Incorporating Reliability Performance Measures into Operations and Planning Modeling Tools: Reference Material (2014)

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7 PART 1 IDENTIFICATION AND ADAPTATION OF EXISTING MODELS CHAPTER 2 Producing Travel Time Variability Using Existing Tools As indicated in Chapter 1, Part 1 of the report contains three key chapters that discuss the general methodologies to produce travel time reliability relations; a description of the adaptation, calibration, and validation of the selected mesoscopic model; and a description of the calibration and validation of the microsimulation model. 2.1 Methodology to Leverage Networkwide Travel Time Variability Relation 2.1.1 Introduction The reliability of experienced service levels is a critical dimension of transportation network performance for users and a major determinant of users’ travel choices. As such, measures of travel time reliability must be increasingly incorporated by operators as a key performance indicator of service quality and included in evaluation studies of contemplated strategic policy- based and operational system interventions. Unreliability results from a variety of factors that affect both the supply and demand for transportation in a system. A convenient categorization of these factors adopted by the second Strategic Highway Research Program (SHRP 2) and the Federal Highway Administration (FHWA) is given in the final report of project National Cooperative Highway Research Program (NCHRP) 3-68. It recognizes seven sources of travel time unreliability—namely, traffic incidents, special events, work zones, weather, day-to-day demand (volume) fluctuations, traffic control devices (railroad crossing, poor signal timing), and “inadequate base capacity” (NCHRP Project 3-68). These categories are helpful both for developing modeling frameworks that incorporate reliability, as well as identifying potential solutions and mitigating interventions. Traffic analysts and policy makers increasingly recognize the importance of travel time reliability as a factor influencing the decision making of individual travelers, particularly when a time constraint is imposed on the trip. The valuation of travel time reliability has been studied comprehensively (Lam and Small 2001; Small et al. 2005; Bates et al. 2001). Experiments have shown that, in some situations, travelers seem to place a higher relative value on reducing travel time variability than on the mean travel time (Bates et al. 2001). Several quantitative measures and proxies for travel time reliability have been proposed (Lomax et al. 2003). Virtually all those measures are related to the properties of the day-to-day distribution of travel time. Four different measures currently used by FHWA are the following (FHWA 2005):

8 1. 90th or 95th percentile travel time: it is the value that 90% or 95% of the travel time will be below it. It is used to estimate how much delay will be on a specific route during the heaviest traffic days; 2. Buffer Index: represents the extra amount of time that needs to be added to the average travel time to ensure on-time arrival for most of the times. It can be calculated as the difference between 95th percentile travel time and mean travel time, divided by the mean travel time; 3. Planning Time Index (PTI): represents how much total time should be allowed to ensure on-time arrival. It can be calculated as the planned total travel time to ensure arrival on time for 95% of the time, divided by the free-flow travel time; 4. Frequency that congestion exceeds some expected threshold: it is usually expressed as the percent of days or time that travel times exceed a certain level or travel speeds fall below a specified value. Methods to characterize the variability of network performance and the factors that influence it, particularly in relation to other measures of system performance, are important to the modeling and evaluation of system interventions. The primary focus is on the variability of the travel time experienced by different users in a network. Ideally, the distribution function, or probability density function (PDF), of travel time should be characterized and monitored using historical data; however, most of the time, this is not possible due to either unavailability or insufficiency of data. Alternatively, the distribution of travel time can be approximated if some of the statistics are known. The two most basic statistics are the mean and standard deviation (SD), with one depicting the central tendency and the other describing the dispersion of the distribution. Usually, the average travel time of a trip is relatively easy to obtain―perceived, measured, or obtained from theory or models; obtaining or predicting the SD is more challenging. In this report, the team shares an approach that traces its intellectual origins to Nobel Laureate Ilya Prigogine and Robert Hermann’s seminal work on the kinetic theory of traffic flow (Prigogine and Hermann 1971). The kinetic theory implies the existence of a relationship between mean travel time and its variance. According to Prigogine and Hermann, the magnitude of the mean travel time and its variability change with traffic conditions. Relatively short mean travel times often prevail together with low SDs; this corresponds to the uncongested traffic condition, when drivers are able to maintain fairly constant and higher speeds. As the congestion increases, the average speed decreases and drivers experience increased interactions with other drivers in the vicinity. These interactions cause increased fluctuation in driving speed and thus affect the travel time. This relationship between mean travel time and its variation is supported by empirical studies. A simple theory was developed by Herman and Lam (1974) to relate the dispersion of travel time distribution to the mean travel time and verified through data collected by circulating vehicles in a network. Subsequently,

