Exploring travel time distribution and variability patterns is essential for reliable route choices and sophisticated traffic management and control. State-of-the-art studies tend to treat different types of roads equally, which fails to provide more detailed analysis of travel time characteristics for each specific road type. In this study, based on a vast amount of probe vehicle data, 200 links inside the Third Ring Road of Beijing, China, were investigated. Four types of roads were covered including urban expressways, auxiliary roads of urban expressways, major roads, and secondary roads. The day-of-week distributions of unit distance travel time were first analyzed. Kolmogorov-Smirnov test, Anderson-Darling test, and chi-squared test were employed to test the goodness-of-fit of different distributions and the results showed lognormal distribution was best-fitted for different time periods and road types compared with normal, gamma, and Weibull distribution. In addition, four reliability measures, that is, unit distance travel time, coefficient of variation, buffer time index, and punctuality rate, were used to explore the day-of-week travel time variability patterns. The results indicated that urban expressways, auxiliary roads of urban expressways, and major roads have regular and distinct morning and afternoon peaks on weekdays. It is noteworthy that in daytime the travel times on auxiliary roads of urban expressways and major roads share similar variability patterns and appear relatively stable and reliable, while urban expressways have most reliable travel times at night. The results of analysis help enable a better understanding of the volatile travel time characteristics of each road type in urban network.
Nowadays, high traffic demand and limited road capacities make people spend much more time on their daily journeys. Travel time reliability (TTR), defined as the level of consistency of travel conditions over time [
Thanks to advanced traffic sensing technologies, various travel time related information can be collected conveniently nowadays. The technologies essentially include station-based traffic state measurement (e.g., loop detector, video camera, and microwave sensor) and point to point travel time collection (e.g., automatic vehicle identification systems, license plate recognition systems, mobile, Bluetooth, and probe vehicles). The acquisition results of station-based devices strongly depend on the spatial layout and fixed position of traffic detectors. In contrast, probe vehicles equipped with the global positioning system (GPS) could travel all over the network and record the travel time and location information of vehicles at a certain interval. These data are known as probe vehicles data, representing the relatively complete operation conditions for urban traffic. With increasing amounts of data available, there has been a surge of literature devoted to the analysis of TTR and TTD in recent years.
Urban traffic times are essentially volatile due to various influencing factors, for example, weather, road types, roadway geometry, traffic control, accident, and varying traffic demands. Martchouk et al. [
It is noteworthy that most studies on travel time analysis employed travel time data from only one road type [
The remainder of this paper is organized as follows. Section
Most existing studies on TTV put significant effort into identifying the best statistical model for fitting travel times. The fitted distributions can be categorized into three main classes, that is, single-mode distributions [
Single-mode distributions apply one kind of standard distributions to characterize travel time, like normal, lognormal, gamma, Weibull, burr, and other distributions that are commonly used. For example, Emam and Al-Deek [
Multimode distribution appears more accurate because more than one component will be used to model travel time states. The components can come from one or multiple standard distributions. For example, Chen et al. [
Truncated distribution enables the travel time to be restricted within a certain limited range. Thus, excessively short or long travel times, which correspond to unreasonable values generated from the tail of standard distribution, can be excluded [
Three main types of TTV can be generally found in the literature, that is, vehicle-to-vehicle variability that characterizes the difference between travel times of different vehicles traveling the similar route at the same time, period-to-period variability that relates to vehicles traveling the similar route at different periods within a day, and day-to-day variability that represents the travel time variations between similar trips at the same time period on different days [
In order to identify the variability patterns of travel time, researchers have defined several reliability measures for quantitative analysis. TTR measures can be generally grouped into three categories, that is, the probability index, the statistical index, and the buffer time type index. The probability index mainly reflects the probability that the travel time can match the specified conditions in the form of probability distribution. The statistical index analyzes TTV based on historical data or real-time information, while the buffer time type index indicates the reservation time to ensure that the probability of reaching the destination is large enough from the travelers’ point of view.
Moylan [
Travel time reliability measures.
