Probability Distribution Of Travel Times Research Articles

Estimating Probability Distributions of Travel Times by Fitting a Markovian Velocity Model

Estimating Probability Distributions of Travel Times by Fitting a Markovian Velocity Model

Study on the Shortest Reliable Path of Stochastic Time-Dependent Transportation Networks considering Waiting Time at Signalized Intersections

Theoretical research is conducted on finding the shortest path with stochastic and time-dependent characteristics of link travel time in urban road networks. Considering the influence of signalized intersections on travel time, the research first presents a function of waiting time at signalized intersections and analyzes its characteristics. Then, considering the reliability of travel time, the travel time model under the min-max theorem is established, and a mathematical proof that the stochastic time-dependent traffic networks can be reduced to a deterministic time-dependent network is presented by using the first mathematical induction. Finally, based on analyzing the characteristics of the shortest path and minimum travel time, which vary with the start time, we propose solving the shortest path problem with a shortest path algorithm based on Dijkstra’s algorithm that takes the waiting time at signalized intersections into consideration. The research results showed that the algorithm proposed in this study does not depend on the acquisition of the probability distribution of travel time compared with the traditional algorithm. The range of uncertain travel time can be derived from historical data and travelers’ experiences, and the obtained shortest path has more robust time reliability.

Modeling conditional dependencies for bus travel time estimation

In the age of Intelligent Transportation Systems it is essential to provide operators and passengers with reliable information. The estimation of probability distributions of public transport travel times is crucial as it directly informs about the reliability of travel times. Thus, the probability distributions of travel times are useful for timetabling and route choice. This work estimates probability distributions of segment (multi-section) bus running and dwell times. We propose a hidden Markov chain framework, which captures the dependency structure of consecutive section running times and includes conditional correlations. The dependency structure of consecutive segment dwell times is modeled as a combination of correlation and operation-specific dependencies. Such a model allows describing the relationship between section-level running/station-level dwell time distributions and segment level distributions. The model is interpretable, as the dependency structure is explicitly modeled. Finally, the proposed model is evaluated on the operation of the trolley bus network of Zurich, Switzerland, and shows an average increase in fitting quality (measured by Wasserstein distance) of 26% for running times and 29% for dwell times compared to an approach not including conditional dependencies, i.e., convolution of link running and dwell times.

Link dynamic vehicle count estimation based on travel time distribution using license plate recognition data

This paper proposes a dynamic vehicle count estimation method for signalized links using license plate recognition (LPR) data considering the recognition errors. The framework contains three sub-components. First, travel time probability density function is estimated based on the kernel density model using the LPR data of matched vehicles; Second, the expected travel time is estimated for unmatched vehicles based on the travel time probability distribution; Third, with the complete arrival and departure information, the cumulative arrival-departure curve of the link is reconstructed, and the link dynamic vehicle count (LDVC) can be calculated correspondingly. To explore how different recognition errors affect LDVC estimation performance, the proposed model is validated through an empirical case of a real LPR dataset and a simulation case with different levels of matched rate. The results show that the model has a high level of estimation accuracy in different levels of matched rate of LPR data.

Service-oriented distributionally robust lane reservation

Most existing lane reservation studies usually consider a static transportation network with assuming constant link travel times. However, in reality the link travel times are highly uncertain due to various factors such as weather, accidents, road maintenance, intersections, etc. Moreover, the precise link travel time probability distribution is usually difficult to be obtained. This paper studies a new stochastic bus lane reservation problem with partial link travel time information, i.e., only the mean and covariance matrix are known. The objective is to maximize the bus service level measured by the probability of the event that all lines are jointly scheduled on time. For the problem, we formulate a service-oriented distributionally robust optimization model. Its complexity is shown to be NP-hard. To solve the problem, a sample average approximation (SAA)-based method is first adapted. Since the SAA-based approach is computational expensive, a new approximated mixed integer second-order cone programming (MI-SOCP)-based approach is developed. Computational results on a real-life case show that the proposed MI-SOCP-based approach can efficiently obtain satisfactory solutions of high quality. Besides, our results indicate that the proposed model and algorithm can provide better solutions with higher service level, as compared with general stochastic models with known distributions and without considering service levels.