9 Taylor (1982) collected data for public transit in Paris and confirmed and extended the theory of Herman and Lam. Taylor also introduced a measure of variability that could be related to the traffic congestion level. Later Jones (1988) verified the relationship between the SD of travel time and mean travel time, both taken on a per unit distance basis, for different link types, based on vehicle probe data in Austin, Texas. Those early studies were limited by data availability to the particular conditions encountered in a small area over a relatively short period. Furthermore, with increasing interest in developing models and methods to incorporate reliability in modeling and evaluation tools, those early works remain relatively obscure, insufficiently understood, and perhaps inaccessible to many researchers and practitioners. This chapter addresses this gap by investigating and extending the scope and scale over which this relation can be expected to hold and shows how it may be leveraged as a first-order approach to incorporate reliability measures for both planning and operational analyses in conjunction with most network performance modeling tools. To establish this approach, the relationship between mean travel time per unit distance and its variability is investigated thoroughly on a multiscale and multilevel basis. A simple yet robust method for estimating the SD of travel time per unit distance is proposed and verified. 2.1.2 Methodology Relationship Between Mean Travel Time per Unit Distance and Its Variability As the kinetic traffic flow theory suggests, there is a strong correlation between the magnitude of mean travel time and its fluctuation; the SD (or variance) of the travel time could be estimated based on the mean travel time, if a sufficiently robust relation is uncovered and an appropriate model is constructed. Herman and Lam (1974) first proposed that when travel times on different links are independent and identically distributed, the SD is proportional to the square root of the average travel time. However, the underlying model assumption is too stringent and is rarely met in reality—that is, link travel times are neither identically distributed nor uncorrelated. Herman and Lam’s empirical observations from work journeys also suggested the square root function fit the data rather well for average travel times up to 30 min, while a linear model offered a better overall fit. One shortcoming of using trip travel time data alone for studying travel time variation is that the variability can come from both the traveling speed and trip distance. So when attempting to draw conclusions about variability involving trip distance, it is uncertain whether it is due to speed or trip distance. To overcome this problem, Richardson and Taylor (1978) suggested using the unit travel time, which is travel time divided by the trip distance, and studied the relationship between congestion level and average unit travel time. Jones (1988) studied trip travel time variability at the link level using commuting data in Texas and confirmed there is a linear relationship between mean trip time per mile and its SD. Following the suggestions in the literature and based on historical data observations, it is reasonable to propose a linear or near-linear relationship between mean and SD of travel times per unit distance. It should be noted that, to exclude the variability in trip distance, the travel time

10 needs to be normalized by trip distance (Equation 2.1), yielding travel time per unit distance (or the inverse of the space mean speed). t t d   (2.1) where t′ = travel time per mile, t = travel time, and d = travel distance in miles. In this chapter, three different model specifications are tested: the linear model, the square root model, and the quadratic model. The linear model is suggested by both the traffic flow theory and the data, which reflects the fact that the variability of travel speed increases as its mean value decreases. The square root and quadratic models are proposed to account for the initial upward trend of the SD versus mean curve and the subsequent downward trend, which is suggested by traffic flow theory though it has not been widely observed. The formulations of these three models can be written as follows: (1) Linear model   )'()'( 21 tEt (2.2) (2) Square root model   )'()'( 21 tEt (2.3) (3) Quadratic model   2321 )'()'()'( tEtEt (2.4) where σ(t′) = SD of t′, E(t′) = mean value of t′, θ1, θ2, θ3 = coefficients, and ε = random error. Simulation Calibrating the above models requires travel time and travel distance information of individual vehicles within a given network over time. Real-world traffic data acquisition is not readily available, though may be achievable with modern technologies such as GPS, albeit at