Category | Measure | Definition |
---|---|---|
Probability index | Variance | The expected value of the square of the deviations of a random variable from its mean value |
Standard deviation | The square root of the variance | |
Coefficient of variation | Standard deviation divided by the average travel time | |
| ||
Statistical index | | |
Skewness statistic | The ratio of the difference between 90th and 50th percentile travel time to the difference between 50th and 10th percentile | |
Width statistic | The ratio of the difference between 90th and 10th percentile travel time to the 50th percentile travel time | |
Travel time index | The ratio of the average travel time to free-flow travel time | |
Planning time index | The ratio of 50th percentile travel time to free-flow travel time | |
| ||
Buffer time type index | Buffer time index | The ratio of the difference between 95th percentile and the average travel time to the average travel time |
Failure/on-time performance | Percent of trips with travel time less than, for example, | |
Frequency of congestion | Percent of time that the travel time is larger than double the free-flow travel time | |
Misery index | The average of the highest five percent of travel times divided by the free-flow travel time |
To sum up, though TTD and variability patterns have been extensively investigated, limited studies shed light on the distinction at fine grained levels of spatial and temporal aggregation, for example, travel times for different road types during typical time periods. To fill this gap, this study conducts detailed analysis of travel time characteristics for each specific road type, for example, urban expressways, auxiliary roads of urban expressways, major roads, and secondary roads, based on a vast amount of proven vehicle data in the urban network of Beijing, China. Period-to-period and day-to-day variability are taken into comprehensive consideration. The results help reflect the travelers’ preference to different road types and further enhance our understanding of traveling behavior in urban networks.
In this study, the probe vehicle data collected in the urban network of Beijing, China, during one week from June 1st (Monday) to 7th (Sunday), 2015, were utilized. About every one minute, taxis equipped with GPS devices uploaded a set of instantaneous information, such as location, direction, speed, and being occupied or not occupied by passengers. Note that only the taxis occupied by passengers were considered, which are supposed to be closer to the behavior of regular vehicles. And enough data (at least five sets of traffic information) was uploaded within two minutes for each link. The data quality is good overall. The probe vehicle data were first preprocessed to remove erroneous information and a Kalman filtering process is utilized to achieve smoothing the vehicle trajectories. Then, by resorting to the map matching method, the travel speed data were further aggregated into every two minutes for each link within the network.
Note that in Beijing several two-way urban expressways, that is, Rings 2–5, enclose the urban area. The lengths of Rings 2–5 are 33, 48, 65 and 98 km, respectively. The urban expressways are surrounded by auxiliary roads, and the auxiliary roads and urban expressways are connected by various types of interchanges with paired entrances and exits for merging or diverging traffic. Besides, major and secondary roads are widely distributed in the urban network.
In order to explore the TTD and variability patterns of urban roads, a large area inside the Third Ring Road in Beijing was selected and in total 200 links covering four road types, that is, urban expressways, auxiliary roads of urban expressways, major roads, and secondary roads, were exploited. Each type of road includes 50 links that are evenly distributed in the road network, and hence TTVs for each type of road are supposed to be represented comprehensively. Figure
Spatial distribution of different types of roads.
Urban expressways
Auxiliary roads of urban expressways
Major roads
Secondary roads
In this section, the probability distributions of travel time during different time periods were analyzed to investigate the characteristics of travel time variation.
As mentioned above, this study considered links from different road types. It is expected that various urban traffic states from different road types in different time periods can be described and compared in detail. For analyzing travel time distributions of each road type, we selected four typical periods of time, that is, peak hours on weekdays, off-peak hours on weekdays, peak hours on weekends, and off-peak hours on weekends. Peak hours represent the most congested periods during the day while off-peak hours represent the smooth periods. They were distinguished for each road type by referring to the average travel time of the day. During each time period, it is assumed that link travel time is independent and identically distributed. For providing recommendations to large-scale network computing, such as path finding problems, four common types of single-mode probability distributions including normal, lognormal, gamma, and Weibull were employed to test whether they can fit the travel times desirably. The parameters of each distribution were estimated by the fitting functions in Matlab, such as
Transform original vehicle speed data collected from probe vehicles into unit distance travel time. In this study, the unit distance was set as one hundred meters and the calculation of travel time was made in the unit of seconds per 100 meters. This step is to eliminate the influence of different road length on travel time analysis.