Grid-Based Anomaly Detection of Freight Vehicle Trajectory considering Local Temporal Window

The security travel of freight vehicles is of high societal concern and is the key issue for urban managers to effectively supervise and assess the possible social security risks. With continuous improvements in motion-based technology, the trajectories of freight vehicles are readily available, whose unusual changes may indicate hidden urban risks. Moreover, the increasing high spatial and temporal resolution of trajectories provides the opportunity for the real-time recognition of the abnormal or risky vehicle motion. However, the existing researches mainly focus on the spatial anomaly detection, and there are few researches on the real-time temporal anomaly detection. In this paper, a grid-based algorithm, which combines the spatial and temporal anomaly detection, is proposed for tracing the risk of urban freight vehicles trajectory by considering local temporal window. The travel time probability distribution of vehicle historical trajectory is analyzed to meet the time complexity requirements of real-time anomaly calculation. The developed methodology is applied to a case study in Beijing to demonstrate its accuracy and effectiveness.

RFID E-plate Data based Travel Time Reliability Estimation of Urban Expressway

Travel time distribution shows the aggregation degree of vehicle travel time in road network, and travel time reliability is the key index to measure the aggregation degree of travel time. In order to quantitatively evaluate the fluctuation of travel time of expressway vehicles, it is necessary to establish an estimation method of travel time reliability of expressway. Travel time reliability estimation needs to obtain travel time data of vehicles. Traditional Global Position System (GPS) and video detection methods are easily affected by external factors such as weather and environment, and the description of travel time distribution is not comprehensive under the condition of low permeability. On the other hand, in the evaluation of travel time reliability, although there are lots of travel time probability distribution models, the correlation of vehicle travel time between links is ignored when calculating travel time reliability. In order to estimate travel time reliability more accurately, based on Radio Frequency Identification (RFID) electronic license plate (E-plate) data, this paper proposes a path travel time reliability evaluation model considering travel time correlation between links. First, the RFID readers installed on the gantry of expressway acquires data from the electronic tags deployed on vehicles. Secondly, a percentile travel time estimation method is proposed based on Cornish-Fisher expansion and the skewness of travel time distribution. Finally, considering the reliability estimation of travel time under the correlation of links, the estimated value is closer to the true value. The experimental results show that the travel time reliability estimation method considering the correlation between links improves the accuracy of estimation in the expressway network environment.

An EV Charging Demand Model for the Distribution System Using Traffic Property

This paper proposes a mathematical model for the spatial-temporal charging demand for electric vehicle (EV). The determination of spatial-temporal charging demand is a key step for the planning of distribution systems with a scalable application of EV. The spatial-temporal allocation of EV is conventionally obtained through a simulation procedure using traffic topology data, which is not suitable for the regions lacking such information. This model converts the problem of travel distance to travel duration so that the requirement of network geographic information can be avoided. Static EV parameters, EV spatial-temporal moving parameters and system charging model parameters are treated as the deterministic factors for the charging demand allocation. A stochastic travel route simulation procedure, which relies purely on statistical data of traffic flow, is also developed to obtain the EV moving parameters by adopting the traffic property information. The designed procedure derives travel time parameters from the vehicle dynamic-location-property (DLP) model and the travel time probability distribution. The DLP model is established using the traffic property matrix and the regional origin-destination matrix. A simple case is presented to illustrate the result of stochastic travel route simulation. Then, a modified eastern China system is used as an example to analyze the EV charging demand under multiple scenarios. The feasibility and versatility of the proposed model in the large complex system are verified by the test results.

Robust Shortest Path Problem With Distributional Uncertainty

Routing service considering uncertainty is at the core of intelligent transportation systems and has attracted increasing attention. Existing stochastic shortest path models require the exact probability distributions of travel times and usually assume that they are independent. However, the distributions are often unavailable or inaccurate due to insufficient data, and correlation of travel times over different links has been observed. This paper presents a robust shortest path (RSP) model that only requires partial distribution information of travel times, including the support set, mean, variance, and correlation matrix. We introduce a concept of robust mean-excess travel time to hedge against the risk from both the uncertainty of the random travel times and the uncertainty in their distributions. To solve the RSP problem, an equivalent dual formulation is derived and used to design tight lower and upper bound approximation methods, which adopt the scenario approach and semi-definite programming approach, respectively. To solve large problems, we further propose an efficient primal approximation method, which only needs to solve two deterministic shortest path problems and a mean-standard deviation shortest path problem, and analyze its approximation performance. Experiments validate the tightness of the proposed bounds and demonstrate the impact of uncertainty on the relative benefit and cost of robust paths.