11 considerable effort/cost. The availability of such data remains limited. To obtain the scale and scope of desired observations, the team relies on computer simulation of traffic network flows. In this paper, trip data generated by dynamic network analysis software (DYNASMART) are utilized to explore the relationship between mean travel time per unit distance and its associated variation. DYNASMART is a discrete time mesoscopic traffic simulation tool, which models traffic flow dynamics at a macroscopic level while at the same time is capable of modeling individual vehicles’ movements with different driver behavioral characteristics under various information guidance systems. One of the motivations of using a mesoscopic traffic simulator is to gain insight into the simulator’s eventual usefulness for reliability studies, as the relation explored in the paper would then readily be applied to the output (Jayakrishnan et al. 1994; Peeta and Mahmassani 1995; Mahmassani and Liu 1999). Multidimensional Analysis To give a complete picture of how well the model can be used to fit the data and estimate the SD of travel time, analyses are performed on a multiscale and multilevel basis. Multiscale analysis refers to analyzing networks of different sizes, ranging from a small network containing hundreds of links only to large networks that consist of tens of thousands of links. Multilevel analysis, on the other hand, is achieved by analyzing the data within a single network but grouping them at four distinct levels: network level, origin–destination (O-D) level, path level, and link level. Moreover, to capture the travel time variation, the following two types of analyses are performed: (1) variation across vehicles (at O-D, path, and link level), and (2) time-of-day (TOD) variation (at network level). For the first, do the following: 1. Perform one simulation run. 2. Extract travel time and travel distance for each vehicle, and group vehicles according to O-D, path, or link. Each O-D, path, or link produces a sample point. 3. Construct the plot of SD of travel time per unit distance versus its mean value based on the sample points obtained in Step 2. For the second, do the following: 1. Perform one simulation run. 2. Extract travel time and travel distance for each vehicle, and group vehicles according to departure time. Each departure time interval produces a sample point. 3. Construct the plot of SD of travel time per distance versus its mean value based on the sample points obtained in Step 2.

12 2.1.3 Experiments and Results Study Areas Three actual road networks are studied for calibrating the model: Irvine, Baltimore–Washington Corridor, and New York City. The snapshots for these three networks are presented in Figure 2.1. The scale of the networks covers a wide range, from hundreds of nodes to tens of thousands of nodes. Detailed descriptive information of the various network components such as links, nodes, and zones are provided in Table 2.1 for each road network. Figure 2.1. Snapshots of study areas (a) Irvine; (b) Baltimore–Washington Corridor; and (c) New York City. (a) (b (c)

13 Table 2.1. Network Configurations Network Irvine Baltimore– Washington Corridor New York City Number of Zones 61 111 3,697 Number of Nodes 326 2,182 28,406 Number of Links 626 3,387 68,490 Number of Vehicles 58,385 15,1973 6,766,805 Demand Duration (h) 2 2 4 Simulation and Data Processing Iterative dynamic traffic assignment (DTA) is performed by DYNASMART on the three selected networks to achieve user equilibrium (UE), using historical data to calibrate time- varying O-D demand. An important output of the simulation is the itinerary associated with each vehicle within the network. The information included in the itineraries (trajectories) includes departure time, origin zone number, destination zone number, all the nodes been visited, node exit times, link travel time, and accumulated stop time. Based on the simulated vehicle trajectories, the total travel time and total travel distance of each vehicle are extracted, and thus the travel time per unit distance is then calculated. As introduced previously, the analyses are conducted at multiple levels—namely, network, O-D, path, and link levels. At the network level, TOD variation analysis is performed, and this is achieved by grouping vehicles by different departure times. In this analysis, 5 min are used as an interval. Therefore, for a 2-h demand, the vehicles within the network are divided into 24 groups. The mean travel time per mile and its SD are then computed for each group of vehicles. At the O-D level, vehicles are grouped according to origins and destinations, and for each O-D pair, the mean travel time per mile and its SD are calculated. Similarly, at path and link levels, for each single path or link within the network the number of vehicles that followed that path or traversed that link is counted. The mean travel time per mile and its SD are then computed for each path or link. It should be noted that since not all vehicles have reached the destinations at the end of the simulation period, only those vehicles that completed the trips are taken into account. For statistical significance reasons, only those groups that contain more than 30 vehicles are considered as sample points. Model Calibration Three different model forms—that is, linear model, square root model, and quadratic model—are used to represent the data after the SD versus mean plots are constructed. It is observed that the data exhibit some heteroscedasticity at the O-D, path, and link levels for all three networks. This

