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Forecasting journey time distribution with consideration to abnormal traffic conditions

Research output : Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

  • Andy H.F. Chow
  • And 5 others

Research Area(s)

  • Adaptiveness to traffic incident, Functional principal component analysis, Parallel computing, Probabilistic nested delay operator, Travel time prediction

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Review article, review on statistical modeling of travel time variability for road-based public transport.

journey time distribution

  • Institute for Transport Planning and Systems, ETH Zurich, Zurich, Switzerland

Accurately modeling the travel time of road-based public transport can help directly improve current passenger service and operating efficiency. Moreover, it paves the way for control of future high technology automated vehicles, which will share the same characteristics of sharing the road infrastructure with other vehicles; carrying multiple passengers; having a non-negligible dwell process; and run not completely demand-responsive, but in general following a schedule or a target frequency. Recent advances in sensor and communications technology, leading eventually to comprehensive vehicle connectivity, have significantly increased the amount and quality of travel time data available, making it possible to better model distributions of current buses' travel time. We assume that the choice of those distributions with regards to transport performance will hold also in the near future. This paper explains definitions of travel time components and explains how they contribute to variability. It focuses on the description of day-to-day variability, and systematically reviews the current state-of-the-art for statistically modeling bus travel, running, and dwell time distributions. It considers statistical distributions developed based on empirical data from the research literature. Statistical distributions are powerful tools, as they can describe the inherent variability in data with a limited number of parameters. The review finds that both spatial and temporal data aggregation have an important influence on the statistics as well as the choice of the most appropriate probability distribution. This influence is still not well-understood and remains a question for further studies. Furthermore, the review finds that mixture distributions provide good fitting performance. However, it is important to improve the description of components in such distributions to get meaningful and understandable distributions. The methodologies for fitting distributions, for proving if a distribution is suited, and for identifying best fitting, robust, and reproducible distribution should be reconsidered. Such a distribution will enable reporting, controlling operations, and disseminating information to operators and travelers. Finally, this review proposes directions for further work.

Introduction

The observed travel time of road-based public transport vehicles, as well as its components (i.e., dwell time and running time), are subject to variability, caused by the stochastic nature of various factors, including traffic congestion. Travel time variability causes uncertainty, thus increasing costs for travelers and operators ( Li et al., 2010 ). For passengers, not knowing the precise arrival time of a planned public transport service and the expected travel time complicates their decisions regarding departure time, route choice, or even mode choice. Research has shown that reducing travel time variability is even more valuable to passengers than reducing travel time ( Bates et al., 2001 ). For public transport operators, this variability reduces on-time performance and increases operating costs, for example by requiring the addition of recovery times to schedules. Many strategic and operational decisions are also affected by variability and impact the cost of service.

Understanding variation in travel times for public transport vehicles is also critical for transport modeling and simulation. Stochastic simulations, which reproduce the inherent variability of realistic situations, require detailed information about the distribution of travel times. Moreover, travel time distributions are used for arrival time predictions and discrete choice modeling in route selection ( Mazloumi et al., 2010 ). By applying meaningful statistical distribution models, public transport operators can improve the performance of probabilistic delay forecasting and can better inform passengers about their planned journey.

The ease in collecting and processing large sets of travel time data, helped by the implementation of automated vehicle location (AVL) systems and accessibility to data via open data platforms, have strongly increased the potential for studying travel times performance. However, recent findings are strongly influenced by the type and the aggregation level of the tested data, as well as by the assumptions used.

This work aims to collect, compare, and contrast previous research on travel time distributions for public transport systems in terms of the data and methods used, levels of aggregation, and the proposed distributions. With an eye on road-based future transport systems such as autonomous cars, we focus on public buses only, i.e., buses running on roads, which can be shared or not with other traffic. The underlying assumptions are that the dynamic of roads will remain as an invariant for a long time, through the evolution of technology and automation, and penetration rate of automated transport systems. We assume the following dynamics will remain: peak phenomena and recurrent congestion at peak hours, dwell time characteristics of vehicles carrying a substantial amount of passengers, and internal interference from closely spaced successive services, related to bunching phenomena.

The first step of this paper is to report on an extensive review of the relevant literature on statistical modeling of bus travel time variability. This search only considered publications that propose statistical distributions of bus travel time components based on empirical findings from case studies. In addition, we make empirically based considerations with the goal of determining recommendations for future research. Those considerations aim to be universal, regardless of the precise vehicle technology used.

The remainder of the paper is structured as follows: section Definition of bus travel time variability describes the variability of travel time of bus operations. Section components of travel times and their variability defines travel time components and explains how they contribute to variability. Section literature Analysis presents the results of the literature review on distributions for characterizing the day-to-day variability. Section Discussion presents a discussion of results. Section Recommendations for further research presents conclusions and recommendations for further research.

Definition of Bus Travel Time Variability

The variability of travel times can be viewed from three different perspectives: the day-to-day variability, the variability over the course of a day, and the vehicle-to-vehicle variability ( Noland and Polak, 2002 ). These perspectives were initially developed for automobile traffic, but can be adapted for public transport travel times ( Kieu et al., 2014 ).

The day-to-day (or inter-day) variability describes the variability between similar trips within the same time period on different days. Figure 1A illustrates day-to-day variation, which shows the travel time of the same bus trip at different working days in a density curve. Day-to-day variation can be caused by travel demand fluctuations, driving behavior, incidents, and weather conditions.

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Figure 1 . Typical representation explaining (A) day-to-day variability, (B) variability over the course of a day, and (C) vehicle-to-vehicle variability.

The second type of variability, variability over the course of the day (also known as inter-period or period-to-period variability), describes variability between vehicles making similar trips at different times on the same day. As shown in Figure 1B , bus travel times are usually longer for a given trip during peak periods compared with off-peak periods. The variability over the course of the day can be caused by short-term changes in congestion, incidents, or weather conditions.

Finally, the third type of variability, vehicle-to-vehicle (or inter-vehicle) variability, describes the variability between travel times experienced by different vehicles traveling at similar times over the same route. For example, Figure 1C compares the travel times of two subsequent vehicles. Vehicle-to-vehicle variability is mainly caused by different delay times at traffic signals, conflicts with pedestrians, or differences in driving behavior. It can also be used as an indicator for detecting bus bunching.

Most studies considering variability of travel times focus exclusively on the day-to-day variability. The reason for this is that commuters travel with a specific service at a specific time on multiple days. Also for operators, the day-to-day variability is of key interest, as it shows the performance on multiple days and represents the starting point for schedule optimizations and allocating slack times.

The reliability of public transport service is a very important factor in passenger satisfaction and is closely linked with travel time variability. While there is no commonly accepted definition of reliability ( Carrion and Levinson, 2012 ), it is generally used in the transport literature to describe the stability, certainty, and predictability of travel conditions ( Mattsson and Jenelius, 2015 ). Four specific measures for bus reliability are travel time variability, headway variability, passenger wait time variability, and punctuality ( Sorratini et al., 2008 ). From the passengers' point of view, the regularity is an important measure of reliability. Especially in the context of high-frequency services, passengers typically arrive randomly at stops without consulting the timetable ( Cats, 2014 ). Continuous statistical distributions are attractive because their statistical parameters are simple, and can describe elegantly the shape and extent of the travel time distribution, and reproduce it for prediction and control purposes.

Travel time variability regarding bus travel can be assessed using empirical standard deviations ( Abkowitz and Engelstein, 1983 ; Mazloumi et al., 2010 ). This feature cannot comprehensively represent the stochastic features of travel times, since some features (e.g., skewness and multimodality), are missing (see the example for cars in Van Lint et al., 2008 ). Therefore, Kieu et al. (2014) proposed two public transport-oriented definitions of travel time variability, one for a corridor-level and the second at a service-level. They suggest the use of lognormal distributions to calculate the proposed indicators of variability.

Components of Travel Times and Their Variability

This section defines the spatial and time components of a public transport route that are used in this review. The spatial definitions of a bus route are illustrated in Figure 2 . As shown, a bus travels from an origin terminal to a destination terminal passing through a set of bus stops along the way. The link between two consecutive stops is called a section. All sections from an origin to a destination form a route. A segment consists of several consecutive sections.

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Figure 2 . Spatial definitions of a bus line, and components of travel, dwell, and running times.

The aforementioned definitions of travel time are seen from an operator's perspective. From a passenger's perspective, the trip travel time also includes the access time, the waiting time, and the transferring time. Modeling all passenger trip time components is, however, beyond the scope of this review, which focuses only on the operational times. While travel time and its components are durations, points in time are also interesting from a passenger and an operational point of view, namely the arrival time and the departure time at bus stops. These times can be directly calculated by summing up travel time components if one point in time is given.

The operating times can be expressed on a section, segment, or route level. The temporal definitions used in this review are illustrated in Figure 2 . We take on this review only the view from the operator and neglect the view of passengers, which would also include access, egress, transfer time, route, and departure time choice. The travel time consists of the dwell time (the time a bus spends stopped at a planned stop) and the running time (the time a bus is not stopped at a stop). The travel time in a segment corresponds to the sum of the running and dwell times of a bus in the segment under consideration.

As shown in Figure 2 , the dwell time can be subdivided into four subprocess components: time spent on opening the doors, boarding, alighting, and other passenger activities such as fare payment, closing the doors, and synchronization time. Synchronization time is the time early-arriving vehicles spend waiting at control points to return to their schedule. Running time can be subdivided into two categories: the free-flow running time (i.e., the minimum time a bus needs to travel from one bus stop to the next) and the excess running time caused by traffic conditions or traffic signals along the route.

Dwell times are highly dependent on the number of passengers served. Additional factors influencing dwell time are the number of passengers on board, vehicle characteristics such as number of doors, passenger behavior, driver behavior, fare collection methods, and bus lift operations ( Dueker et al., 2004 ).

Running times, on the other hand, depend on the infrastructure configuration (e.g., dedicated bus lanes or signal priority) and traffic conditions. Additional factors influencing running time are driver behavior, weather, and schedule quality ( Abkowitz and Engelstein, 1983 ; Levison, 1983 ). All the factors influencing dwell time and running time are stochastic in nature and challenging to quantify (e.g., separating driver behavior from road traffic context).

As dwell times and running times are affected by different factors, it is meaningful to investigate them separately (see Wong and Khani, 2018 ). Mazloumi et al. (2010) explain this in a practical example: Early running buses wait at timing points until their predefined scheduled departure time. This leads to a longer left tail in the travel time distributions compared with the running time distribution. Hence, separating running and dwell times helps to understand the reasons for the variation of these random variables. Investigating travel time distributions has the advantage that the data does not have to be available at the running and dwell time level. Furthermore, passengers are specifically interested in segment-based travel times (the time they spend in the bus; Xue et al., 2011 ), whereas the split in dwell and running times is more of interest for operators.

In this study, we review papers that interpret the variability as a whole. The individual contribution of the influencing factors toward the distribution of travel times is not in the scope of this study. A variety of studies aim to identify and model these factors, as they are vital for describing and predicting travel times. Parametric statistical approaches (e.g., hazard-based approaches) are a comprehensive approach to model these factors ( Anastasopoulos et al., 2017 ).

Literature Analysis

Methodology.

This systematic literature review aims to report the state of the field of modeling statistical distributions to bus travel time and components of bus travel time observations. In this section, we report the steps of the systematic literature review.

