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49 VISSIM software to simulate the incident-induced delay The distribution of travel time under a certain traffic con- (IID). Even though most of the IIDs estimated by the algo- dition corresponds to a specific component distribution in rithm were smaller than the IID obtained from the simulation Equation 3. For instance, travel time in free-flow conditions models, they were reasonably close with an average difference can be reasonably assumed to be generated from a single- of 15.3%. The proposed algorithm was applied to two sample mode distribution. For a given period, multiple traffic condi- corridors: the eastbound section of SR-520 (Evergreen Point tions might exist, and the overall distribution of travel time Bridge) and the I-405 northbound section between mileposts for this period will follow a mixture distribution. The multi- 1.68 and 15.75. The results validated the algorithm, and the states could be a result of differing traffic conditions spatially estimated delay was comparable to field data. (local bottlenecks at different sections), temporally (during the peak buildup or decay), or both. The multistate model has the advantage of better model-fitting in these multiple states Proposed Modeling and provides a novel approach for interpreting the results. Methodology The kth component fk(T k) in Equation 3 represents the Although the introduced existing measures attempt to quan- distribution of travel time corresponding to a specific traffic tify travel time reliability, they fail to distinguish between condition. The parameter vector k determines the charac- congested and noncongested conditions. Consequently, a teristics of the kth component distribution. The parameter more sophisticated model is needed to quantitatively meas- k represents the probability of each state and has a signif- ure travel time reliability and, at the same time, reflect the icant implication in travel time reliability reporting, which underlying traffic conditions that affect travel time reliability. is discussed later. In achieving these objectives, the team proposes the use of a A specific example is a two-component normal distribu- novel multistate travel time reliability modeling framework tion, as shown in Equation 4. With different combinations of to model travel times under complex traffic conditions (15). mean and variance, as can be seen in Figure 6.1, the model This chapter provides an overview of the proposed approach. can theoretically generate any form of distribution that fits According to the model, traffic could be operating in either any specific traffic conditions and travel time distributions. a congested state (caused by recurrent and nonrecurrent (T - 1 )2 events) or an uncongested state. Travel time variability in a 1 2 2 1 noncongested state is primarily determined by individual f (T , 1 , 2 , 1 , 2 ) = e 2 1 driver preferences and the speed limit of the roadway seg- (T - 2 )2 ment. Alternatively, travel time for the congested state (recur- 1 2 22 ring or nonrecurring) is expected to be longer with larger + (1 - ) e (4) 2 2 variability compared with free-flow and uncongested states. The multistate model is used in this research to quantitatively assess the probability of traffic state and the corresponding where is the mixture coefficient for the first component dis- travel time distribution characteristics within each state. A tribution, which is a normal distribution with mean 1 and finite multistate model with K component distributions has standard deviation 1; the probability for the second compo- the density function shown in Equation 3, nent distribution is 1 - , and the parameters for the second normal distribution are 2 and 2. Figure 6.1 (15) shows the K density curves of a two-component normal mixture distribu- f (T , ) = k fk (T k ) (3) tion. The parameters for the two-component distribution are k =1 1 = 10, 1 = 5, and 2 = 35, 2 = 10, respectively. The plot shows the variation in the mixture distribution as a function where T is the travel time; f (T ,) is the density function of of variations in . The model can accommodate multiple the distribution for T, representing the distribution of travel modes as commonly observed in travel time data. It is flexi- time in the corresponding state; = (1, 2, . . . K) is a vector ble enough to capture a wide range of patterns. In theory, the of mixture coefficients and K k =1 k = 1; = (1, . . . , K) is a mixture distribution can approximate any density function. matrix of model parameters for each component distribu- The mixture model is calibrated using the expectation and tion; k = (k1 . . . , kI) is a vector of model parameters for the maximization (EM) method instead of maximum likelihood kth component distribution that determines the characteristics methods because the data have multiple modes. of the kth component distribution; and fk(T k) is the density To verify the multistate distribution of travel time proposed function for the kth component distribution corresponding to above, the team randomly examined data from 10 drivers in a specific traffic condition. Depending on the nature of the the 100-Car Study data set. The home and work addresses pro- data, the component distributions fk(.) can be modeled using vided by drivers were geocoded to a geographic information normal, lognormal, or Weibull distributions. system (GIS) road network database; all the trips made by that

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50 Figure 6.1. Mixture distribution and multimode travel time distribution. driver were mapped to this database to visualize the trips. conditions. In particular, the mixture parameter k in Equa- Figure 6.2 shows the home-to-work trips made by one par- tion 3 represents the probability that a particular travel time ticipant. Travel times were then extracted from the relational follows the kth component distribution, which corresponds database and plotted in histograms. As shown in Figure 6.3, the to a particular traffic condition, as discussed earlier. This pro- distribution of travel time for work-to-home trips by that par- vides a mechanism for travel time reliability reporting. A novel ticipant is double-mode, which is in accordance with the two-step travel time reliability reporting method is thus pro- assumption of the travel time model proposed by the team in posed. The first step is to report the probability of each state the previous section. The start and end points of the trips have as indicated by the mixture parameter k. From a statistical been eliminated from the picture to follow IRB rules. standpoint, k represents the mixture probability of each component distribution. The interpretation of this proba- bility from the transportation perspective depends on the Model Interpretation and Travel Time sampling mechanism. The sampling mechanism refers to Reliability Reporting how trips were selected for analysis. Two types of sampling The multistate model provides a platform to relate the param- schemes--proportional sampling and fixed-size sampling-- eters of the mixture distribution with the underlying traffic could be used, as discussed in this section.

