#### Read Modeling and Simulation of Unconventional Traffic Circles text version

Bartin, Ozbay, Yanmaz-Tuzel and List

MODELING AND SIMULATION OF UNCONVENTIONAL TRAFFIC CIRCLES

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Bekir Bartin Research Associate. Civil and Environmental Engineering Department Rutgers University, New Jersey. Address: 623 Bowser Rd. Piscataway, NJ 08854 Phone: 732-445-3162 Fax: 732-445-0577 E-mail: [email protected] Kaan Ozbay Associate Professor Civil and Environmental Engineering Department Rutgers University, New Jersey. E-mail: [email protected] Ozlem Yanmaz-Tuzel Graduate Student Civil and Environmental Engineering Department Rutgers University, New Jersey E-mail: [email protected] George List Professor Civil and Environmental Engineering Department North Carolina State University E-mail: [email protected]

Submitted to the 85th Transportation Research Board Annual Meeting, August 1, 2005 Word Count: 5,238 words + 4 Tables + 4 Figures = 7,238 words

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ABSTRACT

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Microscopic simulation tools have been gaining popularity among the traffic engineers. Enhanced inputoutput capabilities of these simulation tools allow engineers model and simulate complex transportation networks, and gather results relatively fast. However, extensive validation/calibration efforts are required to verify the credibility of the results of these simulation models. This paper deals with the development of credible and valid simulation models of two unconventional traffic circles in New Jersey, namely the Collingwood and Brooklawn circles. These two circles are modeled in PARAMICS simulation software. The model development and validation / calibration steps are presented in detail. PARAMICS is one of the few off-the-shelf simulation software packages that can model unconventional traffic circles. However, it is shown here that the default capabilities of PARAMICS are not sufficient to model statistically valid simulation models of these two unconventional circles. It is also shown that the use of Application Programming Interface (API) feature of PARAMICS is required to develop realistic models that can accurately represent drivers' facility specific behaviors. A gap acceptance/rejection binary probit model based on the analysis of field data is developed and implemented using PARAMICS API. The differences between the simulation results of the API enhanced model and the default PARAMICS model are presented to demonstrate the importance of extending the existing default models and the careful validation/calibration using real-world data Finally, the sensitivity analyses of the developed simulation models are presented. KEYWORDS Microscopic simulation, traffic circles, gap acceptance, PARAMICS.

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INTRODUCTION

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Traffic circles have been used in the United States since 1905. However, their use has been limited since the 1950s due to the realization that they worked neither efficiently nor safely (1). Recently, there has been an increasing interest in improving existing traffic circles to address these efficiency and safety problems. Several States are in the process of exploring effective operational alternatives for enhancing the safety and efficiency of these traffic circles built in the early periods of the 20th century. Many existing traffic circles that were designed to handle lower traffic volumes than today's volumes fall under this category of traffic circles that need to be improved, because they are faced with increasing congestion and safety problems. Only in New Jersey 30 of the 67 traffic circles built during the 1920s were replaced until 2001. Although the replacement of these traffic circles with more efficient facility types appears to be a viable option, time and money needed for the construction of alternative solutions can be prohibitive. The next best option appears to be the implementation of operational alternatives that can extend the life of these circles until they can be rebuilt in the next 10 to 20 years. Reliable computerized analysis tools can offer invaluable capabilities for assessing the benefits of the operational alternatives for these facilities. Several deterministic and stochastic computer tools now provide extended capacity and delay analysis of roundabouts, and compare various operational alternatives. The choice of the appropriate tool should be based on the scope of the project. The deterministic tools offer efficient and detailed analysis of roundabouts and estimate the capacity and delay at the approaches. Microscopic simulation tools can analyze not only the roundabouts, but also the network surrounding it. These tools can provide the effects of various operational alternatives at the network level. There are various deterministic tools that can analyze the capacity of roundabouts such as RODEL, aaSIDRA and Highway Capacity Software (HCS). These tools are based on empirical equations that were developed using field data collected at various roundabouts in Britain and Australia. An ongoing project conducted by National Cooperative Highway Research Program is aimed at calibrating the model parameters of aaSIDRA and RODEL based on various roundabouts in the U.S (1). However, it should be emphasized that microscopic simulation tools such as VISSIM, SYNCHRO, PARAMICS that are commonly used to model roundabouts are too based on empirical models. Therefore, there is not a clearcut comparison of the reliability of various computer-modeling tools for local conditions. Therefore, regardless of the selected tool, the traffic engineer is responsible for further calibration of the input parameters to ensure the reliability of the predictions of the developed models. S/he should determine the most important variables that affect the efficiency of the facility and improve the model parameters based on actual data, if available. More importantly, it should be emphasized that the commonly used deterministic tools such as aaSIDRA, HCS, RODEL, are used specifically for roundabouts. They are capable of modeling a limited range of geometric and operational designs. Therefore, they cannot be used for traffic circles, especially for the unconventional ones with unusual geometric and operational characteristics1.