Modeling the impacts of speed limits on uncertain road networks

ABSTRACTIn this paper, we examine the network-wide impacts of speed limits on network uncertainty. A multi-dimensional modeling approach is proposed to investigate the impacts of speed limits on link travel time uncertainty profile, travelers' risk-averse route choice decisions, and network flow reallocation under uncertainty. Methodologically, a truncated travel time probability distribution is proposed and its moments are derived to characterize the impact of speed limit on link travel time uncertainty profile. The mean-excess travel time is then adopted as a risk measure to account for travelers' risk-averse route choice adjustment and network flow reallocation induced by speed limits. We find that a speed limit scheme has different impacts on various dimensions of travel time uncertainty. The shift in travel time uncertainty profile of the directly affected links adjusts the traveler's risk-averse route choice decisions, the travel time uncertainty profile of the indirectly affected links, and consequently the network flow reallocation.

Predicting travel time reliability using mobile phone GPS data

Estimates of road speeds have become commonplace and central to route planning, but few systems in production provide information about the reliability of the prediction. Probabilistic forecasts of travel time capture reliability and can be used for risk-averse routing, for reporting travel time reliability to a user, or as a component of fleet vehicle decision-support systems. Many of these uses (such as those for mapping services like Bing or Google Maps) require predictions for routes in the road network, at arbitrary times; the highest-volume source of data for this purpose is GPS data from mobile phones. We introduce a method (TRIP) to predict the probability distribution of travel time on an arbitrary route in a road network at an arbitrary time, using GPS data from mobile phones or other probe vehicles. TRIP captures weekly cycles in congestion levels, gives informed predictions for parts of the road network with little data, and is computationally efficient, even for very large road networks and datasets. We apply TRIP to predict travel time on the road network of the Seattle metropolitan region, based on large volumes of GPS data from Windows phones. TRIP provides improved interval predictions (forecast ranges for travel time) relative to Microsoft’s engine for travel time prediction as used in Bing Maps. It also provides deterministic predictions that are as accurate as Bing Maps predictions, despite using fewer explanatory variables, and differing from the observed travel times by only 10.1% on average over 35,190 test trips. To our knowledge TRIP is the first method to provide accurate predictions of travel time reliability for complete, large-scale road networks.

Solving the Dynamic Vehicle Routing Problem Under Traffic Congestion

This paper proposes a dynamic vehicle routing problem (DVRP) model with nonstationary stochastic travel times under traffic congestion. Depending on the traffic conditions, the travel time between two nodes, particularly in a city, may not be proportional to distance and changes both dynamically and stochastically over time. Considering this environment, we propose a Markov decision process model to solve this problem and adopt a rollout-based approach to the solution, using approximate dynamic programming to avoid the curse of dimensionality. We also investigate how to estimate the probability distribution of travel times of arcs which, reflecting reality, are considered to consist of multiple road segments. Experiments are conducted using a real-world problem faced by Singapore logistics/delivery company and authentic road traffic information.

Multi-class time reliability-based congestion pricing model based on a degradable transportation network

In this study, we present a multi-class time reliability-based user equilibrium model with elastic demand, which is based on a degradable transportation network. Due to the uncertainties in the path travel time induced by random capacity degradation, the model captures the path choice behaviors of travelers in the form of the robust effective path travel time and there is no need for a known travel time probability distribution. By including the heterogeneous degrees of risk aversion among travelers, the equilibrium model can be transformed into an equivalent variational inequality (VI) problem. We also propose a heuristic solution algorithm to solve the VI problem. By taking the VI model as a constraint, a multi-class time reliability-based congestion pricing model is formulated as a mathematical programming with equilibrium constraints problem, which can be solved using the sensitivity analysis-based conjugate sub-gradient projection method. Two numerical examples are provided to illustrate the applications of the proposed models and to demonstrate the efficiency of the solution algorithms.