14 phenomenon is confirmed by performing a Breusch–Pagan test (Margiotta 2010). The statistical test results are presented in Table 2.2, indicating that the null hypothesis of homoscedasticity is rejected at O-D, path, and link levels, while it cannot be rejected at the network level. The results indicate a violation of the assumption of constant variance in error term in classic regression modeling, and therefore this assumption needs to be corrected before further statistical testing of the model is conducted. One possible explanation of such heteroscedasticity is that it results from interactions between drivers and interactions between the physical system and the drivers. At the network level, no heteroscedasticity is observed on either one of the three networks, and ordinary least squares (OLS) estimation is used to calibrate the models; while at the other three analysis levels, weighted least squares (WLS) method is applied to accommodate the presence of heteroscedasticity. In the WLS method, it is assumed the variance of the error term is proportional to the magnitude of average trip time per mile. A weight of one over the square root of the average trip time per mile is used accordingly. The plots, together with the fitted curves, are presented in Figure 2.2 through Figure 2.4. Table 2.2. Breusch–Pagan Test Results Aggregation Level Irvine Baltimore–Washington Corridor New York City test statistic p-value test statistic p-value test statistic p-value Network 10.68 0.0011 1.36 0.2434 1.84 0.1746 O-D 59.96 <0.0001 69.50 <0.0001 80.35 <0.0001 Path 89.46 <0.0001 89.33 <0.0001 14.06 <0.0001 Link 54.25 <0.0001 527.46 <0.0001 51.33 <0.0001

15 Figure 2.2. Network-level SD of travel time per mile versus mean travel time per mile for (a) Irvine; (b) Baltimore–Washington Corridor; and (c) New York City. Figure 2.3. O-D–Level SD of travel time per mile versus mean travel time per mile for (a) Irvine; (b) Baltimore–Washington Corridor; and (c) New York City.

16 Figure 2.4. Link-level SD of travel time per mile versus mean travel time per mile for (a) Irvine; (b) Baltimore–Washington Corridor; and (c) New York City. The calibration results for the three networks at all four different analysis levels are given in Table 2.3 and Table 2.4, which include point estimates of the model coefficients, the 95% confidence interval, t-statistics, associated p-values, and R 2 values. The results show that the quadratic model has the highest overall goodness-of-fit value (R 2 value) in most of the cases. However, the sign of the quadratic term coefficient is not always consistent with the theory—that is, the fitted quadratic curve shows an upward trend as the mean travel time per mile increases in some cases. Moreover, the associated p-values of some quadratic model coefficients are large, which suggests the corresponding coefficients are not significantly different from zero. These facts lead to reject the quadratic model as the best model. Comparing the linear model and the square root model, the p-values show all the coefficients are statistically significant for both models at every analysis level, except for the New York City link-level case, where the intercept term of the linear model is not significantly different from zero. The R 2 value of the linear model is generally greater than that of the square root model in most cases, which suggests the linear model fits the simulation data better. In summary, the linear model is the best model among all the three candidate models, and the simulation data show that it works universally well at different network scales and analysis levels.

