Synthesizing Route Travel Time Distributions from Segment Travel Time Distributions

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Transportation Research Record: Journal of the Transportation Research Board, No. 2396, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 71–81.DOI: 10.3141/2396-09

I. K. Isukapati, G. F. List, and B. M. Williams, Department of Civil, Construction, and Environmental Engineering, North Carolina State University, 208 Mann Hall, 2501 Stinson Drive, Campus Box 7908, Raleigh, NC 27695-7908. A. F. Karr, National Institute of Statistical Sciences, 19 T. W. Alexander Drive, Research Triangle Park, NC 27709-4006. Corresponding author: I. K. Isukapati, [email protected].

(4). Follow-on efforts focused on finding ways to improve reliability (5) and mitigate congestion (6). Margiotta et al. developed a guide to effective freeway performance measurement and created analyti-cal procedures for determining the impacts of reliability mitigation strategies (7, 8).

Vehicle-based sensors in the form of automatic vehicle identifica-tion (AVI) and automatic vehicle location (AVL) systems have made it possible to collect data on individual trips. Although AVI technol-ogy is as old, if not older than AVL, it has only recently been used to monitor travel times. The use of AVI technology gained popularity when toll tags were put into service by agencies such as the New York State Thruway and companies such as Mark IV (9). As with AVL, AVI probes make it possible to monitor point-to-point travel times on both freeways and arterials. Li et al. used AVI data to gain insight into travel time variability and its causes (10). Wasson et al. were first to suggest using media access control (MAC) identification (ID) matches to estimate travel times (11). The advent of Bluetooth sens-ing technology and the increasing density of MAC IDs in the traffic system made this approach more attractive.

The question is how to generate travel time probability density functions (TT PDFs) at the route level on the basis of these AVI and AVL data. If the number of observations for a given origin–destination pair is large, the PDF can be developed directly. If the observations are too sparse, it ought to be possible to construct route-related travel times and travel time distributions on the basis of segment-level obser-vations. But, to do so, correlations between the travel times have to be taken into account. In general, if X and Y are independent random variables, then the distribution h(Z) for Z = X + Y can be constructed by convolving f(X) with g(Y). However, if X and Y are correlated, then convolution cannot be used; a different technique must be employed. This paper shows that those correlations are significant.

Therefore, this study explores an idea for synthesizing route TT PDFs that takes into account the possible correlations between adjacent segments. This paper presents findings from 3 months of Bluetooth data collected on I-5 in Sacramento, California.

Prior Work

The ADVANCE project in Chicago, Illinois, was one of the first to use probe data to study travel times (12, 13). Vehicles equipped with dead-reckoning equipment recorded time stamps at specific locations (Evanston and O’Hare airport) as they traveled across northeast Chicago. These travel times were then used to help motorists deter-mine the best paths to use when traversing this largely arterial-based urban network.

Synthesizing Route Travel Time Distributions from Segment Travel Time Distributions

Isaac Kumar Isukapati, George F. List, Billy M. Williams, and Alan F. Karr

This paper examines a way to synthesize route travel time probability density functions (PDFs) on the basis of segment-level PDFs. Real-world data from I-5 in Sacramento, California, are employed. The first find-ing is that careful filtering is required to extract useful travel times from the raw data because trip times, not travel times, are observed (i.e., the movement of vehicles between locations). The second finding is that significant correlations exist between individual vehicle travel times for adjacent segments. Two analyses are done in this regard: one predicts downstream travel times on the basis of upstream travel times, and the second checks for correlations in travel times between upstream and downstream segments. The results of these analyses suggest that strong positive correlations exist. The third finding is that comonotonicity, or perfect positive dependence, can be assumed when route travel time PDFs are generated from segment PDFs. Kolmogorov–Smirnov tests show that travel times synthesized from the segment-specific data are statistically different only under highly congested conditions, and even then, the percentage differences in the distributions of the synthesized and actual travel times are small. The fourth finding, somewhat tangential, is that there is little variation in individual driver travel times under given operating conditions. This is an important finding, because such an assumption serves as the basis for all traffic simulation models.

