Path Travel Time Distribution Research Articles

Estimation of Path Travel Time Distributions in Stochastic Time-Varying Networks with Correlations

Transportation research has increasingly focused on the modeling of travel time uncertainty in transportation networks. From a user’s perspective, the performance of the network is experienced at the level of a path, and, as such, knowledge of variability of travel times along paths contemplated by the user is necessary. This paper focuses on developing approaches for the estimation of path travel time distributions in stochastic time-varying networks so as to capture generalized correlations between link travel times. Specifically, the goal is to develop methods to estimate path travel time distributions for any path in the networks by synthesizing available trajectory data from various portions of the path, and this paper addresses that problem in a two-fold manner. Firstly, a Monte Carlo simulation (MCS)-based approach is presented for the convolution of time-varying random variables with general correlation structures and distribution shapes. Secondly, a combinatorial data-mining approach is developed, which aims to utilize sparse trajectory data for the estimation of path travel time distributions by implicitly capturing the complex correlation structure in the network travel times. Numerical results indicate that the MCS approach allowing for time-dependence and a time-varying correlation structure outperforms other approaches, and that its performance is robust with respect to different path travel time distributions. Additionally, using the path segmentations from the segment search approach with a MCS approach with time-dependence also produces accurate and robust estimates of the path travel time distributions with the added benefit of shorter computation times.

Incorporating Travel Time Reliability in Equitable Congestion Pricing Schemes for Heterogeneous Users and Bimodal Networks

Congestion pricing is proposed as an effective travel demand management strategy to circumvent the problem of congestion and generate revenue to finance developmental projects. There are several studies focusing on optimal pricing strategies to minimize the congestion level or maximize the revenue of the system. However, with regard to equity issues, benefiting only users with higher value of time is claimed to be the main factor that prevents implementation of such policies in practice. While many studies aimed to tackle the equity issues by certain welfare analyses, most of these studies fail to fully consider realistic features of users’ behavior and the uncertainty in link travel times. Given the variability of travel time in real-world networks and the impacts of pricing policies on path travel time distributions, it is important to consider the users’ reliability valuations, in addition to their travel time valuations. Thus, the goal in this study is to find an equitable pricing scheme that minimizes the total travel time of auto users in a general bimodal network considering heterogeneous users with different values of time and reliability. A particle swarm optimization algorithm is proposed to find self-funded and Pareto-improving optimal toll values. A reliability-based user equilibrium algorithm is embedded into this optimization algorithm to assign travelers to the equilibrated paths for different user classes given toll values. The proposed approach is successfully applied to a modified Sioux Falls network to explore impacts of subsidization, congestion level, and considering travel time reliability on the pricing strategy and its effectiveness.

Bus travel time modelling using GPS probe and smart card data: A probabilistic approach considering link travel time and station dwell time

Path travel time estimation for buses is critical to public transit operation and passenger information system. State-of-the-art methods for estimating path travel time are usually focused on single vehicle with a limited number of road segments, thereby neglecting the interaction among multiple buses, boarding behavior, and traffic flow. This study models path travel time for buses considering link travel time and station dwell time. First, we fit link travel time to shifted lognormal distributions as in previous studies. Then, we propose a probabilistic model to capture interactions among buses in the bus bay as a first-in-first-out queue, with every bus sharing the same set of behaviors: queuing to enter the bus bay, loading/unloading passengers, and merging into traffic flow on the main road. Finally, path travel time distribution is estimated by statistically summarizing link travel time distributions and station dwell time distributions. The path travel time of a bus line in Hangzhou is analyzed to validate the effectiveness of the proposed model. Results show that the model-based estimated path travel time distribution resembles the observed distribution well. Based on the calculation of path travel time, link travel time reliability is identified as the main factor affecting path travel time reliability.

Path Travel Time Estimating Method by Incomplete Traffic Data

This study proposes a method estimating stochastic path travel times by using both traffic counter data and probe car data complementarily. A stochastic traffic demand for each O-D pair in a road network is estimated by maximum likelihood estimation with respect to traffic counter data. Stochastic path travel times are addressed as a prior multivariate path travel time distribution. The estimated stochastic path travel time is updated by applying Bayesian inference using observed probe car data. The updated path travel time can be regarded as a posterior multivariate path travel time distribution. Numerical experiments demonstrate inference processes of path travel time and verify our proposed model.