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51 The number of travel time observations for a given period depends on traffic conditions. Typically, the number of trips per unit time is larger for congested periods when compared with trips in a free-flow state. In a proportional sampling scheme, the number of trips is proportional to the number of trips for any given interval. For example, in a 10%, proportional sampling approach, 10 trips are selected from every 100 trips. For propor- tional sampling, the probability k can be interpreted from both macro- and micro-level perspectives. From the macro-level perspective, this corresponds to the percentage of vehicles in traffic state k; for example, the percentage of drivers that expe- rience congested traffic conditions. This interpretation can be used to quantitatively assess system performance from a traffic management perspective. The k can also be interpreted from a micro-level perspective. Because the relative frequency (per- centage) can also be interpreted as the probability for individu- als, the probability k also represents the probability that a particular traveler will travel in state k in a given period. This is most useful for personal trip prediction. In a fixed-size sampling scheme, a fixed number of trips are sampled for a given period regardless of the total number of Figure 6.2. Home-to-work trip visualization. trips during this period. For example, 30 trips will be sampled every 10 min. The k under a fixed sample scheme represents the proportion of the total duration where traffic is in the kth condition. For example, a value of 80% for the con- gested component implies that, out of 60 min, the traffic is in Figure 6.3. Home-to-work travel time histogram.

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52 a congested state for a total of 0.8 60 min. = 48 min. The The goal of the simulation effort was twofold. The first fixed-size sampling scheme also provides useful information objective was to demonstrate that the model parameters for individual travelers, such as the proportion of time traffic estimated are comparable to the characteristics of each traf- will be in a congested condition. fic state. The second objective was to demonstrate model The multistate model provides a convenient travel time interpretation under two alternative sampling schemes: pro- reliability analog to the well-accepted weather forecasting portional sampling and fixed-size sampling. example. The general population is familiar with the two-step Two O-D demands, a congested state and an uncongested weather forecasting approach (e.g., "the probability of rain state (scaling down from the congested matrix), were used to tomorrow is 80%, with an expected precipitation of 2 in. per simulate temporal variations. Specifically, a database of 1,000 hour"). The same method can be used in travel reliability simulation runs, 500 of them with high demand and 500 with forecasting (e.g., "the probability of encountering congestion low demand, was constructed for various travel demand levels. in the morning peak along a roadway segment is 67%, with The mixture scenarios were generated by sampling from the an expected travel time of 30 min"). Travel time under each 1,000 time units of simulation output. The simulated travel state can be reported by using well-accepted indices such as times were mixed at fixed mixture levels of 10%, 25%, 50%, the percentile and the misery index, which can be readily cal- and 75% of time units in a congested stage. The mixed travel culated from each component distribution. times were fitted to the two-state model, a mixture of two This two-step reporting scheme provides rich information normal distributions. The fitting results demonstrated that for both travelers and traffic management agencies. By know- the two-state model provides a better match to the simulated ing the probability of a congested or incident state and the travel times when compared with a unimodal model, as expected travel time in each state, an individual traveler can shown in Figure 6.4 (16). make better travel decisions. For instance, in the case of an The results also showed that for proportional sampling in important trip in which the traveler must arrive at his or her which the number of trips sampled in a given period is pro- destination at a specific time, the traveler can make a decision portional to the total number of trips in that period, under based on the worst-case scenario and estimate the required high-congestion scenarios (75% of time units in congested starting time from that scenario. For a flexible trip, the trav- state), the model underestimates the true proportion and eler can pick a starting time with a lower probability of overestimates the variance of the travel time in the free-flow encountering a congested state. For traffic management agen- state. The reason for the bias is