To identify relevant publications, we performed a snowball search in the following databases: ScienceDirect, Web of Science, SCOPUS, and Google Scholar. We used multiple databases in order to identify a greater diversity of papers. A combination of keywords to state the model (keywords: “public transport” and “bus”), the considered observation (keywords: “travel time,” “running time,” and “dwell time”), and the model (keywords: “variability,” “reliability,” “distributions,” and “statistical modeling”) was used. Furthermore, common alterations of the keywords were considered, e.g., “transport” and “transportation.” This search for the chosen keywords was directed in the databases, if possible, for the title, abstract, and keywords. We considered snowballing by means of the reference list of the found papers. Since this topic is relevant for public transport systems throughout the world, consequently, a specific geographical constraint was not applied.

Any study included in this review needs to fulfill the following criteria: it needs to be a journal article or conference proceeding published in English that proposed statistical distributions based on empirical findings from case studies. The found papers were reviewed manually to strictly conform to the inclusion criteria.

To standardize the literature information, this paper adopts the matrix method to extract key factors, by which the publications differ or accord. In the following, we first look at the studies that investigate the travel time as a whole. In the subsequent sections, we look at the studies that investigate only the travel time components, i.e., the dwell time and the running time. Various statistical distributions are proposed. For the readers' convenience, we listed the proposed statistical distributions in Appendix A .

Travel Time

In an early work, Taylor (1982) manually collected travel time data from 15 successive daily home to work trips commencing at 8:15 a.m. each day. The research has intensified in the last decade as data has become easier to collect and analyze. Recent studies incorporate larger data sets to analyze and model bus operations: for example, Mazloumi et al. (2010) consider 3,351 bus runs and Dai et al. (2019) consider 2,932 bus runs.

While most data was originally collected by hand ( Taylor, 1982 ), today data can be collected using AVL and GPS systems, which provide precise time and position information for vehicles at stops or in-motion. Still, the massive amount of data is often reduced to the arrival and the departure time. Another method for collecting bus position information is transit signal priority data ( Kieu et al., 2014 ), which provide bus position information at certain locations.

Similar to car traffic, bus travel times vary over the course of a day. Typically, running and dwell times and the variations in these times are higher during peak periods. Given this variation, most of the studies reviewed for this research only consider one period or treat periods of the day separately ( Xue et al., 2011 ; Cats et al., 2014 ; Kieu et al., 2014 ; Durán-Hormazabal and Tirachini, 2016 ; Ma et al., 2016 ; Yan et al., 2016 ; Chen and Sun, 2017 ; Chepuri et al., 2018 ; Rahman et al., 2018 ; Wong and Khani, 2018 ). Some studies further aggregate travel time observations into departure time windows (DTW). Another approach is to only consider specific buses ( Taylor, 1982 ; Kieu et al., 2014 ). Out of all the considered studies, only Dai et al. (2019) aggregated all measured data for the entire day rather than focusing on a specific time period or DTW.

The distributions of travel times are investigated on different spatial aggregation levels. Whereas, Kieu et al. (2014) investigated section travel times, others such as Chen and Sun (2017) focus on segment travel times; other studies, e.g., Mazloumi et al. (2010) , consider route travel times.

The studies tested numerous different distributions for estimating travel time variability and made different recommendations for the best distribution. The normal, lognormal, or log-logistic were frequently proposed to be the best distribution for conventional distributions.

Recent automobile traffic research suggests the use of mixture distributions for fitting travel time distributions ( Guo et al., 2010 ; Susilawati et al., 2013 ). This is because Van Lint and Van Zuylen (2005) identified for private vehicles four phases (free-flow conditions, congestion onset, congestion, and congestion dissolve) that yielded distinctively different shapes of travel time variability. Ma et al. (2016) adapted this approach to bus travel times. They used Gaussian mixture models with up to three components, namely free-flow, recurrent, and non-recurrent traffic state for describing the probability distributions. Similarly, Chen and Sun (2017) fitted mixture distributions to travel time of buses to account for different traffic states. Their results showed that a four-component model worked best at representing the data during peak hours and a two-component model worked best during off-peak hours. However, it remained unclear how the observed components can be allocated to service states or phases.

Table 1 summarizes the key characteristics of the reviewed research. The characteristics shown are data collection method, study area, time of the day when the study was performed, DTW (departure time window, i.e., temporal aggregation level), and spatial aggregation (section, segment, and route). Finally, the proposed distributions and the number of component distributions are specified.

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Table 1 . Compilation of studies examining travel times.

Running Time

In contrast to automobile traffic, only a limited number of studies have been conducted to empirically understand which distributions are suitable for modeling bus running times. The first studies of automobile travel times proposed symmetrical continuous distributions, in particular the normal distribution, to characterize vehicle travel time on a link level. However, further research pointed out that travel time distributions are asymmetric and considerably right-skewed ( Richardson and Taylor, 1978 ). More recent research recommends using the lognormal distribution due to its good fit and the fact that it is characterized by only two parameters ( Xue et al., 2011 ). When buses share the same road space with automobiles and have similar characteristics (e.g., maximal speed), there is a strong similarity between automobile travel time and bus running time. Under these conditions, it is logical that the statistical modeling of both means of transport can also be similar. Therefore, most research recommends the use of lognormal distributions are proposed to statistically model running times of buses ( Uno et al., 2009 ; Mazloumi et al., 2010 ; Xue et al., 2011 ).

However, an early study by Jordan and Turnquist (1979) , which investigated the running time of buses in the morning peak in Chicago, USA, recommended using a shifted gamma distribution, where the shift equals the minimum motion time. Consequently, the gamma distribution essentially represents the delay, whereas the travel time consisted of a deterministic part (minimum motion time) and a stochastic part (excess running time). Table 2 summarizes the key characteristics of the reviewed publications considering bus running times.

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Table 2 . Compilation of studies examining running times.

Most of the literature models the dwell time as a function of the number of boarding and alighting passengers as well as the crowdedness ( Dueker et al., 2004 ; Tirachini, 2013 ). Some works include additional factors such as the bus type. Very few studies directly address modeling statistical distributions of bus dwell times. Jiang and Yang (2014) performed video analysis of three bus stops during peak hours in Shanghai, and proposed using an Erlang distribution to fit the data. Li et al. (2012) studied the dwell times in Changzhou, and proposed using a lognormal distribution. The lognormal distribution was also proposed by Rajbhandari et al. (2003) , who studied dwell times in New Jersey, USA. Koshy and Arasan (2005) proposed using a normal distribution for dwell time data from a curbside bus stop in Chennai City, India.

Khoo (2013) investigated the influence of various parameters on the dwell time distribution. They found that the Pearson 6 distributions yield the best fit to the data for both peak and off-peak conditions. They furthermore investigated the effects of platform crowding level and suggested Weibull distribution for less crowded situations and the Pearson 6 distribution for crowded situations. Rashidi and Ranjitkar (2013) investigated bus dwell time data collected by an AVL system in Auckland, New Zealand and concluded that the data is well-represented by the Wakeby distribution. In addition, they found that lognormal distribution performed satisfactorily while using a normal distribution was not suitable for estimating dwell time. Finally, Wong and Khani (2018) fitted different distributions to dwell times. However, they could not identify which yields the best fit using Cullen-Frey graphs. Table 3 summarizes the key characteristics of the reviewed publications considering bus dwell times.

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Table 3 . Compilation of studies examining dwell times.

Descriptive Analytics: Spatiotemporal Aggregation

Bus operations show remarkable variations over the course of a day, with typically larger running times and dwell times, as well as their variations, during peak conditions. Figure 3 shows a typical graph of the mean and the variation of the route travel time of buses over the course of a day. Hence, the distribution of travel time components depends on the temporal aggregation.

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Figure 3 . Typical changes of the mean (red) and the variation (gray) of route bus travel times over the course of the day.

Mazloumi et al. (2010) investigated the effects of the temporal aggregation of bus travel time data by aggregating the data into departure time windows (DTWs). This allowed them to examine the travel time distributions for DTWs at different times of the day. Their study showed that for narrow DTWs, travel time distributions are best characterized by normal distributions, while for wide DTWs, peak-hour travel times are well-represented by normal distributions, and off-peak travel times seem to follow lognormal distributions. Ma et al. (2016) also concluded that increasing the temporal aggregation of travel times tends to make distributions more asymmetric and to decrease the normality of distributions. In fact, as the temporal aggregation attribute is increased, some of the variability within the course of a day is incorporated into the day-to-day variability.

The spatial aggregation also significantly influences the distribution of travel time components. For example, Ma et al. (2016) investigated section level and route level distributions. They found that section level travel times are often multi-modal, whereas route travel times seem to be rather unimodal. In other words, the multi-modality of the distribution on a section level seems to be eliminated when the data is aggregated on a route level. This can be explained by the ability of the driver to speed up in successive sections to catch up with the timetable, especially if buses have the exclusive right of way, where drivers can precisely control speed.

Rahman et al. (2018) used a slightly different approach to show the influence of spatial aggregation. They worked with GPS-data and defined the term pseudo horizon as the distance from a GPS point to an upstream GPS point on the same route. Then, they analyzed the changes in bus travel time characteristics as the pseudo horizon varies. They found that at a range of 8 km (a boundary probably depending on the topology of the network and operations), there is a significant change in bus travel time characteristics. The travel time distributions of buses converge from a rightly skewed distribution to a more symmetrical distribution from shorter to longer pseudo horizons. They recommend lognormal distributions for pseudo-horizons of under 8 km and normal distributions for longer pseudo-horizons. Xue et al. (2011) contradict this finding by stating that the kurtosis and skewness of the distribution become larger for longer segments, but do not provide an explanation for this behavior. However, generally, with the exception of Xue et al. (2011) , studies agree that smaller spatial aggregation leads to higher skewness values and more variability.

The combination of section travel times into route travel time needs to incorporate correlation effects into successive sections. Spatial regression models have been used for this purpose for automobile traffic, e.g., Hackney et al. (2007) . Additionally, it is also interesting to study how the knowledge of a distribution is affected given some additional information over time (the approach in Corman and Kecman, 2018 ).

Figure 4 illustrates the spatial and temporal aggregation levels of the studies reviewed in this research, which target different spatial and time aggregations, and are therefore hard to compare directly. Studies that systematically investigate either the influence of the temporal or the spatial aggregation are shown with dotted borders. Since none of the studies investigated spatial and temporal aggregation at the same time, it is unknown which type of aggregation has greater effects on the travel time variability and choice of distribution model. Figure 4 furthermore points to a research gap in studying subsequent buses, i.e., descriptive analytics for studying bus bunching.

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Figure 4 . Representations of spatial and temporal aggregation of (A) travel time, (B) running time, and (C) dwell time.

When considering route travel times at peak period, Mazloumi et al. (2010) and Chepuri et al. (2018) propose normal distributions, while Ma et al. (2016) and Chen and Sun (2017) propose Gaussian mixture models, and Yan et al. (2016) propose lognormal distributions (see Figure 4A ). For short segments, skewed distributions such as log-logistic (Durán-Hormazabal and Tirachini, 2016 ) or lognormal ( Kieu et al., 2014 ; Rahman et al., 2018 ) perform better than normal distributions.

Figure 4B presents studies of running times. Only Uno et al. (2009) conducted analysis on different levels of spatial aggregation, demonstrating that the lognormal distribution always provides the best fit. Figure 4C presents studies of dwell times. In this case, however, no systematic study on the influence of temporal or spatial aggregation has been conducted.