Modeling Distribution of Travel Time in Signalized Road Section Using Truncated Distribution

The probability distribution of travel time is the foundation of travel time estimation or modeling travel time variability and reliability. With considering the fact that travel time is distributed in a limited range, we use a truncated distribution to model the distribution of travel time in a signalized road section. Link travel time is decomposed into time-in-motion (the time a vehicle is actually moving) and time-in-queue. Time-in-motion is modeled using a truncated distribution, while time-in-queue is modeled as a mixed distribution consisting of a mass distribution and a uniform distribution. A probability density function for travel time is then derived and parameterized by fraction of queuing vehicles, red signal time, and motion behavior parameters including truncation points. These parameters are obtained from sample link travel times using a maximum likelihood estimator. In the experiment, a four-intersection network is developed in VISSIM, 10% probe vehicles in traffic are assumed in the test network. Time-in-motion and link travel time exhibit significant truncation, and they are better modeled by truncated distributions than by non-truncated distributions. Additionally, the Kolmogorov-Smirnov test is performed to test the estimated models of link travel time distribution from probe vehicle data with various polling frequencies. The results show, the link travel time distribution can be estimated from probe data that polling interval is less than 10s when the significance level is set to 0.01.

Accommodating perceptual conditioning in the valuation of expected travel time savings for cars and public transport

Travel time variability (i.e., random variations in travel time) leads to a travel time distribution for a repeated trip from a fixed origin to destination (e.g., from home to work). To represent travel time variability, a series of possible travel times per alternative (departure time, route or mode) are often used in stated choice experiments. In the traditional models, the probabilities associated with different travel scenarios (e.g., arriving early, on time and late) shown in the experiments are directly used as weights. However, evidence from psychology suggests that the shown probabilities may be transformed (underweighted or overweighted) by respondents. To account for this transformation of probabilities, this study incorporates perceptual conditioning through a non-linear probability weighting function into a utility maximisation framework, within which the empirical estimate of the value of expected travel time savings is estimated. The key advantage of this framework is that the estimated willingness to pay value can be directly linked to the source of utility (i.e., the probability distribution of travel time), while taking into account the perceptual transformation of probabilities.

Measurements of femtosecond temporal speckle field of a random medium

The femtosecond temporal speckle field of a random medium is studied theoretically and experimentally. Femtosecond temporal speckle arises from the interference of multiple randomly scattered electric fields. The femtosecond temporal speckle field is measured with a cross-correlation frequency-resolved optical gating method. The spatial average of the speckle field yields a smooth transmitted profile. The speckle field is a circular complex Gaussian variable because the scattered light beams from different trajectories have no correlation with each other. The field and the intensity profiles of individual speckle spots fluctuate randomly in time. The ensemble average of the temporal intensity profiles converges, thereby yielding the photon travel time probability distribution function.

Runoff generation in karst catchments: multifractal analysis

Time series of hydrological and geochemical signals at two karst springs, located in the Dolomiti del Brenta region, near Trento, Italy, are used to infer how karst catchments work internally to generate runoff. The data analyzed include precipitation, spring flow and electric conductivity of the spring water. All the signals show the signature of multifractality but with different intermittency and non-stationarity. In particular, precipitation and spring flow are shown to have nearly the same degree of non-stationarity and intermittency, while electric conductivity, which mimics the travel time distribution of water in the karst system, is less intermittent and smoother than both spring flow and precipitations. We found that spring flow can be obtained from precipitation through fractional convolution with a power law transfer function. An important result of our study is that the probability distribution of travel times is inconsistent with the advection dispersion equation, while it supports the anomalous transport model. This result is in line with what was observed by Painter et al. [Geophys. Res. Lett. 29 (2002) 21.1] for transport in fractured rocks.