Travel times are an important aspect of transportation systems (1–3). Travelers and shippers want to know how long it will take to get from point A to point B and whether the trip will be completed on time. Users want to select routes that have reliable travel times and, if possible, depart when the likelihood of arriving on time is highest. In addition, system operators want to minimize variations in travel times at the segment and route level. The sources of variability can be traveler behavior or influencing factors such as capacity reduc-tions, traffic control devices, weather, incidents, work zones, special events, and demand fluctuations.

The recent studies of travel time reliability have produced signifi-cant new findings. Seminal work in this regard was conducted by Lomax et al., who studied possible travel time reliability measures

72 Transportation Research Record 2396

Morris et al. reported that trucking companies use AVL technologies to manage their fleets (14). Information about truck movements is retrieved in real time; the dispatcher switches load assignments and reroutes the trucks to maximize quality of service while minimizing the impacts of congestion. Transit operators use AVL technology to track buses, which improves their ability to keep the buses on sched-ule, to alert riders to bus arrival times and locations, and to make it easier for the system to meet rider demands. Quiroga and Bullock developed a methodology for conducting travel time studies using Global Positioning System (GPS) equipment and created guidelines for determining appropriate sample sizes (15, 16).

List et al. used a peer-to-peer system of AVL probes to collect real-time travel times (17). Two hundred probes were deployed in upstate New York. The main focus was on journey-to-work trips for people traveling to a cluster of businesses in a single area. The experiment demonstrated that time data from such probes would help drivers determine the best routes to use through congested and incident-affected networks. The findings from the experiment suggested that probe data could be used to create real-time route guidance systems (17–22).

There has also been experimentation with using transit vehicles and taxis as AVL probes. Chakroborty and Kikuchi developed a model for predicting travel times for automobiles using transit vehicles as probes (23). Hall and Nilesh compared automobile and transit vehicle trajectories to explore alternative methods for detecting congestion on arterials (24). Bertini et al demonstrated that buses can be used as probes to evaluate traffic conditions on arterials (25). Pan et al. indicated that taxis have been used to collect traffic data in Japan, Germany, and Malaysia (26). Ma and Koutsopoulos indicated that in Berlin the taxis served as floating cars and their travel time observations were processed into traffic information that other services then offered to clients (27).

Methodologically, Hellinga and Fu developed insights into reduc-ing the bias of link travel time estimates from probe data (28). Cetin et al. suggested factors that affect the minimum number of probes required for reliable average travel times (29). Yamamoto et al. devel-oped ways to analyze the variability of travel time estimates using probe vehicle data (30). Byon et al. developed GISTT [an integrated GPS–geographic information system (GIS) for travel time surveys] (31). This system enables one to match GPS data with spatial map features using GIS for monitoring traffic conditions on specific links. Dion and Rakha developed ways to construct average travel times using AVL-based sampling (32). Pan et al. proposed a GPS-based methodology for collecting historical travel time data that includes link travel time and information on intersection signal delay for an arterial (26). Furthermore, they developed a posttrip map-matching algorithm to project GPS data onto an arterial network. Berkow et al. developed techniques for constructing the shape of the congested regime in time and space along an urban arterial, combining signal system detectors and buses as probe vehicles (33). Feng et al. explored the ability of GPS-equipped probe vehicles to provide characterizations of arterial travel times and real-time traffic conditions (34).

Researchers also explored statistical ways to represent the distribu-tions of observed travel times. Van Lint et al. suggested using variance as a measure to portray travel time unreliability on freeways (35). Jintanakul et al. experimented with using Bayesian mixture models to estimate freeway travel time distributions on the basis of small samples of probe vehicles on multiple days (36). Guo et al. experi-mented with using a multistate (multimode) model to portray the distributions of travel times being observed (37). Susilawati et al. have experimented with the use of multimodal Burr density functions

(38). This work shows promise and is intriguing because the analytical functions are simple and the density function allows for skew.

