition properly. The accuracy of data is greatly influenced by traffic lights. Our test results suggest that the accuracy of data is more than 85% on closed roads, while it is more than 70% on open roads.
The dataset is incomplete, with some velocities missing. We compensate the missing velocities by considering road network topology [1], where the missing velocity of road equals to the average velocity of its neighboring roads with the same direction. The data availability is constrained by our agreement with data provider of company.
3 Model
Then we remove all links with congested state and calculate strongly connected clusters in the rest of the network. A strongly connected cluster is a set of nodes, where there is a path in both directions between each pair of nodes [27]. In our calculation, we use Tarjan algorithm [28] to identify strongly connected clusters in the functional network.
Here we use \(q_{c}\) as an index [1] to measure the reliability level of city traffic: only cars with relative velocity below \(q_{c}\) can travel the main part of the city, i.e. the giant component of traffic network; otherwise, cars with relative velocity above \(q_{c}\) will be trapped in local isolated clusters. Therefore, \(q_{c}\) indicates the maximal relative velocity that allows one to travel the main part of the city, which reflects the global efficiency of traffic in a network view [1]. For comparison, we also take a widely applied index - congestion delay index (CDI) [15]. In practice, with advanced information processing technology, the trajectory can be precisely positioned on the map based on the data obtained from floating cars and GPS navigation. And the travel time of users can be obtained from GPS time stamp records [29]. In this paper, we use the same data set for both indices. Due to the lack of trip details in our data, we have to sample the origin and destination of trip from the traffic network. For simplicity and generality, 120,000 pairs of nodes are randomly selected as origin and destination (OD) separately.
4 Results
We calculate the two indices in Beijing and Shenzhen for a month and compare trends of different indices. We find the trends of the two indices are similar, while the degree of congestion reflected by different indices is different from each other. Figure 2(c) and (d) show examples that the trends of the two indices are similar. For Beijing on Oct. 29th (see Figure 2(c)), both of the two indices increase in the morning and reach to morning peaks around 7:30. Then indices decrease to relative low values, which correspond to a better traffic condition during noon. Around 18:00, evening peaks appear and afterwards both indices begin to decrease. The same curve trends can be also found in Shenzhen, as shown in Figure 2(d). Since CDI has already been widely used to measure traffic state, similar trends reflected by the two indices better illustrate the basic indicative function of \(q_{c}^{{r}}\).
Example of differences in indices values
Date | Index | Time1 | Value1 | Time2 | Value2 | Change |
---|---|---|---|---|---|---|
Beijing | ||||||
Oct. 1st | \(\mathrm{CDI}^{r}\) | 9:10 | 0.69 | 13:35 | 0.69 | 0 |
\(q_{c}^{r}\) | 1 | 0.6 | −40% | |||
Oct. 28th | \(\mathrm{CDI}^{r}\) | 17:10 | 0.89 | 19:20 | 0.39 | −56.2% |
\(q_{c}^{r}\) | 0.71 | 0.71 | 0 | |||
Shenzhen | ||||||
Oct. 1st | \(\mathrm{CDI}^{r}\) | 11:50 | 0.65 | 22:05 | 0.61 | 6.2% |
\(q_{c}^{r}\) | 0.58 | 0.89 | 53.4% | |||
Oct. 28th | \(\mathrm{CDI}^{r}\) | 6:55 | 0.07 | 15:10 | 0.37 | 427.6% |
\(q_{c}^{r}\) | 0.34 | 0.34 | 0 |
The difference in peak instants may be due to the feature of \(\mathrm{CDI}^{r}\)that extreme values of travel time along frequently visited trips will determine significantly peak instants of \(\mathrm{CDI}^{r}\). With the accumulation of traffic volume, the velocity will become slow and travel time through mostly congested area will become one of the determining factors for \(\mathrm{CDI}^{r}\). This also makes \(\mathrm{CDI}^{r}\)more sensitive to the change of local traffic. During the morning rush hours, road conditions of other parts in the city may not change so sharply and global connectivity of whole functional city may stay stable.
5 Conclusion
The percolation threshold naturally acts as a network reliability indicator, quantifying the operational limit of network traffic. Specifically, percolation theory focuses on connected clusters, which fills up the gap of other indices that rarely consider the macroscopic network congestion behaviors from a network view [21]. We find that \(q_{c}\) can reflect the transition of dynamical traffic network, faced with possible extreme local conditions. These features of \(q_{c}\) make it a useful tool under the variation and absence of complete traffic information, and provide supports for the congestion prediction and mitigation research.
Although we made a comparison between the two indices, we found that each index has its own advantages and limitations under certain situation. For example, \(q_{c}\) reflects traffic condition from managers’ perspective, while is less intuitive for travelers; CDI is less scalable for comparing different cities, while has advantages of being more understandable and easier calculation. These important differences will decide the choice of a given traffic reliability index under specified requirement.
However, it should be noted that we just gave a brief analysis of indices in two cities during limited time span. More data and analyses are needed to summarize the travel characteristics of different cities according to indices. In addition, the city is not homogeneous and the travel velocity depends on the trips length [34]. In our present work, we did not cut off the trip length. The influence of trips lengths on index calculation should be discussed in future research.
Our study focuses more on the quantitative comparison of two reliability indices, while the mechanisms behind these two indices are different. CDI assumes that the user trip information can reflect the overall performance of the traffic network, which incorporates the different OD information with their weights. Meanwhile, percolation concept suggests that the traffic organization over the whole network depends on the instantaneous connected clusters of high-speed roads with free flow, where the weight of each road is their velocity, instead of traffic flow. These underlying differences should be studied in the future, especially its relation with macroscopic fundamental diagram (MFD). Geroliminis and Daganzo [35] found that neighborhoods on the order of 10 km^{2} in cities like Yokohama, Japan, should have a well-defined MFD. This MFD can be used to improve accessibility as measured by the city’s trip completion rate. Both of their and our works discuss the index to measure the performance of traffic network. For \(q_{c}\) in our paper, it distinguishes the phase transition of the dynamical traffic network, where we divide based on the velocity the roads into two categories: free and congested. \(q_{c}\) reflects the real-time variation of traffic reliability. For the work of Geroliminis and Daganzo’s, they used MFD for traffic state monitor, which reflects the relation of density, flow and velocity. Due to the lack of density and flow data, it is hard for us to explore the MFD and compare with percolation index in the current stage. Further analysis should be carried out when data are accessible.
Admittedly, only the comparison cannot reveal the underlying mechanism difference of these two indices. In our future work, we wish to gather the value of CDI from different sources including the mobile phone data, and further compare the fundamental relation between these two indices. Based on big data and other advanced technologies [36–39], we can perform a thorough cause-analysis for indices comparison in the future. Meanwhile, we can develop in the next step traffic optimization method based on percolation index and compare with other methods based on CDI.
Declarations
Acknowledgements
This work is supported by National Natural Science Foundation of China (71621001, 71771009).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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