17 Table 2.3. Network and O-D–Level Calibration Results Network Level O-D Level Irvine Baltimore– Washington Corridor New York City Irvine Baltimore– Washingto n Corridor New York City Linear model θ1 estimate -1.2823 -1.0296 -2.2100 -0.5591 -0.3316 -1.1144 conf. interval [-1.9068, -0.6578] [-1.2736, - 0.7857] [-2.6830, -1.7370] [-0.7224, -0.3958] [-0.3796, - 0.2836] [-1.2560, -0.9729] t- statistic -4.2582 -8.7518 -9.4051 -6.7672 -13.5599 -15.4422 p-value 0.0003 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 θ2 estimate 1.5302 0.9130 1.5954 0.5695 0.3369 0.9145 conf. interval [1.1489, 1.9115] [0.7640, 1.0621] [1.5133, 1.6774] [0.4808, 0.6583] [0.3072, 0.3666] [0.8854, 0.9437] t- statistic 8.3231 12.0729 39.1438 12.6785 22.2663 61.5782 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 R 2 07590 0.8800 0.9709 0.6045 0.3862 0.6229 Square root model θ1 estimate -3.8778 -2.4794 -9.6773 -1.7979 -0.8376 -5.2827 conf. interval [-5.1124, -2.6432] [-2.9749, - 1.9839] [- 10.6077, -8.7468] [-2.1737, -1.4220] [0.9365, - 0.7388] [-5.5520, -5.0135] t- statistic -6.5140 -10.3769 -20.9354 -9.4556 -16.6276 -38.4822 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 θ2 estimate 3.9898 2.3025 7.0141 1.7264 0.8303 4.0185 conf. interval [3.0235, 4.9560] [1.9148, 2.6902] [6.6184, 7.4097] [1.4354, 2.0174] [0.7518, 0.9089] [3.8895, 4.1476] t- statistic 8.5633 12.3158 35.6857 11.7254 20.7344 61.0699 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 R 2 0.7692 0.8733 0.9651 0.5656 0.3478 0.6198 Quadratic model θ1 estimate -13.4179 1.4112 -2.0529 -03721 0.5998 -1.5086 conf. interval [- 19.9304, -6.9055] [-1.0039, 3.8262] [-3.5316, -0.5742] [-0.6930, -0.0511] [0.4620, 0.7375] [-1.7729, -1.2443] t- statistic -4.2847 1.2152 -2.7962 -2.2917 8.5449 -11.1946 p-value 0.0003 0.2378 0.0076 0.0234 <0.0001 <0.0001 θ2 estimate 16.0552 -2.1365 1.5268 0.4018 -0.7790 1.0878 conf. interval [8.2769, 23.8336] [-5.1436, 0.8706] [0.9102, 2.1433] [0.1384, 0.6651] [-0.9378, - 0.6202] [0.9854, 1.1903] t- statistic 4.2925 -1.4775 4.9876 3.0152 -9.6260 20.8281 p-value 0.0003 0.1544 <0.0001 0.0030 <0.0001 <0.0001 θ3 estimate -4.3152 0.9481 0.0067 0.0274 0.3194 -0.0152 conf. interval [-6.6244, -2.0060] [0.0142, 1.8820] [- 0.05288, 0.0662] [-0.0131, 0.0680] [0.2747, 0.3642] [-0.0239, -0.0066] t- statistic -3.8862 2.1112 0.2261 1.3370 13.9941 -3.4600 p-value 0.0009 0.0469 0.8221 0.1833 <0.0001 <0.0001 R 2 0.8598 0.9010 0.9709 0.6050 0.5176 0.6261