The most recent efforts continue to advance these lines of investi-gation. Mahmassani et al. explored ideas about robust relationships for reliability analysis (39). Barkley et al., as a result of involvement in SHRP 2 Project L02, explored the use of multistate models to develop relationships between travel time reliability and nonrecurrent events (40). Ernst et al. developed sampling guidelines for using probe vehicles to characterize arterial travel times (41). Ramezani and Geroliminis explored the use of Markov chains to estimate arterial route travel time distributions (42). Guo et al. extended their inves-tigations of using multistate models to represent travel time dis-tributions by exploring the use of mixed skewed models (43), and Feng et al. explored the use of Bayesian models to construct arterial travel time distributions on the basis of data from GPS-equipped probe vehicles (44).

Field data ColleCtion

For this study, time stamps were collected for Bluetooth MAC addresses at four monitoring stations on I-5 just south of US-50 in Sacramento. Data were collected both in the northbound and south-bound directions for approximately 3 months (January to April) in 2011. These data are very useful in exploring ways of generating route TT PDFs from adjacent segment TT PDFs.

data Source

The data were obtained on a 5-mi section of I-5 in Sacramento. Figure 1 shows where the four Bluetooth readers were stationed.

FIGURE 1 I-5 in Sacramento, California.

Isukapati, List, Williams, and Karr 73

The distances between the stations (going southbound) were 3.6 mi, 1.1 mi, and 0.9 mi, respectively. The Bluetooth data included the MAC address, signal strength, and UNIX time stamps. These data were processed using BluSTATS software developed by TRAFFAX, Inc., to create reader-to-reader observations. The output records include the MAC address, time stamps at the upstream and downstream data collection devices, and the associated estimated trip time (computed by differencing upstream and downstream time stamps) in minutes. Around 150,000 unique vehicle trip times were created in each direction.

Nothing in the data set indicates whether the vehicle made inter-mediate stops. Therefore, even though the times between locations can be observed, there is no guarantee that the times represent travel times. Instead, only some of the observations are travel times; the other observations are trip times when the vehicle made an intermediate stop and so forth. The following section explains how travel times were distinguished from trip times.

Filtering travel times from trip times

Figure 2a shows all of the southbound route trip times for the 5.6 mi. It is obvious that more than just travel times are being captured. The times range up to 240 min, the limit on time stamp connections employed by the processing software, not necessarily the greatest differences in time between the time stamps.

Clearly many observations are in the 4- to 10-min range. This is highlighted in Figure 2b where the upper bound on the times dis-played is 40 min. It is also clear that travel times are embedded in the trip times, ranging from 4 to 7 min, and that for certain situations (when nonrecurring events occur) the times are 10 min or more, up to about 35 to 40 min.

Therefore, an important task is to extract the travel times from the trip times. This task is nontrivial, because legitimate travel times do exist under certain (nonrecurring) conditions that are well within the range of values that would otherwise represent trip times.

The methodology employed here stitches together small batches of legitimate observations, working from one batch to the next. It picks acceptable travel times from batch n and then examines batch n + 1 to find additional acceptable values. The procedure begins by examining all the times to find a reasonable lower bound, technically the 0.1% trip time. This eliminates outliers well below times that are physically feasible. It then works with batches of n data points to see which ones should be kept (in this instance, the procedure uses n = 10). From among the fourth through the seventh of 10 values, it finds the smallest one that is above the 0.1% threshold. It checks the remain-ing nine values to see if any of them should be rejected. In doing so, two tests are performed: (a) Is the value less than a local minimum or (b) is it greater than a local maximum? The algorithm dynamically adjusts its asymmetric acceptance window to accept or reject new observations. If the value is within 10% of the upper bound, the allow-able increment to the upper bound is increased by 20%; otherwise, it is decreased by 10%. Similarly, if the value is within 10% of the lower bound, the allowable decrement to the lower bound is increased by 10%; otherwise, it is decreased by 10%.

The results from applying this procedure can be seen in Figure 2c. Notice that the accepted travel times rise dramatically for a short period of time during a p.m. peak when an incident occurred. It is clear that the values representing noise have been removed and that acceptable observations of travel times have been retained.

Categorizing observations by regimes

The operating conditions that are extant on the segments must be taken into account to develop meaningful travel time PDFs. Therefore, the next step is to label each observation (all 150,000 in this case) with the regime that was operative when the travel time was observed. A regime is a combination of a nominal system loading and a non-recurring event condition. Ultimately, it is for these regimes that the TT PDFs are developed and through which the effect of these conditions is understood.