that a single normal distribu- cies, the proportion of trips in a congested state and the travel tion cannot sufficiently model the travel time in the con- time difference between the congested state and the free-flow gested state when the percentage is high. This problem can be state can provide critical information on the efficiency of the resolved by introducing a third component or by using alter- overall transportation system. This can also provide an oppor- native component distributions (e.g., lognormal or gamma). tunity to quantitatively evaluate the effects of congestion alle- For fixed-size sampling in which a fixed number of trips is viation methods. sampled for any given period, the model does reflect the char- acteristics of travel time under different traffic conditions. The parameters of the component distribution can be esti- Model Testing mated satisfactorily, and the interpretation of the mixture To demonstrate and validate the interpretation of the mix- parameters depends on the sampling scheme. ture multistate model, simulation was conducted using The multistate model was then applied to a data set collected INTEGRATION software along a 16-mi expressway corridor from I-35 near San Antonio, Texas. The traffic volume near (I-66) in northern Virginia. The validation was not conducted downtown San Antonio varied between 89,000 and 157,000 using the in-vehicle data for several reasons. First, the time vehicles per day. The most heavily traveled sections were near stamps in the 100-Car data set were not GPS times and had the interchange with I-37, with average daily traffic counts errors up to several hours. Second, the in-vehicle data do not between 141,000 and 169,000 vehicles, and between the south- provide the ground truth conditions, given that the data are ern and northern junctions with the Loop 410 freeway, only available for the subject vehicle and not for all vehicles with average daily traffic counts between 144,000 and 157,000. within the system. However, the simulation environment Vehicles were tagged with radio frequency (RF) sensors, and the provides similar probe data with additional information on travel time for each tagged vehicle was recorded whenever the performance of the entire system and detailed informa- vehicles passed any pair of AVI stations. A two-component tion on any incidents that are introduced. model and a three-component model were fitted to the morn- An origin-destination (O-D) matrix was developed rep- ing peak travel time. The Akaike information criterion (AIC) resenting the number of trips traveled between each O-D was used to compare these models. The smaller the AIC value, pair using field loop detector traffic volume measurements. the better fitted the model is. The results, shown in Table 6.1,

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53 0.006 bimodal unimodal 0.004 D ensity 0.002 0.000 500 10 00 15 00 2 000 2500 Travel time (s) Figure 6.4. Comparison between unimodal and bimodal models. demonstrate that the three-component model provides sub- The multistate travel time reliability model is more flexible stantially better model-fitting than does the two-component and provides superior fitting to travel time data compared model. The travel time reliability reporting listed in Table 6.1 with traditional single-mode models. The model provides a clearly expresses the travel time information needed by travel- direct connection between the model parameters and the ers and decision makers. It not only reports the probability of underlying traffic conditions. It can also be directly linked to encountering a congested state but also reports the expected the probability of incidents and thus can capture the impact travel time under that state. of nonrecurring congestion on travel time reliability. Table 6.1. Mixture Normal Model-Fitting for Morning Peak Travel Time Two-Component Model Three-Component Model Mixture Standard Mixture Standard Proportion Mean Deviation Proportion Mean Deviation Comp. 1 33% 588 38 0.33 588 38 Comp. 2 67% 1089 393 0.59 981 230 Comp. 3 NA NA NA 0.08 1958 223 Log likelihood -3567 -3503 AIC 7144 7020 Travel time 1. The probability of encountering congestion is 67%. 1. There is a 59% probability of encountering conges- reliability If congestion is encountered, there is a 90% proba- tion. If congestion is encountered, there is a 90% reporting bility that travel time is less than 1,592 s. probability that travel time is less than 1,276 s. 2. The probability of encountering a free-flow state is 2. There is a 33% probability of encountering free-flow 33%. If congestion is encountered, there is a 90% conditions. In this case, there is a 90% probability probability that travel time is less than 637 s. that travel time is less than 637 s. 3. The probability of encountering an incident is 8%. In this case, there is a 90% probability that travel time is less than 2,244 s.