The α-reliable path problem in stochastic road networks with link correlations: A moment-matching-based path finding algorithm

Most existing studies on routing guidance only paid attention to the average path travel time, which failed to consider travel time reliability (TTR) preferences by different travelers. In this study, a moment-matching-based hybrid genetic algorithm (MHGA) is proposed to search the reliable shortest path (RSP) in stochastic road networks with link correlations. First, the goodness-of-fit results based on field data reveal that lognormal distributions are more appropriate for characterizing link travel times. The impact of topological distance (measured by the number of links) and road type on link correlations is also scrutinized. Then, a moment-matching method (MOM) is utilized to determine the parameters of the approximate path travel time distribution (TTD) by accounting for link correlations. A local search algorithm is designed to improve the search ability of the path finding algorithm. In view of travelers’ risk tolerance, the algorithm enables the provision of personalized routing guidance for individual travelers. Furthermore, to support path finding applications in a large-scale network, heuristic constraints are imposed to help reduce the computational workload and accelerate the convergence speed of the search process. Finally, numerical case studies based on synthetic networks and a real road network in Beijing are presented, and the results help demonstrate that the algorithm has good potential to solve RSP searching problems in a large-scale network with desirable efficiency.

Heterogeneous Data Fusion Method to Estimate Travel Time Distributions in Congested Road Networks.

Travel times in congested urban road networks are highly stochastic. Provision of travel time distribution information, including both mean and variance, can be very useful for travelers to make reliable path choice decisions to ensure higher probability of on-time arrival. To this end, a heterogeneous data fusion method is proposed to estimate travel time distributions by fusing heterogeneous data from point and interval detectors. In the proposed method, link travel time distributions are first estimated from point detector observations. The travel time distributions of links without point detectors are imputed based on their spatial correlations with links that have point detectors. The estimated link travel time distributions are then fused with path travel time distributions obtained from the interval detectors using Dempster-Shafer evidence theory. Based on fused path travel time distribution, an optimization technique is further introduced to update link travel time distributions and their spatial correlations. A case study was performed using real-world data from Hong Kong and showed that the proposed method obtained accurate and robust estimations of link and path travel time distributions in congested road networks.

Forecasting journey time distribution with consideration to abnormal traffic conditions

Travel time is an important index for managers to evaluate the performance of transportation systems and an intuitive measure for travelers to choose routes and departure times. An important part of the literature focuses on predicting instantaneous travel time under recurrent traffic conditions to disseminate traffic information. However, accurate travel time prediction is important for assessing the effects of abnormal traffic conditions and helping travelers make reliable travel decisions under such conditions. This study proposes an online travel time prediction model with emphasis on capturing the effects of anomalies. The model divides a path into short links. A Functional Principal Component Analysis (FPCA) framework is adopted to forecast link travel times based on historical data and real-time measurements. Furthermore, a probabilistic nested delay operator is used to calculate path travel time distributions. To ensure that the algorithm is fast enough for online applications, parallel computation architecture is introduced to overcome the computational burden of the FPCA. Finally, a rolling horizon structure is applied to online travel time prediction. Empirical results for Guangzhou Airport Expressway indicate that the proposed method can capture an abrupt change in traffic state and provide a promising and reliable travel time prediction at both the link and path levels. In the case where the original FPCA is modified for parallelization, accuracy and computational effort are evaluated and compared with those of the sequential algorithm. The proposed algorithm is found to require only a piece rather than a large set of traffic incident records.

Estimation of Travel Time Distributions in Urban Road Networks Using Low-Frequency Floating Car Data

Travel times in urban road networks are highly stochastic. However, most existing travel time estimation methods only estimate the mean travel times, while ignoring travel time variances. To this end, this paper proposes a robust travel time distribution estimation method to estimate both the mean and variance of travel times by using emerging low-frequency floating car data. Different from the existing studies, the path travel time distribution in this study is formulated as the sum of the deterministic link travel times and stochastic turning delays at intersections. Using this formulation, distinct travel time delays for different turning movements at the same intersection can be well captured. In this study, a speed estimation algorithm is developed to estimate the deterministic link travel times, and a distribution estimation algorithm is proposed to estimate the stochastic turning delays. Considering the low sampling rate of the floating car data, a weighted moving average algorithm is further developed for a robust estimation of the path travel time distribution. A real-world case study in Wuhan, China is carried out to validate the applicability of the proposed method. The results of the case study show that the proposed method can obtain a reliable and accurate estimation of path travel time distribution in congested urban road networks.