The labeling is accomplished in two steps. The first step labels each observation with the nonrecurring event that was underway when the observation was obtained, including “none.” The second step adds a second label that indicates what congestion level was (or should have been) extant when the observation was obtained.

adding the nonrecurring event labels

The label for the nonrecurring event (including “none” or “normal operations”) should be consistent with FHWA’s ideas about the sources of congestion. For the data used here, the Caltrans Perfor-mance Measurement System was used to identify incidents, weather, special events, and unusually high-demand events. “Insufficient base capacity” was captured by the nominal congestion condition catego-ries described above (i.e., situations in which the demand-to-capacity ratio was high enough that sustained queuing occurred). The high-demand category was equivalent to “fluctuations in demand.” Work zones were not present in any of the networks for which analyses were conducted, but it is clear from the congestion-related analyses that they would have an impact on reliability.

adding the Congestion Condition labels

The congestion-level labels were created by analyzing the obser-vations that remain once the nonnormal observations have been removed. Although average speeds and travel times are often used as the basis for congestion assessment, here a semivariance measure was used instead because the study was focused on travel time reliability, not average delays. A semivariance is a one-sided metric that uses a reference value r (instead of the mean) as the basis for calculating the sum of the squared deviations, and only observations xi that are greater than (or less than) that reference value r are included in the calculation.

nx rr i

i

n12 2

1∑( )σ = −

=

and

x rr r iσ = σ ∃ ≥2

For each 5-min time period in the equation above, xi was the average travel time observed on a specific day and r was the mini-mum of those values. The values of σr

2 were computed for each of the 288 5-min time periods of the day.

The σr2 values were used to classify the 5-min time periods into

one of four categories of operation: uncongested, low, moderate, and high. Uncongested meant that the σr

2 value was low (i.e., vehicle

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FIGURE 2 Vehicle trip times on I-5 southbound in Sacramento: (a) observations of vehicle trip times, (b) observations of vehicle trip times 40 min or shorter, and (c) an example of cleaned up AVI data.


interactions were not creating variations in the average value from one day to another). Low meant that there was some variation in the average travel times. Moderate meant that there was more variation and high meant that there was a lot.

SyntheSizing Route tRavel time PdFs

Because there are likely to be instances when too few route-specific observations are available to support direct development of route travel time PDFs, route-based travel time distributions have to be synthesized from segment-level distributions.

evidence of Positive Correlations Between adjacent Segments

Creating the route-level distributions might not be too difficult if the travel times were highly correlated. Suppose the upstream travel times can be used to predict the downstream travel times. If this were the case, an equation could be developed that predicts the downstream travel time on the basis of the upstream travel time.

τ = α τ + β+ p1ji

ji�

where

τ~ij+1 = downstream travel time (rate) of vehicle i,

τij = upstream travel time (rate) of vehicle i, and

α, β = regression parameters.

To see if this might be possible, regression equations were devel-oped for temporally sequential sets of 50 pairwise observations of upstream and downstream travel times. Figure 3 presents the analysis results. The R2 values tend to be quite high during off-peak periods and drop during the peaks. This trend makes sense in that drivers are able to achieve their desired speeds (travel rates) when the congestion is not high; however, when congestion is high the interactions with other vehicles make it difficult for the desired speeds (travel rates) to be achieved. These analyses demonstrated that the times are correlated.

A broader brush analysis, in which the travel times were grouped on the basis of similar flow rates, led to the same conclusion. The data were classified into one of four groups (on the basis of hourly volumes) and a regression analysis was conducted for each group. (Note that in this case the chronology of observations is not preserved.) The R2 values within each group varied between .4 and .7. Moreover, the R2 value diminished as the flow rate increased.

Evidence of these upstream-to-downstream relationships can also be seen in Figure 4. Correlation coefficients are plotted for non-overlapping sets of 50 sequential combinations of upstream and downstream travel times (paired by vehicle ID). That is, if τi

j is the upstream travel time for vehicle i, then τ i

j+1 is the downstream observation for a vehicle i.