The main reasons for investigating the different aggregation level come from the differing interests of passengers and operators. Passengers are interested in the time they will spend on a bus, hence, they are interested in segment travel times ( Xue et al., 2011 ). Operators, on the other side, are interested in all spatial aggregation levels. Also, concerning the temporal aggregation of travel times, the interest of operators and passengers differs. Passengers are mostly interested in short departure time windows to understand the travel times at a specific hour of the day (e.g., Yan et al., 2016 ) or even of a specific bus (e.g., Taylor, 1982 ). Operators might be interested, depending on their analyses, in period-level or single bus aggregation (e.g., Ma et al., 2016 ). Temporal aggregation on the level of whole days can be used as a benchmark for networkwide analyses (e.g., Durán-Hormazabal and Tirachini, 2016 ) and for reporting to funding agencies.

Descriptive Analytics: Type and Modality of Distributions

Unfortunately, it is not common for the authors to give clear reasons why they test and choose a certain statistical distribution. However, they often clearly state that the distributions can never perfectly model observed distributions, but only approximate them.

A first consideration in identifying an appropriate distribution is to decide whether the process being modeled has bounds. Intuitively, the running time, hence the travel time, has a minimal bound given by the maximum possible speed of a bus traveling from one stop to the next. Jordan and Turnquist (1979) applied this process by shifting their proposed (gamma) distribution by the free-flow travel time buses take to traverse the section. Dai et al. (2019) also applied shifted (lognormal) distributions. Similarly, there is the minimum bound for the dwell time, given as the process time, i.e., the time spent on opening and closing the doors. However, none of the publications investigating distributions of dwell times incorporated a lower bound greater than zero. This should suggest using right-skewed distributions rather than symmetrical (e.g., Gaussian) ones.

One of the most discussed considerations in finding a distribution is determining whether the distribution should be modeled as a single component distribution or a mixture distribution made up by multiple components. Recent publications suggest using mixture models, which are capable of linking the shape of the travel time distribution to the underlying travel time states. For example, Ma et al. (2016) modeled travel times with Gaussian mixture models, where the maximum number of components is set to be three. Unfortunately, it is not clear in this study how many components were actually used. In any case, such a model will always have an equal or even a better fit than a normal distribution. Chen and Sun (2017) further elaborated on the number of components used, related to the observed service states. However, their results showed that using threeå to four components for peak conditions and one to two components for off-peak conditions yield the best fit to the data. Finally, choosing mixture distributions does not answer the question on which particular distributions should be used.

Predictive Analytics: Distribution Fitting and Choice of Reproducible Distribution

A variety of different methodologies for identifying the most suitable statistical distribution to model the travel time and its components were used in the studies reviewed for this research. First, the parameters of candidate distributions have to be estimated. This is mostly done by the maximum likelihood estimation (MLE). Other parametric methods as method of moments or regression methods were not applied. After estimating these parameters, typical hypothesis tests such as Anderson-Darling ( Ma et al., 2016 ), Kolmogorov-Smirnov ( Mazloumi et al., 2010 ), or chi-squared ( Khoo, 2013 ) are used. These tests have different limitations.

A major limitation of Kolmogorov-Smirnov and Anderson-Darling tests is that they are not valid if parameters are directly estimated from the tested data. This limitation can be resolved by using a bootstrap procedure as (presented in Stute et al., 1993 , and used in Kieu et al., 2014 ). Another approach is to divide the measured data into a training set, used to estimate the parameters, and a testing set, which is used to test the fit.

A second major limitation is the interpretation of p-values. Here, the frequently used threshold of α = 5% is not well-founded since a conclusion is not automatically “true” on one side of the threshold nor “false” on the other side ( Wasserstein and Lazar, 2016 ). Furthermore, since the p-value depends on the sample size, it is often possible to find a tiny difference between two results that is statistically significant, but where there is no meaning in this difference within the empirical context. This is especially problematic for bus travel time studies where huge amounts of AVL or GPS data are available. Therefore, it is important to communicate the effect size ( Fritz et al., 2012 ). In other words, it is not most important if a candidate distribution is the “true” distribution, but rather how good the distribution is in giving information about the travel time components.

To evaluate which distribution yields the best fit to a given data, different fitted distributions can be cross-compared. This can be done by choosing the distribution that shows the highest likelihood. The problem with this approach is that distributions with more parameters generally yield a better fit in comparison to distributions with fewer parameters. To overcome this problem, various studies use the Bayesian Information Criterion (BIC) or the Akaike Information Criterion (AIC). Both criteria measure the relative quality of a fit and including the number of parameters used.

Prescriptive Analytics: Controlled Bus Operation

An important application of travel time distributions is in developing predictive applications for improving performance. This pertains to buses running to a fixed schedule, buses running on frequency, and to a certain extent also dial-a-ride services; as far as the vehicles will be relatively large, having multiple stops along their trajectory, and subject to pending traffic conditions, we expect the same dynamics will apply (e.g., Alonso-Mora et al., 2017 ). For example, variations in bus travel time can lead to variation in bus headway (e.g., Newell and Potts, 1964 ). This instability in headways can cause bus bunching since a lagging bus must collect more passengers, and therefore tends to fall further behind. Bus bunching can be analyzed by considering headway variability ( Moreira-Matias et al., 2012 ), bus-to-bus travel or dwell time variability (see Figure 1C ). Furthermore, correlations would provide meaningful insight in helping to identify potential bunching patterns. To detect and reduce bus bunching, information about subsequent buses are needed. Day-to-day travel time variability gives no direct information about bus bunching, as the reported variability might come from very high-frequency dynamics (with a period of two buses, i.e., bunching phenomena, which impacts regularity) or lower frequency dynamics (e.g., periods with high travel times followed by periods with low travel times, such as peaks).

Recommendations for Further Research

Understanding aggregation issues.

The spatial and temporal aggregation of travel time, running time, and dwell time data have an important influence on the statistics and the choice of most suitable probability distribution. However, only a few studies have systematically investigated the influence of spatial or temporal aggregation, nor the underlying correlations structures. Conducting such a study in multiple cities with the same methodology would add significant value to the current state-of-the-art and quantify the soft or hard boundaries between states or characteristics as dependent on city structure, supply, and operations. Moreover, peak and off-peak are commonly accepted boundaries, which in reality represent a continuous transition of states. Similarly, the time and space features by which the spatial correlation of travel times reduces and Gaussian assumption start to hold are mostly a continuous phenomenon. Most of those are probably related to spatial characteristics of city structures and operations. A rigorous determination of such space and time transitions would allow for a more meaningful description of traffic dynamics.

Putting Focus on the Tail of the Distribution

The evaluated studies model the distributions at normal conditions. There is no special focus put on the tail of the distributions, representing extraordinary behavior. But as even extraordinary behavior might reoccur, that could be modeled. It is imaginable that distributions and dependencies for extraordinary events could, and possibly should, be modeled differently as for normal operations.

Meaningful Choice of Distributions

Mixture distributions appear to better fit bus travel time data compared to conventional unimodal distributions, especially during peak hours. However, it is important to ensure that the components (i.e., service states) can be fully explained, which is currently not the case. From that perspective, distributions with simply more parameters would always achieve a better fit, but not necessarily result in physically meaningful parameters. Additionally, since the literature suggests that peak travel time components are skewed, skewed distributions such as a lognormal distribution should be considered in the modeling of mixture distributions.

Reproducible and Robust Distributions

In transport analysis, there is a growing interest in using available data for predictive purposes like supply planning, demand modeling, and timetable planning. Therefore, the proposed distributions should be reproducible with additional data. A good approach to determine the best statistical distribution should be based on a training-testing approach to avoid overfitting the data.

Incorporating Control Strategies

Bus operators use various control strategies to maintain service reliability (see Ibarra-Rojas et al., 2015 ). Most operators set control points along a bus route where bus departure times are subject to regulation, or have specific buffer time. Buses that arrive ahead of schedule wait at these control points to return to the schedule. This reduces the variation of the departure time at the control points. This strategy has a major effect on the day-to-day variation of travel time components, as the travel process at those points includes also a possible waiting time for the scheduled departure time. Further analysis should investigate the variation of bus travel times depending on the control strategy, buffers, and holding points.

Conclusions

This article reports on the modeling of bus running, dwell, and travel time. The importance of this topic is crucial not only for the short term improvement of existing public transport services, but even more for a future of increased automation and connectivity in vehicles. We assume in fact that future road-based transport systems, regardless of them having a driver or not, will have similar operating conditions to current buses. This means that they make use of large vehicles capable of bundling the demand efficiently. They will be partially subject to prevailing traffic dynamics and congestion related to peak hours and they will be stop-based, including multiple stops along a generalized circulation to allow people boarding and alighting. They will have a range of operations from pure schedule-based, to frequency-based, to partially demand-responsive, and they might have a varying degree of reserved infrastructure. To this end, we review the current practice for statistically modeling distributions of bus travel time components, pointing to new approaches and models needed for descriptive, predictive, and prescriptive analytics purposes in the context of bus operations.

Spatial and temporal aggregation of data have an important influence on the statistics as well as on the choice of the most appropriate probability distribution. A graphical summary of the findings from the previous section is reported in Figure 5 , addressing descriptive analytics (left; how to model what) and predictive and prescriptive analytics (right: how to ensure predictive power of future operations; and how to support decision making). This calls for flexible models, which can be used and adapted for different locations and times, are able to give the information on non-stationary probability distributions, and remain with good performance through expected and unexpected technological changes.

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Figure 5 . Graphical summary of results from the literature and recommended research Partlabels: (A) Descriptive Analytics, (B) Prescriptive/Predictive Analytics.

The parameters of the model should be relevant and should explain the conditions of the transport network; in addition, overfitting should be prevented. In general, simple models are better, as the required parameters are then easier to understand. Using mixture distributions for modeling distributions of the travel time and its components is a promising path. However, it is important that the components remain meaningful. The determination regarding the best-fitting distribution could be made using training and testing approaches. These approaches need to be further developed but are especially promising because the most important quality for a distribution is its reproducibility. It is important to seize the opportunity that the era of big data provides us in terms of transport analysis, since data is the best resource we get to approximate the truth.

Author Contributions

BB and FC contributed to the analysis of the current state of the research, discussion of results, and the writing of the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: public transport, travel time variability, travel time distribution, probability distribution, reliability

Citation: Büchel B and Corman F (2020) Review on Statistical Modeling of Travel Time Variability for Road-Based Public Transport. Front. Built Environ. 6:70. doi: 10.3389/fbuil.2020.00070

Received: 11 February 2020; Accepted: 27 April 2020; Published: 10 June 2020.

Reviewed by:

Copyright © 2020 Büchel and Corman. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Beda Büchel, beda.buechel@ivt.baug.ethz.ch

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Advances in Planning for Emerging Transportation Technologies: Towards Automation, Connectivity, and Electric Propulsion

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Modelling travel time distribution under various uncertainties on Hanshin expressway of Japan

  • Ravi Sekhar Chalumuri 1 &
  • Asakura Yasuo 2  

European Transport Research Review volume  6 ,  pages 85–92 ( 2014 ) Cite this article

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Metrics details

Studying travel time distribution or variability in travel time is very much useful in travel time reliability studies of transportation system. The properties of this distribution are described by various uncertainties which are derived from supply side, demand side and other external factors of any road network.