Choose peak and off-peak hours according to the unit distance travel time on weekdays and weekends. For simplicity, one hour was set as the length of the analysis period.
Determine parameter(s) of different distribution types from unit distance travel time. Here, the fitting functions in Matlab were used to estimate the optimal values of the parameter(s).
Check the goodness-of-fit of four probability distributions by K-S test, A-D test, and
Calculate the average acceptance rates of each candidate distribution, and those for each time period and for each road type.
Note that Step
The goodness-of-fit test results of four road types during different periods.
Road types | Time periods | Distributions | Test methods | Average acceptance rates of | Best-fitted distribution | |||||
---|---|---|---|---|---|---|---|---|---|---|
Accepted by K-S (%) | Accepted by A-D (%) | Accepted by | each distribution type (%) | each time period (%) | each road type (%) | |||||
Urban expressways | Weekdays | Peak hours 17:30–18:30 | Normal | 58.4 | 73.2 | 66.4 | 66.0 | 67.1 | 73.6 | Lognormal |
Lognormal | 60.4 | 75.2 | 69.6 | 68.4 | ||||||
Gamma | 58.4 | 73.6 | 69.6 | 67.2 | ||||||
Weibull | 56.8 | 70.4 | 73.2 | 66.8 | ||||||
Off-peak hours 12:30–13:30 | Normal | 72.8 | 80.4 | 77.2 | 76.8 | 76.6 | Lognormal | |||
Lognormal | 77.6 | 84.0 | 79.6 | 80.4 | ||||||
Gamma | 73.6 | 82.0 | 76.0 | 77.2 | ||||||
Weibull | 65.2 | 75.6 | 74.8 | 71.9 | ||||||
Weekends | Peak hours 15:00–16:00 | Normal | 70.0 | 83.0 | 76.0 | 76.3 | 76.9 | Weibull | ||
Lognormal | 72.0 | 84.0 | 75.0 | 77.0 | ||||||
Gamma | 70.0 | 83.0 | 76.0 | 76.3 | ||||||
Weibull | 75.0 | 80.0 | 79.0 | 78.0 | ||||||
Off-peak hours 12:00–13:00 | Normal | 67.0 | 78.0 | 80.0 | 75.0 | 73.8 | Lognormal | |||
Lognormal | 70.0 | 78.0 | 80.0 | 76.0 | ||||||
Gamma | 70.0 | 78.0 | 79.0 | 75.7 | ||||||
Weibull | 61.0 | 72.0 | 73.0 | 68.7 | ||||||
| ||||||||||
Auxiliary roads of urban expressways | Weekdays | Peak hours 17:30–18:30 | Normal | 75.6 | 82.4 | 78.8 | 78.9 | 82.0 | 89.0 | Lognormal |
Lognormal | 83.2 | 88.8 | 87.2 | 86.4 | ||||||
Gamma | 78.8 | 88.4 | 85.6 | 84.3 | ||||||
Weibull | 73.6 | 80.8 | 80.8 | 78.4 | ||||||
Off-peak hours 12:30–13:30 | Normal | 89.2 | 92.8 | 89.6 | 90.5 | 90.8 | Lognormal | |||
Lognormal | 90.8 | 95.2 | 90.8 | 92.3 | ||||||
Gamma | 91.2 | 93.6 | 90.0 | 91.6 | ||||||
Weibull | 88.0 | 91.2 | 86.8 | 88.7 | ||||||
Weekends | Peak hours 10:00–11:00 | Normal | 90.0 | 92.0 | 88.0 | 90.0 | 90.4 | Lognormal | ||
Lognormal | 90.0 | 96.0 | 90.0 | 92.0 | ||||||
Gamma | 90.0 | 95.0 | 90.0 | 91.7 | ||||||
Weibull | 84.0 | 91.0 | 89.0 | 88.0 | ||||||
Off-peak hours 13:30–14:30 | Normal | 94.0 | 95.0 | 89.0 | 92.7 | 92.7 | Gamma | |||
Lognormal | 93.0 | 96.0 | 90.0 | 93.0 | ||||||
Gamma | 94.0 | 96.0 | 91.0 | 93.7 | ||||||
Weibull | 93.0 | 93.0 | 88.0 | 91.3 | ||||||
| ||||||||||
Major roads | Weekdays | Peak hours 17:30–18:30 | Normal | 77.6 | 82.4 | 75.6 | 78.5 | 83.9 | 89.6 | Lognormal |