The correlation coefficient was then computed by the following equation:

rn

j jji

j

j

ji

j

j∑( )

=−

τ − τσ

τ − τσ

+

+ +

+

1

1

ˇ ˇ, 1

1 1

1

where

rj,j+1 = correlation coefficient between segments j and j + 1, n = sample size, τ̌j = mean travel time (rate) of the sample upstream, σj = standard deviation of the sample upstream, τi

j+1 = travel time (rate) of vehicle i downstream, τ̌j+1 = mean travel time (rate) of the sample downstream, and σj+1 = standard deviation of the sample downstream.

Several inferences can be drawn from Figure 4. The first is that there are strong correlations between adjacent segments from midnight to about 3:00 p.m. (please note that the r values are consistently in the range of .8 to .9 for Segments 39-9 and 9-10, whereas the range is between .6 and .8 in the case of Segments 9-10 and 10-11). The second is that the correlation coefficient declined between 3:00 and 6:00 p.m. This decline is caused by drivers’ reduced maneuverabil-ity under highly congested network conditions. Also, notice that the correlation values are again high from 6:30 p.m. onward. Therefore, adjacent segments exhibit strong positive correlations except under highly congested conditions.

(a) (b)

FIGURE 3 Scatter plots representing R2 versus time of day: (a) Segments 39-9 and 9-10 and (b) Segments 9-10 and 10-11 (coeff = coefficient).


Synthesis options

Because correlations do exist, convolution cannot be used to construct route-level travel time density functions. Alternative techniques are necessary.

Two approaches are discussed here. The first approach uses simu-lation. The second approach uses a concept called comonotonicity (45, 46), which is an extreme form of positive, but not necessarily linear, dependence.

Monte Carlo simulation uses conditional sampling of the cumu-lative density functions (CDFs) from one segment to the next to create the route-level density function. Technically, if a random vector T = (τ1, τ2, . . . , τn) is the sum of a set of correlated random variables, then it has a joint CDF:

F P T Tn n nn( ) ( )τ τ τ = ≤ τ ≤ τ( )τ τ τ , , . . . , , . . . ,, ,..., 1 2 1 11 2

Here the joint CDF represents the CDF for the route-level travel time. The critical element is that the correlations must be taken into account. The method of conditional distributions needs to be used. The generation of T is explained below for n = 2.

F P T T

P T PT

T

F F

( ) ( )

( )

( )

τ τ = ≤ τ ≤ τ

= ≤ τ≤ τ≤ τ

= τττ

( )

( )( )

τ τ

τ τ τ

, ,, 1 2 1 1 2 2

1 12 2

1 1

12

1

1 2

1 2 1

To generate (T1, T2), one first generates T1 from Fτ1(•), and then

T1 from Fτ2/τ1(•).

The comonotonicity-based idea adds the travel times at matching percentile values for successive segments. Comonotonicity has been employed in calculating portfolio risk. Random variables X and Y are comonotonic, if there is a third random variable Z as well as increasing functions f and g such that X = f (Z) and Y = g(Z). As one would expect, the concept generalizes to three or more random variables. It implies that individual percentile values from each of the distributions can be added to obtain the percentile value for the distribution of the sum.

I-5 data for successive segments strongly suggest that comonoto-nicity holds. Table 1 provides a comparison of the southbound travel times (for four regime conditions) on I-5 predicted by summing the segment travel times against the overall route travel time.

For example, the second, third, and fourth columns show the percentile travel times for Segments 39-9, 9-10, and 10-11 on the basis of the travel times for each individual segment. The fifth column shows travel times obtained if these percentile values are simply summed. That is, the values in this column do not represent the percentiles of any underlying distribution. They are simply the algebraic sums of the percentile-based travel times shown to their left. The sixth column then shows the percentile travel times for the overall route based on the basis of actual observations of the route travel time. Finally, the last column shows the differences between the percentile value sums and actual observed values. It is clear that the values are nearly identical for the uncongested, low, and moderate congestion conditions. Moreover, when congestion is high, the differences for every percentile are more than 1%, and all percentiles greater than 70% have differences between 2% and 6%.