Present study investigates the development of stochastic response surface of travel time variation under uncertain factors of traffic volume and intensity of rain fall by using Stochastic Response Surface Method (SRSM).

Analysis and Results

This model was applied to a section of Kobe Route (Nishnomiya to Awaza), Hanshin expressway, Japan. Besides hourly traffic volume data, incident data for the entire year 2006 were collected and multiple linear regression analysis for entire year data was initially performed to know the functional and significance relation between the input and output variable. Further SRSM analysis has been carried out for working days data. Results shows that travel time distribution obtained using SRSM model is better than distribution obtained by the regression model.

It was observed from the results that SRSM model is efficient for analyzing the stochastic relation between the response variable and uncertain explanatory variables.

1 Introduction

Travel time distribution or variability in travel time is the most useful indicator to measure the performance and reliability of a transportation system. The properties of this distribution are described by various uncertainties which are derived from supply side, demand side and other external factors of a particular road network. Width in travel time distribution indicates higher uncertainties and lower travel time reliability. The measure of central tendencies of travel time distribution is unable to explain the traveler’s experience. Recently, various empirical travel time reliability studies [ 4 , 11 , 17 ] and Asakura [ 2 ] have extensively used travel time distribution as a tool for developing various reliability indices such as Planning Time (95 % travel time), Buffer Time Index and Planning Time Index [ 14 ]. All these reliability indices are useful to improve regional transportation planning [ 10 ].

When we intend to evaluate the effects of a transport policy on travel time reliability, it is necessary to identify the factors (source of uncertainty) that will affect travel time, and relation between the various sources of travel time. In this section various existing studies related [ 1 ] to sources of travel time variation are reviewed. Very few studies have concentrated on quantifying sources of uncertainties making travel time unreliable. Vander loop identified the main causes of unreliability of travel times for Netherlands urban roads. According to his study, 74 % of unreliability in the travel time is mainly due to internal factors of the traffic. The remaining is due to weather (8 %), road works (14 %), accidents (3to12 %) and combination factors (2 %) [ 18 ].

The US, Federal Highway Administration (FHWA) has identified seven sources of events which cause travel time variation. Further they have categorized into three main events such as traffic influence events (includes traffic incidents, work zones and weather), traffic demand events (includes fluctuations in normal traffic and special events) and physical highways features (includes traffic control devices and bottle necks) [ 3 ]. Ruimin [ 16 ], examined travel time variability under the influence of time of day, day of week, weather effect and traffic accident. In that study, the author quantified sources of travel time parameters with the help of multiple linear regressions with two way interaction models. In another study, Florida department of Transportation (Florida DOT) developed empirical travel time variability models such as function of frequency of incidents, work zones and weather conditions. For this, they have considered regression analysis on combination of different scenarios of uncertainty sources [ 5 ]. Asakura [ 2 ] further categorized the sources of travel time fluctuations in to three factors which are from demand side such as day to day traffic variation, supply side such as road closure due to accidents and external factors such as adverse weather effects and natural disaster. Most of the studies in the literature used deterministic approach to model travel time variation under the influence of various factors from supply side and demand side of the system. Travel Time variation on Hanshin expressway, Kobe route is mainly due to traffic volume, traffic accidents and amount of Rainfall.

The present study is an attempt to model the travel time distribution under various uncertainties. For this, Stochastic Response Surface Method (SRSM) has been adopted. SRSM [ 6 ] is an extension of classical Response Surface Method (RSM) to systems with stochastic inputs and outputs. The motivation for considering this model over the traditional Multiple Linear Regression (MLR) and other deterministic approaches is that both of these models fail to map stochastic behavior between response variable and explanatory variable in the system of uncertainty of travel time variation. In particular, two continuous probabilistic random factors were considered in this paper, one is traffic volume and the other is intensity of rain fall.

Archived continuous supersonic vehicle detectors data of Kobe Route on Hanshin expressway network in Japan were considered in this study. Travel time has been estimated for the study corridor by considering time slice method. Traffic incident data was collected for the same study period to model the travel time variation under various uncertainties. The same data has been considered to develop the traditional statistical model such as regression model and stochastic models. The comparative evaluation was made between these modeling approaches.

2 Study area and data collection

2.1 study area.

Data used in this study were collected from a section of Kobe Route, Route Number 3 of Hanshin expressway, Japan. Kobe route extends between Kobe and Osaka city and the length of this route is about 30 km. For the present study a section of the route from Nishinomiya IC to Awaza a total length 14.9 km data was considered for modeling travel time distribution. Figure  1 represents the study area and Kobe Route of Hanshin Expressway.

Study Area of Kobe Route, Route No.3 of Hanshin Expressway

2.2 Data collection

Supersonic vehicle detectors are installed on Hanshin expressway at every 500 m for observing the traffic volume and time occupancy ratio. For this study archived continuous supersonic vehicular detectors data for every 5 min intervals were collected for the entire year 2006. The section travel time at every 500 mts is estimated. After that path travel time for the study area was estimated by considering the time slice method, this has been explained in the next section. This travel time is influenced by various incidents occurred in the study area during the study period. The iincident data of this study area such as traffic accident data, road works, vehicle break down, road cleaning and other traffic related incident data has been collected from Hanshin Expressway Corporation Ltd. for the entire Year 2006. The number of incidents occurred on the study area for the year 2006 is presented in Fig. 2 . From Fig. 2 , it can be observed that the traffic accidents, vehicle breakdown and road inspection were comparatively more on the study area. Rain fall (mm/hr) data was collected from the official website of the Japan Meteorological Agency (JMA) [ 8 ]. Hourly rain fall data of Osaka was considered for the present study area.

Numbers of Incidents Occurred on Kobe Route for the Year 2006

Route travel time estimation by Time slice method

3 Travel time estimation

From vehicle detector data, spot speed is estimated for every 500 m interval and corresponding travel time for the same sections are calibrated by transforming the speed data of the section. Furthermore, path travel time for the study area (14.9 km) is estimated by using the time slice method, which considers the variation of speed over the time by constructing the vehicle trajectory. Travel time obtained from this method is sufficiently close to the actual travel time [ 2 ]. Conventionally the travel time of an entire route is calculated simply by accumulating the travel times of each section at a given time. It is expressed in Eq. ( 1 )

where t i ( s ) denotes the travel time of section “i” at a given time “s”.

Small sections (500 mts interval) in a route are numbered sequentially towards the downstream direction. This method generates an instantaneous travel time based on the assumption that vehicles instantaneously traverse the route. When traffic condition is stable and travel speed is constant, the travel time can be calculated correctly through this instantaneous method. However, the estimated travel time may not be accurate when traffic flows are unstable. The alternative method of calculating the route travel time is the Time slice method, with which the travel times of each section are accumulated successively with the delay of the section travel time. The route travel time is represented as T(s) and this was explained in Fig.  3 . The estimation of travel time by considering time slice method is expressed in Eq. ( 2 ). Yoshimura and Suga compared two sets of travel time estimated by the instantaneous method and the time slice method using Automatic Vehicle Identification (AVI) data as true values. They found that the instantaneous method caused large errors at both increased and decreased hours of traffic congestion and the time slice method could follow actual travel time fluctuation without delay[ 19 ]. The time slice method is more suitable for offline application rather than online application when the speed varies over time [ 9 ] and also provides better results over the instantaneous method. This path travel time is considered as a dependent variable for travel time distribution modeling.

where τ i ( s )denotes the travel time from section 1to section i-1 and written as

Travel time distribution of the study area for the entire year (sample size 8760) were plotted and presented in Fig.  4 . The probability and cumulative distribution is a visual tool representation of travel time variability over the period. The minimum and maximum travel time for this 14.9 km section varies between 498 and 4,383 s respectively. The mean travel time of the study area is 760 s and the standard deviation of travel time is 346 s.

Travel Time distributions of Study Area

4 Modeling travel time distribution

4.1 multiple linear regression analysis.

Multiple Linear Regression (MLR) analysis is carried to understand the influence of all the incidents on travel time variation. Further to understand the behavior of travel time variation on working days MLR analysis was carried out separately. The estimated MLR model coefficients for the entire year data was shown in Table  1 . The basic test of any model estimation is examination of the sign of model coefficients. From Table  1 it can be observed that the sign of estimated coefficients of all the variables are positive for the entire year of data. This indicates that all the incidents have positive contribution towards the travel time variation and this is more logical since travel time increase with occurrence of incidents. Based on t-static values it can be conclude that traffic volume, traffic accidents, vehicle break down, road cleaning, rain fall and other incidents are significant variables contributing to the travel time variation ( t stat value is greater than the critical value of 1.64 at 5 % level of significance). The higher F value (371.53) and corresponding low probability value ( p  < 0.05) of this model indicates that the model is significant. The corresponding R 2 value explains the 24 % of the total variation.

Similarly MLR analysis carried out for working days data (249 days) to understand the effect of incidents on travel time variation during these days. From observation of t stat values of traffic volume, traffic accident, rain fall and other incidents on working days have high magnitude of significance in travel time variation (t stat value is greater than the critical value of 1.64 at 5 % level of significance). Road works and road cleaning incidents generally taken place on non working days except during emergency cases on Hanshin Expressway. Analysis of Variance (ANOVA) of this model having high F value (275.20) and with very low probability value ( p  < 0.005) demonstrates a very high significance for the regression model. The goodness of fit of the model R 2 value indicates that 25 % of the total variation is explained by this model. From Table  1 , it was concluded that the traffic volume, traffic accidents and rain fall incidents are highly significant for travel time variation on this section of Kobe Route.

Further from these parameters, the continuous random attributes such as traffic volume and rain fall effect was considered in probabilistic analysis for modeling the stochastic behavior of travel time distribution. For this Stochastic Response Surface Method (SRSM) was considered to model the travel time distribution under the uncertainties. The limitation of this model is that, it considers the continuous random variables in the modeling. Before carrying out the SRSM analysis MLR analysis has been carried out for the continuous random variable data considered in the SRSM analysis and the results were presented in Table 2 .

Nonlinear regression analysis was also carried out for developing the relation among the Travel Time , Traffic Volume and Rin Fall Intensity parameter. The functional form of the nonlinear models is given in Eq. ( 3 ). The model coefficients estimated for individual data of study area are presented in Table  3 . Non linear model was found better than the linear model based on R2 value is 0.226 for non linear model whereas this value is 0.22 for linear models As in the case of non linear analysis

Where TV Traffic Volume (veh/hr) and RF is Rain fall intensity (mm/hr)and β 0 to a β 5

Further these model coefficients were considered for estimating travel time for the collocation points (Table  6 ) generated for 2nd order polynomial equations. The following sections discuss the SRSM analysis for working days data.

4.2 Stochastic response surface method

Probabilistic analysis is most widely used method for characterizing uncertainty in physical and social systems, especially when estimates of the probability distributions of uncertain parameters are available. These models can describe uncertainty arising from stochastic disturbances, variability conditions and risk consideration. The main process of probabilistic models comprises of probability encoding of inputs and propagation of uncertainties through models. Probability encoding of inputs involves the determination of the probabilistic distribution of the input parameter and incorporation of random variation. This is accomplished by using statistical estimation technique involves estimating probability distribution from available field data. Figures  5 illustrate the concept of uncertainty propagation of travel time. In this each point of the response surface (calculated output value of travel time) of the model, change in traffic volume and rain fall will be characterized by probability density function (PDF) of these inputs. The methodology for adopting this approach was discussed in the next section.