Lognormal | 87.6 | 92.0 | 86.4 | 88.7 | ||||||
Gamma | 85.2 | 88.0 | 84.4 | 85.9 | ||||||
Weibull | 82.8 | 85.2 | 79.2 | 82.4 | ||||||
Off-peak hours 12:30–13:30 | Normal | 91.2 | 94.8 | 92.8 | 92.9 | 93.0 | Lognormal | |||
Lognormal | 94.8 | 96.8 | 93.6 | 95.1 | ||||||
Gamma | 92.8 | 95.6 | 92.0 | 93.5 | ||||||
Weibull | 87.6 | 91.2 | 93.2 | 90.7 | ||||||
Weekends | Peak hours 15:30–16:30 | Normal | 87.0 | 89.0 | 85.0 | 87.0 | 88.8 | Lognormal | ||
Lognormal | 92.0 | 92.0 | 92.0 | 92.0 | ||||||
Gamma | 89.0 | 92.0 | 88.0 | 89.7 | ||||||
Weibull | 86.0 | 88.0 | 85.0 | 86.3 | ||||||
Off-peak hours 12:30–13:30 | Normal | 94.0 | 97.0 | 89.0 | 93.3 | 92.8 | Lognormal | |||
Lognormal | 94.0 | 96.0 | 92.0 | 94.0 | ||||||
Gamma | 94.0 | 96.0 | 89.0 | 93.0 | ||||||
Weibull | 90.0 | 92.0 | 91.0 | 91.0 | ||||||
| ||||||||||
Secondary roads | Weekdays | Peak hours 17:00–18:00 | Normal | 52.4 | 63.2 | 64.4 | 60.0 | 66.7 | 70.4 | Lognormal |
Lognormal | 68.0 | 77.6 | 76.4 | 74.0 | ||||||
Gamma | 62.8 | 72.8 | 71.6 | 69.1 | ||||||
Weibull | 55.2 | 62.0 | 74.0 | 63.7 | ||||||
Off-peak hours 12:30–13:30 | Normal | 62.4 | 70.8 | 70.0 | 67.7 | 71.6 | Lognormal | |||
Lognormal | 71.2 | 80.4 | 78.0 | 76.5 | ||||||
Gamma | 68.0 | 76.0 | 74.8 | 72.9 | ||||||
Weibull | 62.0 | 69.6 | 76.0 | 69.2 | ||||||
Weekends | Peak hours 16:30–17:30 | Normal | 56.0 | 69.0 | 61.0 | 62.0 | 68.4 | Lognormal | ||
Lognormal | 68.0 | 78.0 | 75.0 | 73.7 | ||||||
Gamma | 66.0 | 75.0 | 74.0 | 71.7 | ||||||
Weibull | 59.0 | 67.0 | 73.0 | 66.3 | ||||||
Off-peak hours 12:30–13:30 | Normal | 69.0 | 75.0 | 74.0 | 72.7 | 74.8 | Lognormal | |||
Lognormal | 79.0 | 84.0 | 78.0 | 80.3 | ||||||
Gamma | 76.0 | 81.0 | 72.0 | 76.3 | ||||||
Weibull | 63.0 | 75.0 | 72.0 | 70.0 |
The following presents the analysis of distribution fitting results investigated from three perspectives, that is, probability distribution types, time periods, and road types.
First, the best-fitted distribution type was analyzed on the whole for altogether 16 time periods through comparing the average acceptance rates of each distribution. It was found that lognormal distribution has the highest average acceptance rates; that is,
In this study, the best-fitted time period was defined as the one whose average acceptance rate of all 12 statistical tests, that is, K-S, A-D, and
In practice, travel times of different road types are supposed to have distinct characteristics. When mixing weekdays and weekends for each road type, the average acceptance rates of urban expressways, auxiliary roads of urban expressways, major roads, and secondary roads are 73.6%, 89.0%, 89.6%, and 70.4%, respectively. It implies that the travel times on auxiliary roads of urban expressways and major roads fitted standard distributions better than the other types of road. The related TTV patterns of each road type will be analyzed in terms of reliability measures in the following.