Therefore, not only does comonotonicity seem to pertain, but it does so even though the density functions are multimodal. Figure 5 shows the density functions for four operating regimes. In all four instances, the density functions are bimodal. Except for highly congested con-ditions, the match is strong between the density function obtained by adding the percentiles and that which was actually observed. This occurs because dependencies between adjacent segments are weak when network operating conditions are highly congested.

To examine the comonotonicity idea more closely, Kolmogorov–Smirnov tests were performed (for α = .01, .05, .1, .15, and .2) to see if statistically significant differences existed between synthesized and actual travel time distributions. The results suggest that the density functions are only significantly different for the highly congested conditions. The results are summarized in Figure 5.

driver Consistency

There is a side issue about driver consistency. A driver would be consistent if he or she chose to target the same travel speeds (perhaps by facility type) when making trips. This is a fundamental principle on which all traffic simulation models are based. Driver consistency

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FIGURE 4 Scatter plots representing r versus time of day: (a) Segments 39-9 and 9-10 and (b) Segments 9-10 and 10-11.


TABLE 1 Comparison of Actual Percentile Travel Times for Given Route with Values Obtained by Summing Travel Times for Same Percentile on Individual Segments

SegmentRoute 39-11 Diff. (%)

SegmentRoute 39-11 Diff. (%)Percentile 39-9 9-10 10-11 Sum Percentile 39-9 9-10 10-11 Sum

Segment Travel Times (Uncongested) Segment Travel Times (Low Congestion)

5 0.810 0.757 0.778 4.450 4.500 1.11 5 0.806 0.743 0.778 4.417 4.467 1.12

10 0.829 0.773 0.814 4.566 4.600 0.74 10 0.824 0.773 0.797 4.534 4.567 0.72

15 0.843 0.788 0.833 4.650 4.667 0.36 15 0.838 0.788 0.814 4.617 4.650 0.71

20 0.856 0.803 0.852 4.733 4.733 0.00 20 0.849 0.803 0.833 4.691 4.717 0.55

25 0.866 0.818 0.861 4.792 4.800 0.17 25 0.861 0.818 0.852 4.767 4.767 0.00

30 0.875 0.834 0.870 4.850 4.867 0.35 30 0.870 0.818 0.861 4.808 4.833 0.52

35 0.884 0.848 0.889 4.916 4.917 0.02 35 0.880 0.834 0.870 4.867 4.883 0.33

40 0.896 0.864 0.908 4.992 4.983 0.18 40 0.889 0.848 0.889 4.933 4.942 0.18

45 0.903 0.871 0.926 5.041 5.050 0.18 45 0.898 0.864 0.908 5.000 5.000 0.00

50 0.917 0.886 0.944 5.125 5.117 0.16 50 0.908 0.879 0.926 5.067 5.067 0.00

55 0.926 0.902 0.972 5.200 5.200 0.00 55 0.921 0.894 0.944 5.150 5.150 0.00

60 0.940 0.916 0.991 5.283 5.300 0.32 60 0.933 0.909 0.972 5.233 5.233 0.00

65 0.958 0.939 1.000 5.383 5.433 0.92 65 0.949 0.925 0.991 5.326 5.350 0.45

70 0.977 0.955 1.019 5.484 5.550 1.19 70 0.968 0.939 1.009 5.424 5.467 0.79

75 0.995 0.977 1.037 5.591 5.625 0.60 75 0.986 0.962 1.019 5.525 5.567 0.75

80 1.012 1.000 1.056 5.692 5.700 0.14 80 1.002 0.985 1.037 5.624 5.650 0.46

85 1.028 1.015 1.074 5.784 5.783 0.02 85 1.019 1.000 1.064 5.725 5.733 0.14

90 1.046 1.038 1.092 5.892 5.875 0.29 90 1.037 1.030 1.092 5.849 5.817 0.55

95 1.074 1.075 1.130 6.067 6.017 0.83 95 1.060 1.061 1.130 6.001 5.950 0.86

Segment Travel Times (Moderate Congestion) Segment Travel Times (High Congestion)