Estimation of trip travel time distribution using a generalized Markov chain approach

The increasing availability of opportunistic and dedicated sensors is transforming a once data-starved transport field into one of the most data-rich. While link-level travel time information can be derived or inferred from this data, methods for estimation of trip travel times between an origin and a destination pair are still evolving and limited, especially in the context of probability distribution estimation. This paper proposes a generalized Markov chain approach for estimating the probability distribution of trip travel times from link travel time distributions and takes into consideration correlations in time and space. The proposed approach consists of three major components, namely state definition, transition probabilities estimation and probability distribution estimation. A heuristic clustering method, based on Gaussian mixture models, has been developed to cluster link travel time observations with regard to their homogeneity and underlying traffic conditions. A transition probability estimation model is developed as a function of link characteristics and trip conditions using a logit model. By applying a Markov chain procedure, the probability distribution of trip travel times is estimated as the combination of Markov path travel time distributions weighted by their corresponding occurrence probabilities. The link travel time distribution is conditioned on the traffic conditions of the current link that can be estimated from historical observations. A moment generating function based algorithm is used to approximate the Markov path travel time distribution as the sum of correlated link travel time distributions conditional on traffic conditions. The proposed approach is applied in a transit case study using automatic vehicle location data. The results indicate that the method is effective and efficient, especially when correlations and multimodal distributions exist.

Path Finding in Stochastic Time Varying Networks with Spatial and Temporal Correlations for Heterogeneous Travelers

This paper studies the path finding problem in stochastic networks with known travel time distributions on each link of the network, in the presence of spatial and temporal travel time correlations. As a result of traffic dynamics and the decisions of network users, distributions with different characteristics can be observed for the link travel time at different time intervals. Considering correlations in a dynamic stochastic network and applying simulation-based realistic travel time distributions and correlations in the path finding problem in stochastic networks are the main contributions of this study. Two reliability-based optimal path finding problems were considered: the shortest path problem with on-time arrival probability and the minimum travel time budget path problem. For each problem, user heterogeneity can be considered in the developed approach, as users are allowed to have different preferences for travel time reliability. Solution algorithms for both problems were implemented and tested on a large-scale network in Chicago, Illinois, with networkwide link travel time distributions and correlations obtained from a traffic simulator. Numerical results demonstrate the impacts of correlations in a dynamic stochastic network on the optimal path travel time distribution. The developed approach can be used in the reliability-based user equilibrium problem to solve the main subproblem, the path finding problem, in dynamic stochastic networks.

Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach

This paper studies a mean-standard deviation shortest path model, also called travel time budget (TTB) model. A route’s TTB is defined as this route’s mean travel time plus a travel time margin, which is the route travel time’s standard deviation multiplied with a factor. The TTB model violates the Bellman’s Principle of Optimality (BPO), making it difficult to solve it in any large stochastic and time-dependent network. Moreover, it is found that if path travel time distributions are skewed, the conventional TTB model cannot reflect travelers’ heterogeneous risk-taking behavior in route choice. This paper proposes to use the upper or lower semi-standard deviation to replace the standard deviation in the conventional TTB model (the new models are called derived TTB models), because these derived TTB models can well capture such heterogeneous risk-taking behavior when the path travel time distributions are skewed. More importantly, this paper shows that the optimal solutions of these two derived TTB models must be non-dominated paths under some specific stochastic dominance (SD) rules. These finding opens the door to solve these derived TTB models efficiently in large stochastic and time-dependent networks. Numerical examples are presented to illustrate these findings.