The temporal variation of travel times for each road type was analyzed by exploring the variability patterns, which is beneficial for traffic management agencies to make sophisticated strategies and for travelers to make smart route choices. To this end, four reliability measures were employed, that is, unit distance travel time, coefficient of variation, buffer time index, and punctuality rate. The detailed analysis results are provided below.
As mentioned before, we transformed the original travel time data from probe vehicles into unit distance travel time. According to the state-of-the-art studies [
Figure
Unit distance travel time for different types of road through the day.
Urban expressways on weekdays
Urban expressways on weekends
Auxiliary roads of urban expressways on weekdays
Auxiliary roads of urban expressways on weekends
Major roads on weekdays
Major roads on weekends
Secondary roads on weekdays
Secondary roads on weekends
On weekdays, the morning and afternoon peaks, which look like two bumps (camel shape) and highly skewed distributions, can be obviously distinguished for each road type. Note that the duration of peak hours on urban expressways is comparatively longer than the other types of road, indicating that urban expressways bear higher traffic demand in accordance with its designed function. Moreover, the TTDs over the four hierarchical road types seem to follow a similar and predictable pattern over the day. On the other hand, on weekends all types of roads did not have obvious morning and evening peaks as on weekdays and the unit distance travel times were shortened on the whole to a large extent. It is also interesting to notice that secondary roads tend to have larger unit distance travel times in daytime on both weekdays and weekends. These events may be due to the limited traffic capacity of secondary roads, where it is easier to reach saturation than other road types.
More detailed analysis shows that the morning peak on weekdays appeared around 8:00–9:00 and afternoon peak around 18:00–19:00, which well agrees with the commuting time periods. On weekends, the peak hours through the day appeared around 11:00–12:00 or 16:00–17:00, indicating distinctly different traveling behavior and variability patterns compared with weekdays. Furthermore, during the night periods, for example, 0:00–6:00 and 20:00–0:00, the 15th percentile, average, and the 95th percentile travel times of all road types were short and close to each other. In particular, the 15th percentile travel times of all road types were almost unchanging through the day on both weekdays and weekends. While, during the daytime, for example, 6:00–10:00, the average and the 95th percentile travel times increased significantly. Especially during peak hours, the 95th percentile travel time was about two to three times of the average travel time on weekdays and weekends and as much as six to nine times of the 15th percentile travel time on weekdays and three to five times on weekends.
Although the standard deviation is one of the most commonly used variability measures, the absolute value of the standard deviation may yield questionable inferences about the variability [
The CVs of travel times for each road type are shown in Figure
Coefficient of variation through the day.
On weekdays
On weekends
On both weekdays and weekends, the CVs for urban expressways in daytime were relatively smaller and had a narrower fluctuation range compared with major roads and secondary roads. It indicates relatively stable traffic states of uninterrupted flow and reliable travel times on urban expressways. It should be pointed out that, during the periods of 1:00–3:00, the CVs for urban expressway fluctuated significantly. One possible reason is that during such periods the vehicle speed data collected from probe vehicles were scarce and related to greater randomness of driving behavior.
Furthermore, on weekdays the maximum CVs of urban expressways and auxiliary roads of urban expressways through the day were around 1. The secondary roads had a large value of the maximum CV around 1.6 and the major roads around 1.8. On weekends, the maximum CV of urban expressways through the day was about 1. The values for auxiliary roads of urban expressways and secondary roads were around 1.4, and major roads still had the largest maximum CV of around 1.8. The interrupted nature of travel flows on major roads makes travel times more variable. Due to interactions between volatile traffic regimes and signal control strategies, the resulting travel times on major roads often present various distributions under different levels of congestion [
The buffer time index (BTI) represents the additional time that travelers have to spend on the basic of average travel time when they would like to arrive at their destination on time [
The BTIs of travel times for each road type are shown in Figure
Buffer time index through the day.