5 0.810 0.743 0.778 4.434 4.500 1.47 5 0.861 0.773 0.814 4.683 4.767 1.76

10 0.829 0.773 0.797 4.550 4.583 0.72 10 0.884 0.803 0.833 4.816 4.867 1.05

15 0.838 0.788 0.814 4.617 4.650 0.71 15 0.903 0.818 0.852 4.917 4.967 1.01

20 0.847 0.795 0.833 4.675 4.700 0.53 20 0.919 0.818 0.870 4.991 5.050 1.17

25 0.858 0.803 0.842 4.729 4.767 0.81 25 0.938 0.834 0.889 5.092 5.150 1.13

30 0.866 0.818 0.852 4.784 4.817 0.69 30 0.958 0.848 0.908 5.200 5.250 0.95

35 0.875 0.834 0.870 4.850 4.867 0.35 35 0.977 0.856 0.926 5.292 5.367 1.40

40 0.884 0.841 0.889 4.908 4.917 0.18 40 1.000 0.864 0.926 5.383 5.483 1.82

45 0.894 0.848 0.908 4.967 4.967 0.00 45 1.026 0.879 0.944 5.509 5.583 1.33

50 0.903 0.864 0.926 5.033 5.033 0.00 50 1.051 0.886 0.963 5.625 5.700 1.32

55 0.917 0.879 0.944 5.117 5.117 0.00 55 1.086 0.894 0.981 5.774 5.783 0.16

60 0.928 0.894 0.963 5.192 5.200 0.15 60 1.120 0.909 1.000 5.933 5.883 0.85

65 0.942 0.916 0.981 5.283 5.317 0.64 65 1.153 0.925 1.019 6.084 6.000 1.40

70 0.961 0.939 1.000 5.391 5.450 1.08 70 1.185 0.939 1.037 6.233 6.117 1.90

75 0.981 0.955 1.019 5.500 5.550 0.90 75 1.218 0.955 1.074 6.400 6.233 2.68

80 0.999 0.970 1.037 5.595 5.633 0.67 80 1.259 0.970 1.092 6.583 6.378 3.21

85 1.014 1.000 1.074 5.717 5.700 0.30 85 1.301 0.993 1.130 6.792 6.550 3.69

90 1.033 1.015 1.092 5.817 5.800 0.29 90 1.352 1.023 1.176 7.049 6.750 4.43

95 1.060 1.061 1.139 6.009 5.950 0.99 95 1.431 1.075 1.259 7.466 7.100 5.15

Note: Diff. = difference.


Crit D Max D

0.01 1.63 0.020568 0.014576 Yes

0.05 1.36 0.017161 0.014576 Yes

0.1 1.22 0.015394 0.014576 Yes

0.15 1.14 0.014385 0.014576 Yes

0.2 1.07 0.013501 0.014576 Yes

Passed KS TestAlpha K-alpha

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Travel Time (Min)

SynthesizedActual

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Crit D Max D

0.01 1.63 0.018528 0.010933 Yes

0.05 1.36 0.015459 0.010933 Yes

0.1 1.22 0.013868 0.010933 Yes

0.15 1.14 0.012958 0.010933 Yes

0.2 1.07 0.012163 0.010933 Yes

Alpha K-alpha

Low

Passed KS Test

0

0.02

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Crit D Max D

0.01 1.63 0.016957 0.010495 Yes

0.05 1.36 0.014148 0.010495 Yes

0.1 1.22 0.012691 0.010495 Yes

0.15 1.14 0.011859 0.010495 Yes

0.2 1.07 0.011131 0.010495 Yes

Alpha K-alpha

Moderate

Passed KS Test

0

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Pro

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Travel Time (Min)

SynthesizedActual

Crit D Max D

0.01 1.63 0.017009 0.057092 No

0.05 1.36 0.014192 0.057092 No

0.1 1.22 0.012731 0.057092 No

0.15 1.14 0.011896 0.057092 No

0.2 1.07 0.011166 0.057092 No

Alpha K-alpha

High

Passed KS Test

FIGURE 5 Comparisons of the rate TT PDFs synthesized from individual segment percentiles and the observed values: (a) uncongested conditions, (b) low-congestion conditions, (c) moderate-congestion conditions, and (d) high-congestion conditions [crit = critical; max = maximum; D = Kolmogorov–Smirnov (KS) test statistic].


would not have to exist across days for the results presented earlier to pertain. Drivers could be randomly selecting desired speeds on a per-trip basis and then following those desired speeds for the duration of the trip. This would produce the results presented above. However, intuition suggests that drivers might be consistent more generally about their desired speeds.