OBJECTIVES This paper describes the development of simulation models of Collingwood and Brooklawn circles in New Jersey for the morning peak period (7-9 a.m.). It should be emphasized that these circles are not roundabouts, but they are traffic circles with unusual operational and geometric designs. Namely, the priority rule that governs the roundabout traffic operations i.e. yield at entry rule does not always apply at these circles. Furthermore, the traffic movements into the circles are largely governed by the traffic signals located in the vicinity of the circles.

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At roundabouts, the approaching traffic yields the circulating traffic; whereas, traffic circles do not necessarily follow this rule.

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The effectiveness of traffic operations at circles and roundabouts are highly affected by the gap rejection/acceptance behavior of drivers. In this paper, it is shown that extensive ground truth data are necessary to realistically model vehicles' gap acceptance/rejection behavior. PARAMICS is selected as the microscopic simulation tool. It has the capability of microscopically modeling the vehicle-following and lane-changing behavior of individual vehicles. PARAMICS allows users to customize many features of the underlying simulation model through Application Programming Interface (API). Users can modify the default simulation routine and test their own models using PARAMICS API. Binary gap acceptance/rejection models are developed for each selected yield and stop controlled intersections at these circles based on the field data. These models are then tested using PARAMICS API. The outputs of the enhanced simulation models are compared to the default PARAMICS outputs. It is shown that the default capabilities of PARAMICS are not sufficient to build statistically valid models of these two unconventional traffic circles. This paper is intended to present the additional statistical effort of validating the model outputs and calibrating the input parameters to confirm the reliability of the model for the local conditions.

DECSRIPTION OF THE STUDY CIRCLES Collingwood Circle Figure 1 shows the Collingwood circle. It has a rather unusual geometric and operational design. In modern roundabouts, circulating traffic has the right-of-way. However, in the case of the Collingwood circle, the traffic flow on route 33 WB, route 34 NB and route 33/34 EB have the priority over the circulating traffic. There are 4 yield-control intersections and 1 cross one-way stop-control intersection in the circle. However, only 3 of them (as shown in Figure 1) are selected for validation purposes due to their significant roles in the efficiency of the circle. During the morning rush hour, the traffic flows heavily on 34 NB. Because the traffic on the 33 WB direction has the priority over 34 NB, the circle experiences a queue backup before the yield sign at location 3. This queue blocks the circulating traffic that comes from the 33/34 EB direction and waits for acceptable gaps at location 2. Moreover, the queue backup before location 2 blocks the traffic flow to the 34 SB direction. Another unconventional feature of the circle is the one-way stop controlled junction at location 1, where traffic comes to a full stop to exit the circle into 547 SB direction. Particularly during the afternoon rush hours when the flow on the 33/34 EB direction is high, drivers waiting for acceptable gaps at location 1 form a queue, which then builds up, and blocks the traffic flow to 33/34 WB direction. It should be noted that the circle links and approach links are two-lane wide except location 1. Brooklawn Circle Brooklawn circle is located at the intersection of HWY 130 and NJ 47 in Camden County in New Jersey. Besides these major routes, several other roads converge, making this circle a cross point of the northsound corridor in southern NJ. Brooklawn circle comprises two circles (East and West circles) as it can be seen in Figure 1. HWY 130, NJ47, Creek road, and Hannevig Avenue intersect at the East circle. The West circle is the extension of the East circle, where HWY 130 negotiates though both. New Broadway Avenue and HWY 130 intersect at the West circle. Neither of the circles have a common operational design. Priority movements in the circles do not follow modern roundabout operational restrictions. For example, HWY 130 NB approaching the East circle at location 1 and Broadway Avenue approaching the West circle at location 4 yield the circulating traffic; whereas, at other locations the circulating traffic yields to the approach traffic (See Figure 1).

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The majority of traffic is carried by HWY 130, NJ 47 and New Broadway road. During the morning peak period, the traffic flows towards northbound on HWY 130 at the East circle. During the afternoon peak period, the traffic regime is the opposite. Long backups at HWY 130 southbound at the East circle are observed.