Schematic Representation of Propagation of Travel Time Uncertainty

Evaluation of SRSM: Probability distribution and Cumulative Probability Distribution

4.2.1 Methodology

Stochastic Response Surface Method (SRSM) [ 6 , 7 ] is an extension to the classical deterministic response surface method (RSM). RSM is a collection of mathematical and statistical techniques that are useful for the modeling and analysis of problems in which response of interest is influenced by several variables [ 13 ]. RSM also quantifies relationship among the measured responses and the input factor. The main difference between RSM and SRSM is the way the input parameter are supplied. SRSM is one of the ideal conventional sampling based method for uncertainty analysis and this is accomplished by approximating both inputs and outputs of the uncertain system through stochastic series of well-behaved standard random variable ( srv ). The series expansion of the outputs contains coefficients that can be calculated from the results of limited number of model simulations.

The srv’s are selected from a set of independent, identically distributed (iid) unit random variable ( i = 1, 2….n). Where “n” is the number of independent inputs and each ξ i having a zero mean and unit variance. The following steps are involved in the application of the SRSM to the uncertain analysis of a model with random inputs and random outputs.

Step1 Representation of stochastic model inputs: For each uncertain input, corresponding srv is assigned and the input random variable is expressed in terms of the srv. If the input random variables are mutually independent, the uncertainty in the i-th input variable X i is expressed as a function of the srv.

Step2 Functional approximation of model output: Each model output is expressed as a series of expansion in terms of srv as a multidimensional hermit polynomial with unknown coefficients. A Second order polynomial approximation is generally recommended in the literature. Also this approximation can be refined further using higher order terms depending on the accuracy needs. In this study second order polynomial function with two independent variable ξ 1 andξ 2 were considered and the mathematical expression was presented at Eq. ( 5 )

Step 3: Estimation of unknown coefficients in functional approximation: The unknown coefficients in Eq. ( 2 ) are estimated by equating model outputs with the corresponding polynomial expansions at a set of possible collocation points. Preferably next higher order of functional approximation routes to be considered for the generation of collocation points [ 15 ].

Step 4: Calculation of the statistical properties of model outputs: The model outputs are estimated followed by the estimation coefficients. The statistical properties of the outputs such as probability density function, moments of “y” can be readily calculated. This can be accomplished by generating large number of the srv s and the calculation of the values of inputs and the outputs from the transformation of Eqs. ( 4 ) and ( 5 )

4.2.2 SRSM analysis

Out of 5675 working days sample data, 4180 sample data were incidents free. During this time period travel time varies only due to fluctuation in traffic volume and effect of rain fall. SRSM method was applied to model the travel time distribution due to the effect of these continuous random variable. Table  4 shows the uncertainty ranges of model parameters and sampling strategy considered for transforming the uncertain variables for SRSM model. Also the statistic parameters of input and response variable are presented in Table  4 . Goodness of fit test between the observed frequencies (from data ) and the theoretical fitted frequencies was done by considering Kolmogorov-Smirnov test and Chi-square goodness of fit for the two fitted distributions such as lognormal distribution for travel time and exponential distribution for rain fall. The results indicates that at the 5 % level of significance the decision is to reject the null hypothesis, indicates that no difference between empirical and theoretical cumulative distributions. Therefore lognormal distribution and exponential distribution was considered for generating the random data for traffic volume and rainfall respectively in SRSM model.

Second order SRSM model was considered to approximate the response of travel time (Eq.  5 ). In order to solve for the second order polynomial expansion, the roots of the third order hermit polynomial, + 3 , − √ 3 and zero are used. The points are selected such that each srv takes the value of either 0 or one of the roots of the polynomial. Therefore there are nine possible collocation points they are 0 0 , √ 3 , 0 , 0 , √ 3 , − √ 3 , 0 , 0 , − √ 3 , 3 , − √ 3 , − √ 3 , √ 3 , √ 3 , √ 3 and − 3 , − √ 3 .

Set of model input points for traffic volume and rainfall at the points were generated by using transformation technique and presented at Table 5 . For lognormal distribution exp( μ  +  σξ 1 ) and for exponential distribution − 1 λ log 1 2 + 1 2 erf ξ 2 2 was considered [ 6 ] where erf is a error function.

The unknown coefficients in Eq.  5 considered for SRSM model are solved by using singular value decomposition method and the corresponding coefficients (a0, a1…a5) are presented at Table 6 . The Eq.  5 is well fit for the points which were considered in Table  5 . The highest R 2 value (0.99) of this model indicates that this model is significant for the 2nd order polynomial equation. The highest t-statistic values of this model indicate that coefficients of linear term, quadratic term and interaction term is significant.

Once the coefficients are estimated the travel time distribution can be fully described by random generation of a large number of samples. In this study the 4180 random samples (same size original data) are generated for SRSM analysis. All this procedure was implemented in MATLAB environment [ 12 ]. Travel Time estimated by SRSM model and MLR models are compared against with actual travel time is presented at Fig.  6 . From this figure it can be observed that SRSM probability distribution is uni-modal (having one maximum at 625 sec), asymmetrical and similarly follows the actual travel time distribution. Whereas travel time distribution obtained by MLR models are bimodal frequency curves having two peaks, one maximum at 625 s and the other maximum at 925 s. Even travel time distribution estimated by MLR model by considering all the uncertainty parameters (Table  1 ) also follows bimodal frequency. From this it can be conclude that the MLR models are overestimating beyond the average actual travel time (783 s). It can also be concluded from the Fig.  5 that even if more uncertainty parameters are considered for modeling travel time MLR models are unable to follow the actual travel time distribution. Further, from travel time distribution it can be observed that travel time obtained by SRSM model is well distributed between travel time, 542 s to 2,302 s. Whereas MLR models estimated travel time distribution varies between 493 to 1,363 s. From this it can be concluded that MLR models are unable to map the worst case scenarios, this we can observe from tails of the probability distribution of travel time (Fig.  5 ).

From the above discussion of results, it was observed that SRSM models are capable to analyze the stochastic behavior of uncertain variable and also these models are performs better than the conventional regression model to model travel time distribution. The algebraic expressions in terms of standard random variable (srv) are smooth and continuous could efficiently model the tails of the probability distributions of the outputs (Fig.  5 ). This explains that the SRSM models are capable to model the worst case scenarios. Further the observable difference between the estimated distribution of SRSM model and actual distribution can be improved by increasing the number of uncertainty parameters in the model.

To validate the distributions obtained by both the models, chi-square non-parametric statistical goodness of fit have been carried out between actual travel time considered as observed frequency and travel time estimated by SRSM and MLR model considered as expected frequency and 30 s travel time intervals have been considered for frequency estimation. From the results it can be concluded that MLR models have higher estimated chi-square value (6415) than the SRSM models (2182). This emphasizes that MLR models have grater discrepancy between actual distribution and estimated distribution than SRSM model.

5 Conclusions

Travel time distribution is the most useful indicator to measure performance of any transportation system and properties of this distribution was influenced by various uncertainties which are derived from supply side, demand side and other external factors of any transportation system. Regression analysis between travel time and various uncertain parameters were considered to develop the functional relationship among them. From the t-statistic value it was observed that the effect of traffic volume, traffic accidents and amount of rain fall influence is quite significant on Hanshin Expressway study area. Further, SRSM models were applied in this study to resolve a probabilistic analysis. The uncertain parameters considered in this analysis are traffic volume and rain fall intensity for modeling travel time distribution. The travel time distribution obtained by SRSM model was compared with regression models and observed that SRSM model is better than the regression model and also following the actual travel time distribution. Further the difference between the estimated distribution by SRSM model and actual distribution may be improved by increasing the number of uncertainty parameters in the model.

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Acknowledgments

The authors would like to express their great appreciation to the Han-Shin Expressway Public Corporation for providing the data and other required information. Authors are also thankful to the Dr. Purnima Parida, Head, Transportation Planning division and S. Gangopadhyay, Director of CRRI to publish this paper.

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Chalumuri, R.S., Yasuo, A. Modelling travel time distribution under various uncertainties on Hanshin expressway of Japan. Eur. Transp. Res. Rev. 6 , 85–92 (2014). https://doi.org/10.1007/s12544-013-0111-3

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Data analytics approach for travel time reliability pattern analysis and prediction

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Travel time reliability (TTR) is an important measure which has been widely used to represent the traffic conditions on freeways. The objective of this study is to develop a systematic approach to analyzing TTR on roadway segments along a corridor. A case study is conducted to illustrate the TTR patterns using vehicle probe data collected on a freeway corridor in Charlotte, North Carolina. A number of influential factors are considered when analyzing TTR, which include, but are not limited to, time of day, day of week, year, and segment location. A time series model is developed and used to predict the TTR. Numerical results clearly indicate the uniqueness of TTR patterns under each case and under different days of week and weather conditions. The research results can provide insightful and objective information on the traffic conditions along freeway segments, and the developed data-driven models can be used to objectively predict the future TTRs, and thus to help transportation planners make informed decisions.

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1 Introduction

Travel time reliability (TTR) is an important measure which has been widely used to represent the traffic conditions on freeways. Accurately modeling TTR is very important to both transportation agencies and road users. Different definitions of TTR have been developed in different studies. The Federal Highway Administration (FHWA) [ 1 ] gave a formal definition of TTR, i.e., the consistency or dependability in travel times, as measured from day-to-day and/or across different times of the day. In a Strategic Highway Research Program 2 (SHRP 2) project conducted by Vandervalk et al. [ 2 ], TTR was defined as the variability in travel times that occur on a facility or for a trip over the course of time and the number of times (trips) that either “fail” or “succeed” in accordance with a predetermined performance standard or schedule. Table  1 provides a summary of existing TTR definitions in chronological order.

Different types of measures have also been widely applied to assess traffic performances and reliability in recent years. There are four most widely used TTR measures recommended by FHWA and they are 90th/95th percentile travel times, buffer index (BI), planning time index (PTI), and frequency of congestion (FOC). Other important measures include the standard deviation [ 11 ], coefficient of variation (COV) [ 12 ], present variation [ 5 ], skew and width of travel time distribution [ 13 ], and misery index [ 5 ]. Table  2 provides a summary of the TTR measures that were developed and used in past studies in alphabetical order.

Nowadays, anonymous vehicle probe data have been greatly improved in both data coverage and data fidelity, and thus have become a reliable source for freeway TTR analysis. However, in most cases, TTR data are analyzed in the short term, which may not be able to account for the TTR variability characteristics for all the segments of a corridor in the long term. Therefore, the detailed analysis of the TTR patterns in different cases which can represent different traffic conditions will be helpful.

The purpose of this study is to develop a systematic approach to illustrating how TTR distributes and varies with respect to the time of day (TOD), day of week (DOW), year, and weather. Case studies are conducted to present different TTR patterns under different conditions. The analysis of TTR conducted and the prediction model developed can greatly help decision makers plan, design, operate, and manage a more efficient highway system.