Because the data were available, the authors elected to see if there was any evidence of consistency in desired speeds for specific drivers across all available observations. To be clear, the data set included the MAC IDs for all the vehicles that were observed. So what follows is an analysis in the consistency of travel rates for specific MAC IDs (whatever they happen to pertain to, not drivers per se.)

The analysis involved examining consistency in the travel rates that were observed. The travel rate observations were broken down into regimes and then grouped by MAC ID. Where a specific MAC ID appeared more than once for a given regime condition, its average travel time (τ~i

n), standard deviation of travel times (σ~ in), and coefficient

of variation Civ = τ~ i

n/σ~ in were computed, where τ~i

n is the average of n observed travel times for a specific MAC ID i, σ~ i

n is the standard deviation of n observed travel times for the corresponding MAC ID i, and Ci

v is the coefficient of variation for the corresponding MAC ID i.Each dot in the Figure 6 represents a specific MAC ID, its x-value

represents average travel rate in seconds per mile, and the correspond-

ing y-value represents coefficient of variation. Note that variation in individual driver travel times under normal and uncongested condi-tions and normal and low-congestion conditions is almost negligible. The variation in travel times grows as network operating conditions grow worse.

ConCluSionS and reCommendationS

This paper has examined the distributions of travel times for individual vehicles and segments with a clear focus on how those distributions could subsequently be used for synthesizing route travel time PDFs. Four observations have been highlighted. The first observation is that careful filtering is required to extract the travel times from the raw data because it is trip times that are observed, not travel times (i.e., the movement of vehicles between locations). The second observation is that significant correlations might exist between the travel times for adjacent segments. Two analyses are done in this regard: (a) upstream travel times (rates) are used to predict downstream travel times (rates) and (b) correlation analysis is done to check for relationships in travel times (rates) on upstream-to-downstream segments. Results suggest that there are strong positive correlations between adjacent segments. The third finding is that

(a)

0

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40 45 50 55 60 65 70 75

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40 60 80 100 120 140 160 180

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(d)

FIGURE 6 Average versus coefficient of variation plot for different MAC IDs under various regimes: (a) normal + uncongested, (b) normal + low congestion, (c) normal + moderate congestion, and (d) normal + high congestion.


the concept of comonotonicity, or perfect positive dependence, can be assumed when generating route travel time PDFs from segment- level observations. Kolmogorov–Smirnov tests are conducted to check for statistical differences between synthesized and actual travel time distributions. Results suggest that these CDFs are statistically different only under highly congested conditions. However, the concept of comonotonicity can still be used even under these con-ditions because the percentage errors between the synthesized and actual percentile travel times are small. The fourth observation is that there is little variation in individual driver travel times under given operating conditions. This is an important and pleasing finding, given that an assumption of driver consistency serves as basis for traffic simulation models.

Recommendations from these findings are twofold. First, careful filtering is required to extract travel times from the trip times reported by AVI- and AVL-equipped vehicles. Simple cutoff criteria are inadequate, especially when one has an interest in exploring the impacts of nonrecurring events and other abnormal conditions. Second, the strong correlations among travel times on adjacent segments strongly suggest that synthesis of route-level travel times needs to be predicated on methodologies that explicitly take into account those correlations. Adding the percentile travel times for adjacent segments appears to have promise in reproducing the empirically observed route travel time distribution.

aCknoWledgmentS

The authors acknowledge the support and encouragement provided by the SHRP 2 staff during Project L02, Establishing Monitoring Programs for Mobility and Travel Time Reliability, which forms the basis for much of the material presented here. Of special note is William Hyman, who served as the contract manager through much of the development work. Also of note are the many other individuals who were part of the L02 project team and whose insights contributed to the materials presented here.

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The Freeway Operations Committee peer-reviewed this paper.

Documents

Synthesizing Route Travel Time Distributions from Segment Travel Time Distributions