Runoff generation in karst catchments: multifractal analysis

Time series of hydrological and geochemical signals at two karst springs, located in the Dolomiti del Brenta region, near Trento, Italy, are used to infer how karst catchments work internally to generate runoff. The data analyzed include precipitation, spring flow and electric conductivity of the spring water. All the signals show the signature of multifractality but with different intermittency and non-stationarity. In particular, precipitation and spring flow are shown to have nearly the same degree of non-stationarity and intermittency, while electric conductivity, which mimics the travel time distribution of water in the karst system, is less intermittent and smoother than both spring flow and precipitations. We found that spring flow can be obtained from precipitation through fractional convolution with a power law transfer function. An important result of our study is that the probability distribution of travel times is inconsistent with the advection dispersion equation, while it supports the anomalous transport model. This result is in line with what was observed by Painter et al. [Geophys. Res. Lett. 29 (2002) 21.1] for transport in fractured rocks.

Inverse modeling to estimate LNAPL plume release timing

Methodologies are presented for dating releases of light nonaqueous phase liquids (LNAPLs) using an inverse modeling approach with simple analytical models. Models for LNAPL plume migration are presented to predict LNAPL plume velocity in the unsaturated and saturated zones as a function of basic soil and fluid properties. A relative mobility factor is introduced for LNAPL movement at the water table that depends primarily on the van Genuchten n parameter (related to the breadth of the soil pore size distribution) and the magnitude of water table fluctuations. Estimated LNAPL plume velocities compare reasonably with more rigorous numerical models, which may be used in cases where data availability warrant the greater effort entailed. Two methods of estimating release timing and its uncertainty are investigated. A direct estimation method is described that determines travel time for a single observed travel distance based on estimated soil and fluid properties. Release date uncertainty may be determined using the first order (FO) or Monte Carlo (MC) methods. The second method for estimating release date involves nonlinear parameter estimation utilizing distance vs. time measurements and other data. A case study is presented for a field site where independent estimates of release timing were obtained from a numerical modeling analysis. Release timing estimates based on direct inversion of the analytical timing model agree well with the numerical analysis. Results for a second field site indicate that release date confidence limits estimated by the FO method, assuming log-normally distributed travel times, are close to values determined by the MC method, which makes no assumption regarding the form of the travel time probability distribution. Results for a hypothetical problem indicate that LNAPL velocity and travel time may be accurately estimated if sufficient data on travel distance vs. time are available. Incorporating prior information on relevant soil and fluid properties into the objective function reduces the uncertainty in release date if prior estimates are accurate. However, biased prior estimates may lead to over- or underestimation of release date uncertainty. Simultaneous estimation of soil and fluid properties and release date is possible if prior information is available to condition the parameter estimates.

Probability Distribution of Solute Travel Time for Convective Transport in Field‐Scale Soils Under Unsteady and Nonuniform Flows

This study addresses the development of probability distributions of travel times for one‐dimensional (vertical) solute transport in soils. The field‐scale soils are considered heterogeneous, with stationary fluctuations of soil hydraulic properties in the horizontal direction but nonstationary fluctuations of these properties in the vertical direction due to layering of the soil, which induces nonstationary heterogeneity. Approximate ensemble probability distribution functions of conservative solute travel time for vertical convective solute transport were derived directly from the convective transport stochastic partial differential equation, under both deterministic and stochastic soil surface water flux (infiltration rate) and under unsteady and nonuniform soil water flows. General depth‐varying initial and time‐varying boundary conditions were used in these derivations. The magnitude of the approximation in the theoretical probability distribution functions of travel time is quantified mathematically. Utilizing the soil water content data from a University of California, Davis, field site, it is shown that the mathematical condition for this approximation is satisfied for this field. The spatial heterogeneity is represented through a nonstationary soil water content random field which covaries both in time and in space. Dispersion emerges naturally in the derived ensemble probability distribution functions of solute travel time, owing to the stochasticity of soil water content at field scale. Then the theoretical expression for mean solute concentration over a field is derived, by means of the theoretical solute travel time distribution, as a function of time and soil depth, under vertical transport with rectangular pulse solute loading for the upper boundary condition. Comparisons of theoretical probability density functions of solute travel time against their empirical counterparts, obtained from field experimental observations under steady but nonuniform soil water flow, show good agreement. Comparisons of theoretical mean solute concentrations, as they evolve with time and soil depth, against field experimental observations also show good agreement. However, further field experiments under unsteady flow conditions are required for the comprehensive validation of the developed theory.