2 Literature review

Basically, TTR can be analyzed using travel time distribution data, including both single mode distribution [ 16 ] and multimode distribution [ 17 , 18 ]. Eliasson [ 19 ] investigated the relationship between travel time distribution and different TOD periods. The result showed that travel times are approximately normally distributed under severe congestion conditions. However, the travel time distribution was skewed under low levels of congestion conditions. Emam and Ai-Deek [ 16 ] employed the Anderson–Darling (AD) goodness-of-fit statistics and error percentages to evaluate the performances of different distribution types. The results indicated that the lognormal distribution provided the best model fit and that it was more efficient to use the same day of the week (e.g., Mondays) in the estimation of TTR on a roadway segment than to use mixed data (i.e., data collected across multiple weekdays), because of the significant differences between traffic patterns across multiple weekdays. In addition, the researchers also noticed that the new reliability estimation method showed higher sensitivity to geographical locations. Chen et al. [ 18 ] explored the travel time distribution and variability patterns for different road types in different time periods. The authors compared the goodness-of-fit results of several distribution types and analyzed TTR patterns with BI and COV.

However, to investigate the impact of nonrecurring congestion on TTR, different sources of travel time variability (including traffic incidents, inclement weather, and work zones) were studied by different researchers around the world. With the consideration of incident, Hojati et al. [ 20 ] developed a method to quantify the impact of traffic incidents on TTR on freeways. The authors conducted a Tobit regression analysis to identify and quantify factors that affect extra buffer time index based on the Queensland DOT and STREAMS Incident Management System (SIMS) database. The results indicated that changes in TTR as a result of traffic incidents are related to the characteristics of the incidents. Charlotte et al. [ 21 ] modeled TTR on urban roads in the year 2010 in the region of Paris, France. The 90th percentile of the travel time distribution was modeled with linear regression using explanatory variables including the number of lanes, mean value of the travel time distribution, travel direction, time of the day, number of accidents, and roadworks. With the consideration of weather condition, Martchouk et al. [ 22 ] studied the travel time variability using the travel time data on freeway segments in Indianapolis collected with the help of anonymous Bluetooth sampling techniques. The effects of adverse weather were discussed in the study. The results showed that the travel time increased during adverse weather period, and the variance in travel times during the same time period also increased. Li et al. [ 23 ] conducted a study which focused on studying the weather impact on traffic operations. Different rainfall intensity data in 72 sample days in Florida were incorporated into the TTR model along with the historical speed database. Different scenarios for each hour (under clear weather, light rain, and heavy rain conditions) were created and applied to the respective roadway segments. The results showed that speed reductions on arterials were 10% for light rain and 12% for heavy rain. Kamga and Yazici [ 24 ] conducted a study via merging taxi trips’ GPS records and historical weather records of New York City from 2009 to 2010 and then calculated the descriptive statistics of travel time under different TOD, DOW and various weather conditions. The authors used the classification and regression trees (C&RT) model to extract the travel time distribution information under each DOW-TOD-Weather category. With the analysis results of COV, the authors pointed out inclement weather may not only result in increased travel time but also result in higher TTR. With the consideration of multiple influencing factors, Tu [ 25 ] developed a TTR model with the consideration of four influencing factors including road geometry, adverse weather, speed limits, and traffic accidents. The model was validated using traffic data from urban freeways in Netherlands in the year 2005. The results of road geometry impacts indicated that there was a threshold value L for the length of ramp/weaving section. If the actual length was less than L , the TTR would decrease with the decreasing length of ramp/weaving sections. If the actual length was larger than L , the length would have far less impact on TTR. Javid and Javid [ 26 ] developed a framework to estimate travel time variability caused by traffic incidents based on integrated traffic, road geometry, incident, and weather data. A series of robust regression models were developed using the data from a stretch in California’s highway system in 2 years. The results of the split-sample validation showed the effectiveness of the proposed models in estimating the travel time variability. In conclusion, for incidents occurring on weekends, the highway clearance time would be shorter. Shoulder existence and lane width would adversely impact downstream highway clearance time. Kwon et al. [ 27 ] developed a linear regression model to study the TTR with the consideration of factors under three categories: traffic influencing events (traffic incidents and crashes, work zone activity, weather, and environmental conditions), traffic demand (fluctuations in day-to-day demand and special events), and physical road features (traffic control devices and inadequate base capacity). The model was tested using the data collected from San Francisco Bay Area in the year 2008 and used to identify how each variable contributes to the TTR. The results of this study provided useful insights into predicting the TTR. Schroeder et al. [ 28 ] presented a methodology to conduct the freeway reliability analysis based on freeway data in North Carolina. The variability impact considerations included TOD, DOW, month-of-year differences, and various nonrecurring congestion sources (such as weather, incidents, work zones, and special events). The freeway scenario generator (FSG) was used and resulted in 2508 scenarios based on freeway facility data in North Carolina. The resulting travel time distribution was presented, and a sensitivity analysis was conducted to explore the relationship between weather and incidents and the overall reliability of the facility. Table  3 provides a summary of several previous studies on TTR analysis and modeling in chronological order.

Although useful, most of the studies did not consider the long-term variation of TTR. This study focuses on the long-term TTR analysis step-by-step in order to present data analysis results. The developed data-driven models can be used to objectively predict the future TTRs. This study is novel since it identifies the segments with various TTR patterns at different locations of a corridor, which can help reveal the travel time characteristics of these locations under different traffic conditions. The research results can provide insightful and objective information on the traffic conditions along freeway segments to help the transit planners make informed decisions.

3 Descriptive analysis of TTR

3.1 data description.

This study focuses on the travel time data gathered from the Regional Integrated Transportation Information System (RITIS) web site and uses the collected data to conduct the TTR analysis. A series of major freeway segments are selected for the case study: Interstate 77 (I-77) southbound (Fig.  1 ) is one of the most heavily traveled Interstate highways in Charlotte area and runs from north to south. A total of 32 roadway segments of I-77 southbound are selected in this study, which start from the intersection with Harris Oak Blvd and end at the interchange with I-485 (Exit 2) at the south part of the city. The total length of the selected segments is 19 miles. In this study, travel time and speed data are obtained from the RITIS web site which gathered information about roadway speeds and vehicle counts from 300 million real-time anonymous mobile phones, connected cars, trucks, delivery vans, and other fleet vehicles equipped with GPS locator devices. Travel time data from January 2011 to December 2015 aggregated at 15-min intervals are used in the present study. A sample of raw travel time data utilized in this study is shown in Table  4 , which contains the following information:

figure 1

Selected I-77 southbound segments

TMC_Code The RITIS Probe Data Analytics Suite uses the TMC (traffic message channel) standard to uniquely identify each road segment. This field indicates the segment ID.

Measurement_tstamp This field indicates the time stamp of the record.

Speed This field indicates the current estimated harmonic mean speed for the roadway segment in miles per hour.

Travel_time_seconds This field indicates the time it takes to drive along the roadway segment.

3.2 Study location identification

The historical weather data aggregated at 1-h intervals near the Charlotte Douglas International airport can be found at the www.wunderground.com web site. From previous studies, it is widely accepted that only severe weather events can cause a significant impact on speeds and travel times. Due to the weather characteristics in the Charlotte area and the distribution of each weather category, detailed weather conditions are further classified into three groups including normal, rain, and snow/fog/ice. Table  5 presents the detailed classification of the weather conditions. Conditions such as overcast or mostly cloudy are assumed to be no different from clear conditions due to no obvious impact on traffic conditions. These conditions are classified into normal. All the conditions such as rain or thunderstorm are categorized as rain. In order to ensure the acceptable sample size, snow, fog, ice pellet, and other similar conditions are combined together due to their rate of occurrence.

There are four most widely used TTR measures in previous studies and they are BI, PTI, COV, and FOC. However, BI and COV have a limitation since their values depend on the average travel time, which may change over time [ 30 ]. Therefore, the PTI is chosen and used as the primary measure of TTR in this study. It is calculated as the 95th percentile travel time divided by the free-flow travel time so as to represent the percentage of extra travel time that most people will need to add on to their trip in order to ensure on-time arrival. For example, a PTI value of 1.5 at 5 p.m. means that for a 20-minute trip in light traffic, 30 min should be planned at 5 p.m. to make sure that he or she is on time. The equation of PTI is provided below:

where \({\text{PTI}}_{i}\) is the planning time index of segment I ; \(T_{i95}\) is the 95th percentile travel time on the TMC segment i during the study period across multiple days (e.g., a month) or a year; \({\text{FFTT}}_{i}\) is free-flow travel time on TMC i during the same observation period as mentioned above.

For each roadway segment, the free-flow travel time is computed by dividing the length of segment by the free-flow speed, which is defined as the 85th percentile speed during overnight hours (10 p.m. to 5 a.m.) [ 10 , 30 , 31 ].

The first step to identify the study segments is to plot the two-dimensional PTI matrix for each road segment along the corridor. This would provide a straightforward and visualized tool for decision makers to grasp the average traffic conditions along a corridor. The long-term (in one-year period) PTI values of each segment from 2011 to 2015 were calculated and shown in Fig.  2 . Note that in this figure, the horizontal axis denotes the time of day and the vertical axis represents TMC segments along the selected section on I-77 southbound. Each cell represents the PTI value. The darker the color, the higher the PTI.

figure 2

PTI heat maps of I-77 segments in 5 years

In order to select the sections which can represent different traffic conditions, the qualitative ratings for each freeway segment in the study area are conducted and further classified into different categories/levels based on the qualitative criteria of a previous study [ 32 ]. The ratings, based on the PTI values, are given as: (1) reliable (PTI < 1.5); (2) moderately to heavily unreliable (1.5 < PTI < 2.5); and (3) extremely unreliable (PTI > 2.5).

Based on the rating criteria mentioned above, eight segments (shown in Fig.  3 ) which contain four PTI rating cases are selected as the sample study segments. The four cases are

figure 3

Location of selected I-77 segments

Case 1 (p.m. peak only): The average PTI during a.m. peak period is reliable and during PM peak period is unreliable/extremely unreliable. The selected segments are 125-04779 and 125N04780.

Case 2 (a.m. peak only): The average PTI during a.m. peak period is unreliable/extremely unreliable and during PM peak period is reliable. The selected segments are 125N04788 and 125-04788.

Case 3 (Double peak): The average PTI during both a.m. and p.m. peak periods are unreliable/extremely unreliable. The selected segments are 125N04784 and 125N04785.

Case 4 (No peak): The average PTI during both a.m. and p.m. peak periods are reliable. The selected segments are 125-04790 and 125N04791.

3.3 TTR variability patterns at different study locations

3.3.1 ttr variability pattern under all conditions.

The PTIs of each segment from 2011 to 2015 are shown in Fig.  4 . The TTR variability pattern in each case can be categorized as follows:

figure 4

TTR variability pattern of each segment in 5 years

Case 1 These two sections are located at the south part of the Charlotte downtown area. The volume of outbound traffic during p.m. hours is high and therefore contributes to the frequent congestion under p.m. peak condition. In more detail, in the year 2015, these two segments had obviously higher PTI values during peak hours than those in the years of 2011–2014. The condition like this may be attributed to different factors such as the traffic volume, weather condition, and accidents. One potential reason behind this could be the traffic volume on the segments of case 1 from 2011 to 2015 (annual average daily traffic (AADT): 15,300, 15,200, 15,400, 15,900, and 15,900, respectively). The correlation values between the AADT and average daily PTIs of these two segments are 0.86 and 0.83, respectively, which means that they are highly correlated. Therefore, the traffic volume may be a primary reason of the TTR distribution characteristics. Based on the historical weather data, the frequency of adverse weather in the year 2015 is higher than that in the years from 2011 to 2014. In order to eliminate the possible influence of adverse weather, the TTR distributions under only normal conditions during each year are also tested and the average daily PTI of 2015 is reduced a little bit (from 2.1 to 2.0) but is still higher than those of years 2011–2014. With respect to traffic accident, no detailed historical crash information about I-77 is found. However, the number of total crashes in Mecklenburg county in each year had been increasing from 2011 to 2015 (15,476, 15,915, 16,790, 19,847, and 21,096, respectively) [ 33 ]. This can also be another potential reason that contributes to the worsening of the traffic condition in the year 2015.