West Circle

Collingwood Circle

Brooklawn Circle

East Circle

FIGURE 1. Geometric and operational design of the Collingwood and Brooklawn circles (2)

SIMULATION MODEL DEVELOPMENT IN PARAMICS Model Verification Model inputs related to infrastructure such as number of lanes, speed limits, signposting distances, barred turns; jug handles were collected during various site visits. The models are simulated at varying levels of traffic volumes and using different random number streams to observe if vehicles behave realistically (i.e. correct turns, accepting priority movements at junctions, proper lane changing, using correct lanes while exiting the roadway, etc). This step is crucial in detecting any obvious errors in the simulation model. Data Description Collingwood and Brooklawn circle traffic data were collected on October 24, 2003 and April 21, 2004 using a portable tower video surveillance system (POGO) with two dome cameras and two infrared cameras; and a portable mast with a Sony camcorder and an omni-directional camera; and two camcorders at circle approaches. The recorded traffic data were then extracted in Rutgers Intelligent Transportation Systems Laboratory. The extracted data include (1) vehicle counts per hour at each selected traffic count locations, (2) vehicle inter-arrival times at priority approaches, (3) vehicle wait

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times before yield signs at the selected locations, and (4) gap acceptance/rejection times at the selected locations. Model Validation and Calibration The efficiency of the study circle is directly related to (a) traffic volume levels in and around the circle, (b) interarrival times of vehicles at approach, and (c) vehicles' gap acceptance / rejection behavior. Gap acceptance/rejection data at locations 1, 2 and 3 at the Collingwood circle (See Figure 1) and locations 1, 2, 3, 4, and 5 at the Brooklawn circle (See Figure 1) are used for the analysis carried out in this paper. These locations are selected based on their effect on the overall operational efficiency of these circles. Origin-Destination (O-D) Matrix Origin-destination matrices for each hour are extracted by using the observed traffic volumes at the selected locations of the circle. These locations are depicted by Roman letters in Figure 1. There are various methods for estimating time-dependent O-D matrices for traffic networks. See (1) for a detailed survey of these methods. However, the study networks modeled here are relatively smaller and most O-D pairs have a single connecting path. Thus, the trial and error method is sufficient to determine an accurate O-D matrix for the study network. The hourly counts obtained from independent simulation runs are assumed to follow Gaussian distribution. The simulation models are then run with different random number seeds. At each location, the hourly traffic counts observed from each replication are independent and identically distributed. Thus, the sample mean is an unbiased estimate of the true mean. Since the variance of the traffic volume is unknown, Student t-distribution is used to find a confidence interval of the mean. The trial-and-error approach of determining the O-D matrix can be summarized as follows: 1. Start with an initial O-D matrix M 2. Run the simulation model with matrix M for n number of seeds 3. Construct a 95% confidence interval. on the traffic count at location i, xi ± t0.95 si 2 n , where x i is

2 the average of traffic counts and si is the sample variance at location i.

4. If the confidence interval covers the observed count at location i - stop. Otherwise update matrix M and go to step 1. At each simulation run, the traffic counts at the selected locations are obtained from the simulation. The O-D matrix is updated until the 95% confidence interval of the traffic volume at each location covers the observed traffic count.

Interarrival Times at Approaches

The histograms of the interarrival data of the priority approaches are plotted. (In Figure 1, these approaches correspond to the traffic flows depicted by II, VI and IX in the Collingwood circle, and by II, X and XIV in the Brooklawn circle). The interarrival time histograms at the approaches resemble exponential probability distribution, however with a rather longer tail. Chi-square and Kolmogorov-Smirnoff tests are used to test the goodness-of-fit of the data. However, the Ho hypothesis is rejected even at %10 level of significance. It is then realized that the inter-arrival times of vehicles approaching the circle are largely affected by the traffic signals located in the vicinity of these circles. Vehicles arrive at the circles in batches and in a

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certain order. Their arrival times are governed by the timing plans of the upstream traffic signals and it is not possible to model vehicle arrivals using regular statistical distributions. Therefore, the modeling of the upstream traffic signals is needed to capture batch arrivals. The actual signal timing plans of the two signalized intersections around the Collingwood circle and the six signalized intersections around the Brooklawn circle are then coded by using actuated signals feature in PARAMICS. The simulation models of the Collingwood and Brooklawn circles are shown in Figure 2

Collingwood Circle

Brooklawn Circle

FIGURE 2. Collingwood and Brooklawns Circle simulation models developed in PARAMICS

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Gap Acceptance/Rejection Models

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The gap acceptance behavior of vehicles at uncontrolled intersections is investigated using binary choice model. There are two choices available to the driver waiting for an acceptable gap: (a) accept the gap and enter to the intersection, or (b) reject the gap and wait for the next available gap. To estimate the gap acceptance probability, a binary probit model is considered. The utility obtained from accepting/rejecting a gap can be decomposed into observed and unobserved parts (4).

