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Vehicles travelling on urban streets are heavily influenced by traffic signal controls, pedestrian crossings, and conflicting traffic from cross streets, which would result in bimodal travel time distributions, with one mode corresponding to travels without delays and the other travels with delays. A hierarchical Bayesian bimodal travel time model is proposed to capture the interrupted nature of urban traffic flows. The travel time distributions obtained from the proposed model are then considered to analyze traffic operations and estimate travel time distribution in real time. The advantage of the proposed bimodal model is demonstrated using empirical data, and the results are encouraging.

Travel time is an important piece of information for transportation planners, traffic operators, and road users. It has been widely used in the studies of route choice, origin-destination (OD) flow estimation, and transportation system reliability [

Loop detectors are the most common data source for travel time estimation, particularly on freeways. Since loop detectors provide traffic information, such as volume, speed, and occupancy, at fixed locations, additional assumptions need to be made to estimate vehicle travel times [

Travel times are traditionally modeled with unimodal distributions, such as normal, lognormal, Gamma, and Burr distributions [

However, recent studies have revealed that travel time distribution tends to be bimodal and even multimodal. Jintanakul et al. [

Travel time distributions on urban streets are more complex than those on freeways. Vehicles travelling on urban streets are heavily influenced by traffic signal controls, pedestrian crossings, and conflicting traffic from cross streets. The interrupted nature of urban traffic flows would likely result in large fluctuations in observed travel times. For example, a vehicle passing a signal at the end of the green would experience quite different travel time than the vehicle following behind it that must make a stop for the red, although they traveled next to each other. Taylor and Somenahalli [

This paper develops methodologies to analyze urban link travel times from probe vehicles. Unlike previous travel time studies, this paper not only develops methodologies to capture important characteristics of the travel time distributions, but also explores potential applications of the travel time distributions resulting from the developed model. Specifically, the resulting travel time distributions are used to analyze traffic operations and estimate travel time distribution in real time. The advantage of the developed methodologies is demonstrated using empirical data.

The rest of the paper starts with the travel time model and the algorithms to estimate model parameters. Then we introduce the dataset used in the empirical study. Next we demonstrate how to use the travel time distributions to analyze traffic operations and estimate travel time distribution in real time. Finally, we conclude the paper with a summary of the key findings and possible future research.

Probe travel times collected on a given urban link in a given time-of-day period over multiple days are considered in this study. A hierarchical Bayesian bimodal model is developed to fit travel time distributions. Travel time distributions on urban streets are affected by various factors. Some of the factors, such as geometric characteristics (length, lane width, speed limits, etc.), are time-invariant, while others, such as traffic volume, traffic mix, and signal timing, could vary over days. Therefore, it is natural to model travel times hierarchically, with observed travel times modeled conditionally on certain parameters, which themselves are given a probabilistic specification in terms of further parameters, known as hyperparameters. The associations among travel times in different days are captured by using a joint probability distribution for model parameters in different days.

Given that travel times are positive and their distributions are positively skewed and that urban link travel time distributions tend to be bimodal, it is assumed that each component of the bimodal travel time distribution follows log normal distribution. That is, the model is fitted to the logarithms of the travel time observations. Specifically, the log travel times in a day

Weak prior distributions (i.e., the prior densities are diffuse) are used in this paper such that the posterior distributions are dominated by the data. The prior distribution of the mixture probability

Given the distributional assumptions of (

Even though the proposed model is developed based on standard distributions, the marginal posterior distributions of the model parameters are analytically intractable [

We adopt the “rjags” (stands for just another Gibbs sampler in R) package in the statistics software “R” to simulate the marginal posterior distributions of the model parameters [

Real-time estimation of travel time distribution is an important component in advanced traveler information system. Conditional on the real-time observed travel times, we develop a methodology to estimate the parameters of the travel time distribution for a given period of a given day. Historical data are considered in the methodology through the prior distribution of the model parameters.

The parameter set (

The posterior distributions of the model parameters obtained based on historical data are used as the prior distributions in (

The MCMC algorithm can be used to produce the posterior distributions of the model parameters. Nevertheless, the MCMC algorithm may not be suitable for real-time application due to its relatively high computational requirement. Therefore, we developed an expectation conditional maximization (ECM) algorithm [

Start with some (likely crude) estimates of the model parameters

For

where

CM-steps:

update the probability

update the expected parameter

update the expected parameter

update the variance

update the variance

The methodologies proposed in Section

The probe bus data are provided by the Campus Area Bus Service (CABS) at the Ohio State University (OSU). The CABS serves approximately four million passengers annually on seven routes on and in the vicinity of the OSU Campus. GPS-based AVL systems have been used on all CABS buses since 2009. The AVL system records bus statuses (e.g., location and velocity) at a frequency of 1 Hz. This study considers AVL data collected by buses serving the Campus Loop South (CLS), Campus Loop North (CLN), and North Express (NE) bus routes. The advertised headways of the CLS, CLN, and NE bus routes are 9, 9, and 5 minutes, respectively.