Case 2 These two sections are located at the north part of the Charlotte downtown area. The volume of inbound traffic during a.m. hours is high and therefore contributes to the frequent congestion under a.m. peak condition. Similar to case 1, in the year 2015, these two segments had obviously higher PTI values during peak hours than those of years from 2011 to 2014. The condition like this may also be explained by the potential factors such as traffic volume (with the correlation values being 0.83 and 0.89, respectively), adverse weather and accident that contribute to the worsening of the traffic condition in the year 2015, as presented in case 1.

Case 3 These two sections are located adjacent to Charlotte downtown area. The volume of inbound traffic during a.m. hours and outbound traffic during p.m. hours are both high and therefore contributes to the frequent congestion under double peak conditions. Similar to cases 1 and 2, in the year 2015, these two segments had obviously higher PTI values during peak hours than those in the years of 2011–2014. However, the correlation values between traffic volume and average daily PTIs are not statistically significant (i.e., 0.56 and 0.71, respectively). Therefore, the condition like this may be explained by other potential factors (such as adverse weather and accident) that contribute to the worsening of the traffic condition in the year 2015.

Case 4 These two sections are located far away from Charlotte downtown area. The traffic volumes during both AM and PM hours are low and therefore contribute to the no peak condition. The variation of PTIs throughout the day of each year does not change significantly (from 1.02 to 1.13, and 1.04 to 1.15, respectively).

3.3.2 TTR variability patterns for different days of week

The PTIs of each segment from Monday to Sunday are shown in Fig.  5 , and the average PTIs are shown in Table  6 . The TTR variability patterns for different days of week in each case can be categorized as follows:

figure 5

TTR variability pattern of each segment for different days of week in 5 years

Case 1 For the segments showing the p.m. peak characteristics, the travel time on Friday is least reliable. This result is consistent with a previous study [ 29 ]. The TTR variability patterns on weekends are significantly different from weekdays. There are no PM peak characteristics of the TTR of these two segments on weekends as the PTIs throughout the day do not change significantly. The maximum (and average) PTIs on weekends of these two segments are 1.28 (1.10) and 1.24 (1.08), respectively. The results indicate that traffic congestion on weekends becomes less frequent and also travel demand on weekends is perhaps much lower than that on weekdays, which is consistent with previous studies [ 18 , 34 ].

Case 2 For the segments showing the a.m. peak characteristics, the travel time on Tuesday is the least reliable. Similar to case 1, there are also no a.m. peak characteristics of the TTR of these two segments on weekends as the PTIs throughout the day do not change significantly.

Case 3 For the segments showing the double peak characteristics, the travel time on Friday is least reliable. Similar to case 1, there are also no a.m. peak characteristics of the TTR of these two segments on weekends as the PTIs throughout the day do not change significantly.

Case 4 For the segments showing no peak characteristics, average PTIs of each DOW do not change significantly (from 1.05 to 1.07 and 1.07 to 1.09, respectively). The results indicate that the traffic congestions on these two segments are not frequent on both weekdays and weekends.

3.3.3 TTR variability patterns under different weather conditions

The PTIs of each segment under different weather conditions are shown in Fig.  6 . The TTR variability patterns in each case under different weather conditions can be categorized as follows:

figure 6

TTR variability pattern of each segment under different weather conditions

Case 1 The TTR variability patterns of these two segments under normal and rain conditions are similar, and the pattern is unique under the snow/ice/fog condition. In more detail, the PTIs under rain condition have obviously higher values than those under normal condition throughout the day. This probably suggests that rain can cause several travel problems (such as visibility issues) while driving a vehicle. Heavy rainfall may lead to hydroplaning, slippery surfaces for tires, and road flooding. Therefore, the values of PTIs under rain condition also increase and the traffic congestion becomes more frequent. This result is consistent with other studies [ 23 , 35 ]. The PTIs under snow/ice/fog conditions are also higher than those under normal condition throughout the day because of the influence of road surfaces and visibility problems [ 36 ]. Results clearly show that snow/fog/ice can contribute to unexpected condition on the roadway anytime throughout the day, resulting in the unique TTR variability pattern under the snow/fog/ice conditions. This result is also consistent with a previous study [ 37 ]. In particular, there is an extremely high PTI value at noon. Since the geometric design characteristics of all the segments are similar, the potential reason behind this unique pattern could be the nonrecurrent condition such as the incidents happened during snow condition on the case segments. This hypothesis should be checked in the future if more detailed data are available.

Case 2 and Case 3 Similar to case 1, the PTIs under rain condition have obviously higher values than those under normal condition throughout the day. PTIs under the snow/ice/fog condition are also higher than the PTIs under normal condition throughout the day and demonstrate unique variability pattern.

Case 4 For segments 125-04790 and 125N04791, the PTIs under rain condition have higher values than those under normal condition but do not increase significantly. However, the PTIs under the snow/ice/fog conditions are much higher than the PTIs under normal condition throughout the day. This result shows the adverse weather (such as snow, fog, and ice) can significantly affect the traffic condition of the segment, and the traffic congestion becomes more frequent no matter when.

4 TTR prediction

4.1 time series-based ttr prediction methodology.

Exponential smoothing (ETS) model is a commonly used method in time series analysis and has been widely adopted in traffic forecasting for decades. The ETS model is an intuitive forecasting method that weights the observed time series unequally [ 38 ]. Recent observations are weighted more heavily than remote observations. The ETS equation [ 39 ] is shown as follows:

where \(S_{t}\) is the output of the exponential smoothing algorithm; α is smoothing factor, 0 ≤ α ≤ 1; \(x_{t}\) is the raw data sequence.

Based on the historical travel time data, the PTIs from Monday to Sunday in each year and the PTIs of each month can be calculated. Those values can be used as the input to the exponential smoothing model. The ETS model is utilized in this study to predict the PTIs from Monday to Sunday and the PTIs in each month in the year 2015. Note that the TTRs under different weather conditions are not predicted due to its limited sample size in a single year. The PTIs from Monday to Sunday and the PTIs in each month from 2011 to 2014 are used as the raw input in the model to predict the PTIs in 2015. In order to select the best smoothing factor α, the grid search method is adopted in this study (with an accuracy level of 0.1). The comparison results indicate that α with the value of 1 can provide the best prediction result. Therefore, the smoothing factor α with the value of 1 is utilized in this study to minimum prediction error.

The mean absolute percentage error (MAPE) is used in this study to evaluate the prediction results. The MAPE equation is shown as follows:

where M is the absolute percentage error; n is the number of predicted points; \(A_{t}\) is the actual TTR value; \(F_{t}\) is the predicted TTR value.

4.2 TTR prediction results

4.2.1 ttr prediction results considering dow.

Table  7 shows the MAPE of prediction results from Monday to Sunday. The result shows that for the segments showing PM peak characteristics, the prediction model can provide reliable prediction results for each DOW with the MAPEs being 8.42% and 8.18%, respectively. The prediction model can provide most reliable prediction results on Monday with the MAPEs being 6.54% and 5.72%, respectively. For the segments showing AM peak characteristics, the prediction model can provide reliable prediction results for each DOW with the MAPEs being 9.38% and 7.91%, respectively. The prediction model can provide most reliable prediction results on Monday with the MAPEs being 8.07% and 7.55%, respectively. For the segments showing double peak characteristics, the prediction model can provide prediction results for each DOW with the MAPEs being 17.33% and 21.21%, respectively. The prediction model can provide most reliable prediction results on Monday with the MAPEs being 11.52% and 18.06%, respectively. For the segments showing no peak characteristics, the prediction model can provide prediction results for each DOW with the MAPEs being 2.83% and 2.73%, respectively. The prediction model can provide most reliable prediction results on Sunday with the MAPEs being 2.20% and 2.24%, respectively.

4.2.2 TTR prediction results considering MOY

Table  8 shows the average prediction error from January to December. The result shows that for the segments showing PM peak characteristics, the prediction model can provide reliable prediction results with the MAPEs being 7.68% and 8.83%, respectively. The prediction model can provide most reliable prediction results on April with the MAPEs being 3.90% and 7.03%, respectively. For the segments showing AM peak characteristics, the prediction model can provide reliable prediction results with the MAPEs being 8.07% and 7.29%, respectively. The prediction model can provide most reliable prediction results on June with the MAPEs 5.67% and 4.99%, respectively. For the segments showing double peak characteristics, the prediction model can provide prediction results with the MAPEs being 16.10% and 17.94%, respectively. The prediction model can provide most reliable prediction results on May with the MAPEs being 13.03% and 13.44%, respectively. For the segments showing no peak characteristics, the prediction model can provide reliable prediction results with the MAPEs being 2.77% and 3.15%, respectively. The prediction model can provide most reliable prediction results on March with the MAPEs being 1.85% and 2.15%, respectively.

5 Conclusion

With the analysis of the TTR of eight typical segments on the I-77 southbound corridor in Charlotte, NC, the TTR variability patterns could be identified under different conditions. Based on the historical TTR data (2011–2014), the TTR for specific DOW and the TTR of each month in the year 2015 are also predicted. The information gathered out of this study can be concluded as follows.

In general, the TTR variability patterns of different segments along the corridor are different. Different cases including PM peak only, AM peak only, double peak and no peak are analyzed separately since they demonstrate different results. The TTR prediction result also indicates that the TTR of a year could be predicted accurately based on the long-term historical TTR data.

With respect to DOW, the TTR analysis results show that for the segments with noticeable peak hour trend, the TTRs on weekends are much lower than those on weekdays. The TTR prediction results also show that the prediction errors on weekends are lower than those on weekdays. For the segments with no peak hour, the TTRs on weekends are similar to those on weekdays. The TTR prediction results show that the prediction errors on weekends are a little higher than those on weekdays. In particular, for the segments under cases 1 and 3 (PM peak only and double peak, respectively), the TTR on Friday is the highest. For the segments under case 2 (AM peak only), the TTR on Tuesday is the highest. For the segments under case 4 (no peak hour), the TTR on each DOW does not significantly change.

With respect to weather conditions, the TTR analysis results show that the PTIs under rain condition have obviously higher values than those under normal condition throughout the day. The PTIs under snow/ice/fog conditions are also higher than those under normal condition throughout the day with unique variability patterns.

In most cases, TTR data are analyzed at the segment level in the short term, which may not be able to account for the TTR variability characteristics for the whole section in the long term. This study aims to develop a systematic approach to analyzing TTR of roadway segments with different variability patterns along a corridor in the long term. However, with the limited amount of data, the impacts of accidents and roadworks on TTR are not discussed in this study. In the future, the impacts of these variables will be studied when the data can be made available. Spatial relationships between segments along the corridor and their impacts on the TTR will also be investigated. Furthermore, the TTR analysis will be conducted at a network level and relevant characteristics will be examined in detail.