18 Table 2.4. Path- and Link-Level Calibration Results Path Level Link Level Irvine Baltimore– Washington Corridor New York City Irvine Baltimore– Washington Corridor New York City Linear model θ1 estimate -3.185 -0.2600 -0.8439 -0.2400 -0.6440 -0.1640 conf. interval [-0.4101, -0.2269] [-0.2861, - 0.2338] [-0.9646, -0.7231] [-0.3285, -0.1516] [-0.6735, - 0.6145] [0.4088, 0.00807] t- statistic -6.8626 -19.5140 -13.7230 -5.3328 -42.7886 -1.3160 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 0.1886 θ2 estimate 0.3206 0.2544 0.5261 0.4668 0.6416 1.0358 conf. interval [0.2719, 0.3692] [0.2379, 0.2709] [0.4940, 0.5582] [0.4278, 0.5057] [0.6242, 0.6590] [0.9432, 1.1283] t- statistic 12.9986 30.1931 32.1925 23.5288 72.2741 21.9740 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 R 2 0.5500 0.5494 0.6537 0.4078 0.7315 0.4300 Square root model θ1 estimate -1.0007 -0.6430 -3.0602 -1.6077 -1.9968 -3.6563 conf. interval [-1.2189, -0.7824] [-0.6974, - 0.5885] [-3.3183, -2.8021] [-1.7975, -1.4180] [-2.0753, - 1.9183] [-4.1264, -3.1862] t- statistic -9.0496 -23.1691 -23.2795 -16.6480 -49.8986 -15.2698 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 θ2 estimate 0.9634 0.6285 2.2510 1.7408 1.9206 4.2867 conf. interval [0.7943, 1.1325] [0.5846, 0.6725] [2.1070, 2.3950] [1.5955, 1.8860] [1.8571, 1.9841] [3.9370, 4.6364] t- statistic 11.2478 28.0857 30.7000 23.5479 59.3067 24.0682 p-value <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 R 2 0.5455 0.4947 0.3755 0.5710 0.3485 0.4955 Quadratic model θ1 estimate 0.0710 0.2067 -0.7892 -0.3463 -0.5420 -0.7437 conf. interval [-0.0879, 0.2300] [0.1349, 0.2784] [-1.0101, -0.5682] [-0.4684, 0.2242] [-0.5884, - 0.4956] [-1.0495, -0.4387] t- statistic 0.8820 5.6519 -7.0131 -5.5708 -22.9049 -4.7735 p-value 0.3790 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 θ2 estimate -0.0200 -0.3182 0.4992 0.5531 0.5535 1.4878 conf. interval [-0.1461, 0.1061] [-0.4026, - 0.2338] [0.4029, 0.5956] [0.4741, 0.6321] [0.5179, 0.5890] [1.3137, 1.6620] t- statistic -0.3137 -7.3991 10.1701 13.7582 30.5183 16.7712 p-value 0.7541 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 θ3 estimate 0.0503 0.1655 0.0022 -0.0080 0.0113 -0.0340 conf. interval [0.0329, 0.0677] [0.1415, 0.1895] [-0.0053, 0.0098] [-0.0143, -0.0016] [0.0073, 0.01535] [-0.0451, -0.0228] t- statistic 5.7026 13.5322 0.5805 -2.4646 5.5646 -5.9606 p-value <0.0001 <0.0001 0.5618 0.0141 <0.0001 <0.0001 R 2 0.3812 0.3585 0.3460 0.4180 0.2621 0.4788