On weekdays
On weekends
Pu [
In this study, the punctuality rate (PR) was defined as the probability that the travel times were less than or equal to 1.1 times as much as the average travel time of road type
The PRs of travel times for each road type are shown in Figure
Punctuality rates through the day.
On weekdays
On weekends
It is worth noting that during the daytime after 7:00 the PRs of urban expressways began to decline rapidly and remained the lowest among four road types. On the other hand, the PRs of auxiliary roads of urban expressways and major roads began to rise slightly and remained the top two among four road types. For secondary roads, the PRs remained relatively stable through the day, and even higher than urban expressway during the daytime period of 7:00–19:00 on weekdays and 9:00–19:00 on weekends. It implies that, with the increasing commuting traffic demand, most of travelers might prefer to choose urban expressways first and they increased the possibility of traffic congestion on urban expressways. Meanwhile, the PRs of auxiliary roads of urban expressways and major roads became higher. Furthermore, it is interesting to notice that auxiliary roads of urban expressways and major roads have similar changing tendency of PRs as well as unit distance travel time and BTI and thus had similar travel time variation patterns.
In order to investigate the volatile characteristics of travel time in an urban network, this study explored the urban TTD and the variability patterns of different road types by using the probe vehicle data collected inside the Third Ring Road in Beijing. Different from the previous studies, in total, 200 links covering four road types, that is, urban expressways, auxiliary roads of urban expressways, major roads, and secondary roads, were exploited to provide more detailed analysis of travel time characteristics for each specific road type. The major achievements are summarized below, which may be beneficial to travelers for making reliable route choices and to traffic engineers for deploying sophisticated management and control. Link TTDs are characterized using probe vehicle data. Four common probability distributions, including normal, lognormal, gamma, and Weibull, are subjected to standard statistical tests such as K-S, A-D, and chi-squared test. The periods of interest were divided into four groups, that is, peak hours on weekdays, off-peak hours on weekdays, peak hours on weekends, and off-peak hours on weekends. Lognormal distribution is superior compared with other three types of distribution. The average acceptance rates on weekdays were always less than on weekends for each road type, and those during peak hours were always less than during off-peak hours within each time period except for the weekends of urban expressway. The travel times on auxiliary roads of urban expressways and major roads fitted standard distributions better than the other types of road. It can be implied that travel time characteristics vary from different traffic states and road types. Moreover, fitting results of link TTDs can provide guidance for determining path TTDs and network modeling. Four reliability measures, that is, unit distance travel time, CV, BTI, and PR, were used to quantify the day-of-week TTV patterns of different road types. In general, various indicators can reflect the change trend of traffic states collaboratively. However, some reliability measures may be inconsistent in their depictions of TTV, such as the case of BTI that may remain constant for different values of CV. As a result, it is more reasonable and accurate to consider the changes of multiple indicators comprehensively when measuring the road performance. On weekdays, the morning and afternoon peaks could be distinguished easily for urban expressways, auxiliary roads of urban expressways, and major roads, all of which appeared around 8:00–9:00 and 18:00–19:00, respectively. However, on weekends, all types of roads did not have obvious morning and afternoon peaks, and the peak through the day appeared around 11:00–12:00 or 16:00–17:00, indicating distinctly different traveling behavior and variability patterns compared with weekdays. Moreover, travelers can attempt to avoid traveling during peak hours to save time, and traffic management departments may target peak hours for improving traffic conditions. Auxiliary roads of urban expressways and major roads had similar changing tendency of PR as well as unit distance travel time and BTI, thus sharing similar TTV patterns. The travel times of auxiliary roads of urban expressways and major roads appeared more stable and reliable than other road types in daylight, and urban expressways had the most reliable travel times at night. It implies that with the increasing commuting traffic demand, most of travelers might prefer to choose urban expressways first and increase the possibility of traffic congestion on urban expressways.
In the future work, to enhance the effectiveness and reliability of the results, it is expected to adopt larger scale probe data in the time-space dimension or to integrate the data from different sources, for example, fusing probe data with other data collected by traffic sensors such as loop detectors and camera [
The authors declare that they have no conflicts of interest.
The authors acknowledge the National Nature Science Foundation of China (U1564212 and 51508014) and the Fundamental Research Funds for the Central Universities for kind support of this research.