Two links are considered in the empirical demonstration. The lengths of Links 1 and 2 are 248.1 and 216.8 meters, respectively. Link 1 contains a four-way signalized intersection and Link 2 contains a four-way signalized intersection and a pedestrian crossing. To capture the total time losses due to vehicle acceleration and deceleration caused by signal controls or pedestrians crossing the street in the travel times, links are defined such that the intersection and pedestrian crossing are located inside the links. In addition, although it is possible to eliminate the increase in travel time due to stopping at bus stops for passengers alighting and boarding [

Travel times collected in a.m. (7:30 a.m.–7:45 a.m.) and p.m. (4:30 p.m.–4:45 p.m.) periods of 40 weekdays are considered for the empirical demonstration. Summary statistics of the travel time observations and the signal information for the corresponding travel direction are provided in Table

Summary statistics for the two links of interest.

Link 1 | Link 2 | |||
---|---|---|---|---|

a.m. period | p.m. period | a.m. period | p.m. period | |

Number of observations | (1, 4.2, 6) | (2, 3.3, 5) | (1, 4.2, 7) | (2, 3.2, 5) |

Mean travel time (second) | 66 | 64 | 47 | 29 |

Cycle length (second) | 120 | 120 | 100 | 100 |

Green phase (second) | 35 | 49 | 37 | 64 |

As shown in Table

Figure

Empirical distributions of the log travel time for each link and period combination.

Two Markov chains are run for each link and period combination to evaluate the convergence of the MCMC algorithm. Each chain is run for 20,000 iterations and the first 10,000 iterations are discarded as burn-in. The rest of the sequence is thinned by keeping samples of every 10th iteration to reduce the correlations between samples, resulting in 2,000 samples (1,000 samples for one chain) to represent the posterior distributions of the model parameters. The MCMC samples of some model parameters are used in the following to analyze traffic operations.

Figure

Posterior distributions of the mixture probability.

Figure

Posterior distributions of the expected delay (in seconds).

Table

Comparisons of signal timing and model parameters.

Link | Link 1 | Link 2 | ||
---|---|---|---|---|

Period | a.m. | p.m. | a.m. | p.m. |

Red ratio | 0.71 | 0.59 | 0.63 | 0.36 |

Mixture probability | 0.69 | 0.57 | 0.79 | 0.48 |

Red phase (second) | 85.0 | 71.0 | 63.0 | 36.0 |

Expected delay (second) | 56.2 | 52.6 | 31.8 | 18.3 |

Figure

Posterior distributions of the expected travel time (in seconds) for the “fast bus” and “slow bus” groups.

On Link 2, the expected travel times for the “fast” group in a.m. period are slightly higher than those in p.m. period, and the expected travel times for the “slow” group in a.m. period are much higher than those in p.m. period. The higher expected travel times for the “slow” group in a.m. period are due to the greater value of the red ratio in a.m. period, as discussed when presenting Table

As demonstrated above, the proposed model provides useful information for analyzing traffic operations on urban streets. Such information cannot be obtained when travel times are assumed to be unimodally distributed. The results presented in Figures

To evaluate the performance of the proposed methodologies in estimating travel time distribution, travel times on Link 1 in a.m. period of a hypothetical day are simulated. The true model parameters for the hypothetical day are randomly selected from the MCMC samples. Based on the “assumed” true model parameters, various numbers of travel time observations are simulated using (

The model proposed in [

The estimated travel time distributions produced by the two methods are presented in Figure

Estimated travel time distributions in a hypothetical day on Link 1 in a.m. period.

Furthermore, we carry out the simulation for 100 times. In each time, the mean, lower quantile (i.e., 25% quantile), and the upper quantile (i.e., 75% quantile) of the travel time distributions produced by the bimodal and unimodal models are computed. The simulated distributions of the three summary statistics for 5 and 100 numbers of observations are presented in Figure

Simulated distributions of the mean, lower quantile, and upper quantile of the travel time distributions produced by the bimodal and unimodal models.

Empirical studies have revealed that the prevalent travel time distributions on urban links are bimodal, thanks to the presence of traffic signals. A hierarchical Bayesian bimodal model is developed to fit observed vehicle travel time data on urban streets. Since the bimodal travel time distributions capture the interrupted nature of traffic flows on urban streets more accurately, it offers a better model for a variety of applications. We demonstrate empirically that the bimodal model provides useful information for analyzing traffic operations and that it provides higher estimation accuracy of the travel time distribution than does the unimodal model.

This study can be extended in several directions. For example, this study considers probe data alone to estimate travel time distributions. Considering traffic volume and traffic signal timing as covariate variables in the model would improve the accuracy of the travel time distribution estimation. The effect of some other observable factors, such as bad weather and accidents, could also be considered in an extension of the proposed model.

In addition, this study uses probe bus data to demonstrate the advantage of the proposed methodologies. It would be valuable to evaluate the proposed methodologies using other types of probe vehicle data, such as probe taxi data. Lastly, some other valuable applications based on the proposed model are worth pursuing in the future, such as congestion identification, accident detection, evaluation of coordinated signal control systems, and travel time reliability.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (51308410) and the Fundamental Research Funds for the Central Universities of China (1600219210). The work was also supported by Program for Changjiang Scholars and Innovative Research Team in University. The authors are grateful to Srah Blouch and Chris Kvitya of the OSU Campus Area Bus Service for providing the datasets used in this study, to Mr. Danie E. Moorhead and Mr. Gary J. Holt at the city of Columbus for providing the signal timings at the two intersections of interest, and to Mr. Xudong Hu for conducting a site survey. The authors also appreciate the technical support provided by the Campus Transit Lab (CTL) at the OSU.