Finding most reliable paths on networks with correlated and shifted log–normal travel times

There is a growing interest in modeling travel time uncertainty in transportation networks in addition to optimizing the reliability of travel times at the path and network level. This paper focuses on the analysis and optimization of travel time (including stopped delays) Reliability on the Urban Road Network in Chennai. Specifically, two objectives are investigated. The first objective involves the quantification of travel time reliability at the link and path level. In particular, the distribution of link travel times is quantified for the Chennai Urban road network using empirical data. The results indicate that the shifted log–normal distribution (SLN) reasonably represents link travel time for all facility types and relevant facility wise distribution parameters are estimated. Further, the resulting path travel time distribution is approximated by a SLN distribution, which is computationally less expensive than traditional Monte-Carlo estimation techniques with an acceptable compromise on accuracy. The second objective addresses the optimal reliability path problem on a network with SLN link travel times with general correlation structure. For this problem, it is shown that the sub-path optimality property of shortest path problems does not hold making traditional label-setting/label correcting algorithms inapplicable. Consequently, a sufficient optimality condition based on reliability bounds is established and a new network optimization algorithm is proposed and proof of correctness is presented. The convergence rate of the algorithm was shown to increase at every iteration under some mild conditions. The computational performance of the proposed algorithm is investigated using synthetic and real-world networks and found to be reasonably accurate.

Finding Reliable Shortest Paths in Dynamic Stochastic Networks

A reliable a priori shortest path (RASP) offers the least time budget and ensures that a traveler can arrive at the destination on time with the desired probability. A RASP is equivalent to enumerating all nondominated paths under the first-order stochastic dominance rule. Compared with the problem in static networks, the RASP problem becomes more complex in dynamic networks because it is more difficult to compute path travel time distributions. Two modules of process are the keys to solving the RASP problem. One module is the convolution scheme (how to compute a path travel time distribution from its member links' travel time distributions), and the other module is the stochastic dominance scheme (how to determine nondominated paths). This study aims to find an efficient solution algorithm for this problem. An alternative convolution method is developed on the basis of the adaptive discretization approach, which was originally proposed to solve the RASP problem in static networks. In contrast, the higher-order stochastic dominance rule can be shown to reduce the number of nondominated paths; this rule promises to improve computational efficiency. Numerical experiments show that the second-order stochastic dominance rule offers good approximations, and the central processing unit time is reduced by at least 50%.

Reliable route guidance: A case study from Chicago

Reliable route guidance can be obtained by solving the reliable a priori shortest path problem, which finds paths that maximize the probability of arriving on time. The goal of this paper is to demonstrate the benefits and applicability of such route guidance using a case study. An adaptive discretization scheme is first proposed to improve the efficiency in computing convolution, a time-consuming step used in the reliable routing algorithm to obtain path travel time distributions. Methods to construct link travel time distributions from real data in the case study are then discussed. Particularly, the travel time distributions on arterial streets are estimated from linear regression models calibrated from expressway data. Numerical experiments demonstrate that optimal paths are substantially affected by the reliability requirement in rush hours, and that reliable route guidance could generate up to 5–15% of travel time savings. The study also verifies that existing algorithms can solve large-scale problems within a reasonable amount of time.

Application of Discrete Fourier Transform to Find Reliable Shortest Paths

Reliable shortest paths, in this paper, are paths that ensure a desired probability of arriving on time or earlier with the least time budget. Finding reliable shortest paths requires the evaluation of path travel time distributions through numerical convolution, which is a computationally intensive procedure. This paper develops several convolution methods based on the discrete Fourier transform (DFT) and compares them with a direct convolution method implemented with adaptive discretization. Two shortcomings in the conventional DFT-based convolution are recognized, namely, the incompatibility with dynamic programming and the computational inefficiency associated with support points that have negligible probability mass. Remedies are proposed to overcome these shortcomings. However, numerical experiments suggest that, even with these improvements, the DFT-based convolution methods are still not as computationally competitive as the direct convolution method, contrary to the promise of the theoretical complexity analysis.

Essentially best routes in dynamic and stochastic transportation network

Routing in dynamic and stochastic network is to provide adaptive 'best' route guidance where the link/route travel times are modelled as a time-varying stochastic process. Typical approaches for routing when travel times are both dynamic and stochastic assume that link travel times are independent and travellers are seeking for expected shortest paths. It is observed in many applications that providing the single least-expected travel time path is not adequate and appropriate to help diverse travellers make travel decision. Given the stochastic link attributes, the accumulated uncertainty of those attributes over any given route can easily make many routes statistically indistinguishable from the expected shortest path. In this paper, we investigate the dependences between link and path attributes, derive the exact path travel time distribution from link travel time distributions with consideration of their interrelationship, and employ an efficient simulation method to determine a set of adaptive 'best' paths. Given certain reference function of travel time, the individual turning decision at each branching point is made accordingly.