19 Though the linear model is a regression model based on observations, its coefficients have some physical interpretations. The magnitude of the slope term (θ2) tells how much the SD will increase when the average travel time per mile increases by one unit. A greater θ2 value means travel time variability is introduced more easily as the mean travel speed decreases. Based on calibration results, New York City has the greatest θ2 value at all four analysis levels. Irvine ranks the second, and Baltimore–Washington Corridor has the smallest slope value except for the link level. The value of the slope term ranges from 0.25 to 1.60. The y-intercept term (θ1) alone may not have a physical meaning; however, the negative ratio between θ1 and θ2 (–θ1/θ2)—that is, the x-intercept—represents maximum speed of the network. The value of x-intercept ranges from 0.84 to 1.39 min/mi for the three studied networks, which corresponds to 43 to 72 mph. It is found that the Irvine network has the smallest x-intercept value, which represents the largest maximum speed among the three networks. New York City has the largest x-intercept, while Baltimore–Washington Corridor has the value in between. When applying the linear model to predict SD, given the mean travel time per mile, if the mean travel time per mile value is smaller than –θ1/θ2, the SD of travel time per mile is assumed to be zero, to avoid getting a negative value, which is not meaningful. When comparing the calibration results among different analysis levels, it is found that the network-level linear model has the greatest slope, while the path-level model has the smallest slope. This finding is consistent among all three networks. For Irvine and Baltimore–Washington Corridor, the link-level model has a steeper slope than the O-D–level model, while for New York City, it is vice versa. It can be concluded that from network level to O-D level and then to path level, the linear model becomes less and less steep. This is mainly because from network to path level, individual trips are grouped into finer groups, and thus less variability is involved in the model. On the other hand, the reason why the link-level model shows a steeper slope than the path-level model can be because the variabilities within the link level are canceled out when added up together at the path-level analysis. 2.1.4 Validation Validation of the proposed model is a difficult task, since the available real-world vehicle trajectory data are scarce. Some studies in the literature have used real trip data to show there is high correlation between SD of travel time per mile and mean travel time per mile; however, the results are limited. Jones (1988) used manually recorded travel time data in the Texas area to show the correlation and performed the analysis at the link level only, though based on different facility types and link exit conditions. Margiotta (2010) used loop detector data collected on U.S. 101 southbound in California to study travel time reliability and showed the SD of travel time per mile increases with mean travel time per mile. The analysis was also only performed at the link level. Although it is difficult to obtain individual trip data for the same study areas as the simulations, vehicle probe data from other networks can be used to verify the model correctness, if not its accuracy. In this document, data from the Traffic Choices Study project

20 (http://psrc.org/transportation/traffic) in the Seattle area are used for model validation purpose. The project was conducted by the Puget Sound Regional Council (PSRC) to study travel behavior in response to variable charges for road use. The data consist of GPS probe data from more than 400 participating vehicles during year 2005 to 2007. The network consists of over 600 zones and 6,000 links and is comparable in size to the Baltimore–Washington Corridor network in the team’s simulation study. Before performing the travel time analysis, individual trips are extracted from the project database. Trip information includes vehicle ID, trip starting time and location, trip ending time and location, miles traveled, and the location and time stamps of when the vehicle entered each link. These trips are then mapped onto the Seattle area map using geographic information system (GIS) software, to find out the trip starting and ending traffic analysis zones (TAZs). Examples of extracted trips are shown in Figure 2.5. A total number of 549,624 trips were identified and used in the latter analysis. The same methodology is applied for analyzing the trip data. Trips are grouped together at three different levels—namely, O-D, path, and link. Trip travel times are normalized by travel distance. The mean and SD of travel time per unit distance are then computed for each O-D pair, path, and link. Figure 2.6 presents SD of travel time per mile versus mean travel time per mile plots, with the fitted linear regression model. Figure 2.5. Examples of extracted trips shown on map of Seattle.

21 Figure 2.6. Standard deviation of travel time per mile versus mean travel time per mile for Seattle network at (a) O-D level; (b) path level; and (c) link level. The network-level analysis performed on the simulation data is not applicable here for the GPS probe data because of an insufficient number of trip records within 5-min departure intervals. The probe data exhibit a pattern that is consistent with the simulated trajectory data— that is, the SD of travel time per mile is positively correlated to the mean travel time per mile, though some degree of heteroscedasticity exists. The calibration results of the linear model are also qualitatively consistent with those obtained from simulation data, as shown in Table 2.5, with the O-D–level model having the steepest slope and the path-level model having the mildest slope. The number of samples in the table represents the number of O-D pairs, paths, and links involved in the analysis. It can be concluded, with the support of probe data, that the linear regression model is capable of representing the relationship between travel time variability and mean travel time at different analysis levels. Table 2.5. Calibration Results of Linear Model Using GPS Probe Data Number of Samples θ1 θ2 R 2 O-D level 3,193 –1.6728 1.0665 0.5770 Path level 1,735 –1.1286 0.7323 0.3861 Link level 5,415 –0.4736 0.9936 0.6675