## 2 Literature Review  
Research related to travel time discrepancy and variability in literature usually addresses four main questions:  
* What are the usual travel time distributions? * Which indicators are the most suitable for describing travel time discrepancy and variability, and what trends do they show by time or distance? * What are the optimal data aggregations for observing the discrepancy and variability? * Which predictors are most significant for travel time? Previous statistical analyses of travel time by periods during a day frequently showed a normal distribution [15], and some also showed positive skewness [16], often with very long upper tails [17]. Different distributions were used to describe travel time during different periods during a day: gamma-distribution for the morning commuter period [18], normal distribution for any period [19], log-normal distribution for any period [20], and log-logistic distribution for any period [21]. [PERSON] et al. [4] compared commuter and other periods and showed that normal distribution is better for describing travel time in commuter periods, while log-normal distribution is better for other periods. [PERSON] et al. [22] tested five different types of distributions for different distances and concludedthat log-normal distribution is mostly suited for distances less than 7 km, and for longer distances, normal distribution becomes dominant. [PERSON] et al. [23] reported the highest accuracy of the results for a generalized extreme value distribution. When travel time variability is described as a statistical indicator, the usually used ones are the absolute ones in minutes and the relative ones in percentages [24]. The absolute ones are as follows:  
* Standard deviation \(\sigma\)--a usual indicator to describe the entire sample width;
* Difference between 90 th and 10 th percentile (\(T_{90}-T_{10}\)) to describe the entire width of the sample; in same fashion as standard deviation;
* Difference between the median and 10 th percentile to describe early vehicles;
* Difference between 90 th percentile and the median to describe late vehicles. The relative ones are as follows:  
* Coefficient of variation (standard deviation over the average): a usual indicator to describe the entire sample width;
* Difference between 90 th and 10 th percentile divided by the median (\(\Delta_{90/10}\)): to describe sample width, in same fashion as coefficient of variation. A typical travel time distribution implies an increase with the median travel time, average travel time or distance travelled, such as in Figure 2. Studies that used standard deviation for travel time variability frequently showed that it increases linearly with the average travel time [25]. [PERSON] et al. [26] showed over several days that, with the average travel time, \(\Delta_{90/10}\) is inverse or constant, and \(T_{90}-T_{10}\) has a linear trend. And although [PERSON] et al. [4] showed that both \(\sigma\) and \(T_{90}-T_{10}\) increase linearly with the average travel time, standard deviation showed a better fit. On the other hand, the research by [PERSON] et al. [27] concluded that the distribution function and the median are more precise as indicators than the arithmetic mean. According to [28], there are three types of data aggregations when observing travel time variability: on the same line in the same period, on the same line in different periods, and on the same lines in same periods in different days. In general, travel time variability can be observed from different spatial and temporal data aggregations, where  
Figure 2: An example of transit travel time distribution by traveled distance [1]. the combination of space and time in data aggregations usually provides better results. The research by [PERSON] et al. [29] confirmed that day-of-the-week and segment scale aggregations provided the best insights into the data.