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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.