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Acknowledgements

The authors want to express their deepest gratitude to the financial support by the United States Department of Transportation, University Transportation Center through the Center for Advanced Multimodal Mobility Solutions and Education (CAMMSE) at The University of North Carolina at Charlotte (Grant Number: 69A3551747133).

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Chen, Z., Fan, W. Data analytics approach for travel time reliability pattern analysis and prediction. J. Mod. Transport. 27 , 250–265 (2019). https://doi.org/10.1007/s40534-019-00195-6

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Canadian prime minister Justin Trudeau has joined those to express their concern about Israel's planned assault on the southern Gaza city of Rafah.

Mr Trudeau was speaking with Israeli war cabinet member Benny Gantz on Monday.

A statement from Mr Trudeau's office said he had "shared his concern" around the planned offensive "and the severe humanitarian implications for all civilians taking refuge in the area".

"He underscored the need to increase the volume of life-saving humanitarian aid for civilians and to ensure aid reaches all those in need, safely and without delay."

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Israeli prime minister Benjamin Netanyahu said during the weekend that the assault was still necessary to "eliminate the remaining terrorist battalions in Rafah".

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Technology continues to make travel easier for millions of people around the globe. New apps offer travelers fresh ways to research a destination, connect with fellow globetrotters, communicate with locals, and track their flights. As a full-time digital nomad , I’ve learned the importance of having the right tools at your disposal when on the road.

Over the past few years, I’ve visited dozens of countries, flown hundreds of thousands of miles, and tested countless apps to find the ones that prove to be the most helpful. And while tools like Meetup, Google Maps, Google Translate, currency converters, and vacation rental , airline, and hotel–specific apps are in fact indispensable on most trips, I find that many of the best travel apps are still undiscovered by my fellow travelers.

Below, seven travel apps that make life on the go markedly easier (and more fun), from real-time flight trackers to photo-sharing gems—all of these apps are free, but some have paid pro versions worth considering. Don’t think these apps are only for full-time travelers—they prove useful for every type of traveler, whether you're in a brand-new destination or back for a return visit.

Discover the best apps for travelers:

Apps for flying, apps for exploring, apps for photo sharing.

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My favorite app on this list is Flighty, a must-have for any frequent flier . The free version is fantastic, but I recommend splurging for the premium version, which gives you even more useful information and comes at a reasonable cost of $48 per year. The functionality it provides more than justifies the price.

The free version allows you to input all future flights plus past journeys you've taken within the last year. It provides updated information if any schedule changes occur to upcoming flights and allows you to track all the miles you’ve flown in one place.

The other main benefit is that it allows you to share your flight information with friends and family members who are also on Flighty. They will receive real-time updates about your flight without you having to text them. My parents love to make sure I’m safe so they appreciate getting detailed information about my travels—even while I’m in the air.

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Flightradar24 is a favorite app among fellow aviation geeks. It lets you track almost any aircraft you see in the air throughout the day or night. You can track your own flight, a friend's flight, or any plane you see in real-time. It shares plenty of cool data, like the altitude, speed, arrival and departure airports and times, and tail number.

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There are tons of eSIM apps on the market, but Airalo is probably my favorite, with digital SIM cards available for dozens of countries. I find it to be the most reliable and offers the best data packages. It can be a little pricey, but it is the best option if you’re only taking short trips. Holafly is another eSIM app that offers unlimited data and works well, but I find the app to be much more glitchy than Airalo and would recommend using the desktop version.

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This one is for the planner of the group. Wanderlog merges the best of a bunch of different apps into one. Imagine an app that lets you plan a trip with ease: collaborate with friends as if you’re on Google Docs, manage expense tracking like Splitwise, and have a central place for your reservations, checklists, and more. Tripit is a similar travel planner app, so you can download both and get a feel for which interface you like better.

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Been allows you to track the countries you’ve visited and make a wish list of places you can’t wait to cross off your travel bucket list . It gives you information like the percentage of the globe you’ve been to and shows you a world map of your trips. It’s always fun to add in a new country whenever I reach my destination.

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This app was introduced to me by fellow nomads I met at a cafe in Montevideo, Uruguay . It automatically tracks your route across the world, allowing you to quickly upload pictures throughout your travels that can be viewed by anyone who has access to your profile. It’s an excellent way to share important moments with loved ones without needing to send photos to each person individually. Users can also create easily accessible guides filled with up-to-date information to help you plan your own trip.

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Steller is another app that a friend recently introduced me to, and I’ve quickly fallen in love with it. It uses a combination of AI and user experiences to help you plan trips around the globe. From recommendations of things to see, like the Recoleta Cemetery in Buenos Aires or the Royal Palace of Naples, to full guidebooks created by users, you can get inspired, book experiences, and discover hidden gems within this app.

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We are thrilled to announce the launch of Journey by Mediavine , a new ad management solution for sites starting at around 10,000 monthly sessions.

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IMAGES

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  3. Frequency distribution of journey time for S1 and S2.

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  4. Distribution of total journey distance and time for the participants

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  5. The Travel Time Distribution is the Basis for Defining Reliability

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  6. Distribution of patient journey times.

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  4. 🇬🇧🇺🇸JUNE'S JOURNEY TIME RUSH SCENE SHIFT 💥 30JAN24💥🇬🇧🇺🇸

  5. June's Journey TIME RUSH COMPETITION, 25/26 December 2023 updates

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COMMENTS

  1. Analyzing travel time distribution based on different travel time reliability patterns using probe vehicle data

    Travel time distribution (TTD) has been widely used to represent the traffic conditions on freeways and help to analyze travel time reliability (TTR). The goal of this study is to develop a systematic approach to analyzing TTD on different types of roadway segments along a corridor.

  2. Exploring Travel Time Distribution and Variability Patterns ...

    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.

  3. Learn Travel Time Distribution with Graph Deep Learning and Generative

    Instead of a determined value, the travel time within a future time period is a distribution. Besides, they all use grid structure data to obtain the spatial dependency, which does not reflect the traffic network's actual topology.

  4. Learning Travel Time Distributions with Deep Generative Model

    In this paper, we develop a deep generative model - DeepGTT - to learn the travel time distribution for any route by conditioning on the real-time traffic. DeepGTT interprets the generation of travel time using a three-layer hierarchical probabilistic model. In the first layer, we present two techniques, amortization and spatial smoothness ...

  5. Travel Time Variability—The Case of Two Public Modes

    This paper describes some results obtained from a study of travel time reliability for a work trip using two alternative and directly comparable public transport modes, bus and underground rail (metro), in Paris, France. Data were collected to permit the study of the distributions of variations in daily travel times, relationships between ...

  6. Estimation of lane-level travel time distributions under a connected

    Travel time distribution estimation is fundamentally important for the evaluation of travel time variability and reliability.

  7. Forecasting journey time distribution with consideration to abnormal

    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.

  8. Learning Travel Time Distributions with Deep Generative Model

    A deep generative model to learn the travel time distribution for any route by conditioning on the real-time traffic, which produces substantially better results than state-of-the-art alternatives in two tasks: travel time estimation and route recovery from sparse trajectory data. Travel time estimation of a given route with respect to real-time traffic condition is extremely useful for many ...

  9. Frontiers

    The observed travel time of road-based public transport vehicles, as well as its components (i.e., dwell time and running time), are subject to variability, caused by the stochastic nature of various factors, including traffic congestion.

  10. Modelling travel time distribution under various uncertainties on

    Studying travel time distribution or variability in travel time is very much useful in travel time reliability studies of transportation system. The properties of this distribution are described by various uncertainties which are derived from supply side, demand side and other external factors of any road network. Present study investigates the development of stochastic response surface of ...

  11. Modeling distributions of travel time variability for bus operations

    Travel time distribution fitting is the preliminary preparation for reliability analysis. The GMM model can provide much detailed travel time information for both management agencies and individual travelers. For agencies, the GMM model provides a flexible and superior distribution fitting than its alternatives, which enables accurate and ...

  12. Forecasting journey time distribution with consideration to abnormal

    Journey time reliability analysis is then conducted using the skewness of dynamic journey time distribution. An empirical study is carried out by fusing the bus transit date of No. 261 bus route ...

  13. Characterizing corridor-level travel time distributions based on

    The distribution of the estimated route-level travel times for each 5-min interval, calculated over the three time periods with available data (4:00-4:15 pm, 5:00-5:15 pm, and 5:15-5:30 pm) are plotted in Figure 13 with the ground truth route-level travel time obtained directly from the NGSIM data. As can be observed, the distribution of ...

  14. (PDF) Analyzing distributions for travel time data collected using

    First, travel time data are shown to have a complex statistical structure: the travel time distribution is in general peaky, multi-modal, and skewed to the right, which cross validates findings ...

  15. Complete Estimation Approach for Characterizing Passenger Travel Time

    Compared to the on-train time, travel times at stations, including walking time and waiting time, have been receiving less attention and therefore become more difficult to analyze. A common method to analyze the travel time at a rail transit station is to directly assume a distribution function and to further fit the distribution.

  16. How Random Incidents Affect Travel-Time Distributions

    How Random Incidents Affect Travel-Time Distributions Abstract: We present a novel analytical model to approximate the travel-time distribution of vehicles traversing a freeway corridor that experiences random quality of service degradations due to non-recurrent incidents.

  17. Modeling distributions of travel time variability for bus operations

    Bus travel time reliability performance influences service attractiveness, operating costs, and system efficiency. Better understanding of the distribution of travel time variability is a prerequisite for reliability analysis. A wide array of empirical studies has been conducted to model distribution of travel times in transport.

  18. Journey time distributions

    The distribution of journey times in the synthetic data is shown in Figure 6. As expected, the journey times for passengers during the peak are longer than in the off-peak because of the ...

  19. Rail Transit Travel Time Reliability and Estimation of Passenger Route

    With this information, the journey time distribution of any path can be established, and methods were proposed for inferring route choice proportions. After data preparation, the methods were applied to two typical origins and destinations from the Beijing Metro. Key values concerning travel time reliability, such as PET, PET-Trans, travelers ...

  20. Data analytics approach for travel time reliability pattern analysis

    Basically, TTR can be analyzed using travel time distribution data, including both single mode distribution and multimode distribution [17, 18]. Eliasson investigated the relationship between travel time distribution and different TOD periods. The result showed that travel times are approximately normally distributed under severe congestion ...

  21. Four Keys To Understanding Travel And Hospitality In 2024

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    Many properties offer deals in January, following the holiday travel surge. The Best Time To Visit Mexico's Cities. Depending on which city you want to visit, there's a best time of year for ...

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    2. Hollywood Road, Hong Kong: This busy street in Hong Kong is number two on Time Out's list. Time Out spotlights the street's fun vibe and Tate Dining Room, a Michelin-starred restaurant.

  25. Israel-Hamas war latest: Hamas number three killed, US says

    Israel raided the hospital for the second time during the war, accusing Hamas of using it as a base. Israel said it had killed more than 20 gunmen in the operation.

  26. 7 Best Travel Apps Worth Downloading Before Your Next Trip

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  27. Introducing Journey, A New Mediavine Ad Management Solution for Growing

    What is Journey? Journey is an entirely new ad offering designed and built from the ground up as a solution for smaller publishers. Powered by Mediavine's first-in-class ad technology, Journey by Mediavine will offer publishers with smaller websites full access to all of our first-party data and identity solutions. The beauty of Journey is that it is a completely self-service offering.