U nj = Vnj +

nj

(1)

where, n = Index for driver, j = Index for gap acceptance behavior (1 if gap is accepted and 0 if gap is rejected), Unj= Total utility function, Vnj = Observed part of the utility function, nj= Unobserved part of the utility function (random error components) In binary probit model n is assumed to be normally distributed with a mean vector of zero, and a covariance matrix " " which can depend on variables faced by driver n. Then, the probability of accepting a gap can be formulated as (5):

Pn1 = Prob(Vn1 +

Pn 1 =

n1

> Vn0 +

V n0 )

n0

)

(2) (3)

[(V n1

]

where, Pn1=Probability of driver n accepts the available gap and =Standardized cumulative normal distribution The variables of the probit model are (a) Accept : Dummy variable for acceptance behavior (1 if the gap is accepted, 0 otherwise) and (b) Gap: Time between consecutive vehicles at the approach (milliseconds). Gap acceptance/rejection data are extracted in the form of time stamps of vehicle actions at each yield or stop controlled junctions. As a vehicle in the minor traffic stream arrives at a yield or stop sign, its time is recorded using a simple computer script. The same is repeated as the vehicles in the major traffic stream pass the location, and the vehicle in the minor traffic stream leaves the location. Different keys are assigned to each of these vehicle actions. A MATLAB code is developed to read the data in this format and extract each vehicle's acceptance and rejection time, wait time, waiting lane index, approaching lane index, etc. The estimation of the probit model using the extracted data is performed in STATA statistical package (6). Table 1 demonstrates the results and the model performance of the binary probit models for the Collingwood and Brooklawn circles, respectively. The number of observations for each location is also given in Table 1. It should be mentioned that the probability of accepting the null hypothesis that the parameter coefficient is equal to zero is less than 0.01 for all the coefficients in Table 1.

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TABLE 1. The results of the binary probit model during the morning peak period COLLINGWOOD CIRCLE Location 1 Variable Constant ( ) Gap (g) Constant Gap Constant Gap Constant Gap Constant Gap Coefficient -3.25858 0.00588 -2.52357 0.00596 -2.76297 0.00454 -2.85311 0.00829 -3.96964 0.00897 Number of Observations 1170 Std. Err. 0.1664 0.0003 0.1410 0.0004 0.1209 0.0003 0.2489 0.0009 0.2655 0.0007 z -19.58 17.71 -17.90 12.95 -22.85 16.48 -11.46 8.85 12.41 -14.95 Log Likelihood -204.4 Pseudo R2 0.68

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Vn1 = 3.25858 + 0.00588.g

2 - Inner Ln 806 -171.6 0.61

Vn1 = 2.52357 + 0.00596.g

2 - Outer Ln 1372 -188.7 0.70

Vn1 = 2.76297 + 0.00454.g

3 - Inner Ln 363 -79.5 0.59

Vn1 = 2.85311 + 0.00829.g

3 - Outer Ln 1064 -99.0 0.77

Vn1 = 3.96964 + 0.00897.g

BROOKLAWN CIRCLE Location 1 Variable Constant ( ) Gap (g) Constant Gap Constant Gap Constant Gap Constant Gap Constant Gap Coefficient -1.35782 0.00657 -1.77330 0.00648 -1.47133 0.00699 -2.66259 0.00826 -3.33663 0.00949 -2.82885 0.00629 Number of Observations 2027 Std. Err. 0.1134 0.0004 0.1170 0.0004 0.0766 0.0003 0.1082 0.0004 0.2361 0.0007 0.1731 0.006 z -11.97 15.87 -15.16 15.11 -19.20 20.73 -24.61 21.66 -14.13 13.81 -14.48 12.13 Log Likelihood -339.2 R2 0.58

Vn1 = 1.35782 + 0.00657.g

2 1158 -368.8 0.51

Vn1 = 1.77330 + 0.00648.g

3- Inner Ln 1785 -834.8 0.32

Vn1 = 1.47133 + 0.00699.g

3- Outer Ln 1642 -557.6 0.47

Vn1 = 2.66259 + 0.00826.g

4 962 -159.4 0.74

Vn1 = 3.33663 + 0.00949.g

5 574 -206.1 0.43

Vn1 = 2.82885 + 0.00629.g

Note 1: Log-likelihood values represent the likelihood value calculated at convergence. Note 2: Inner lane of the minor stream vehicle refers to the lane that is closer to the circle

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It should be mentioned that during the data extraction, it was often difficult to discern the lane indices of vehicles at yield signs and at the approaches for certain locations. There were other variables that were incorporated in the probit model such as the approach lane index, wait time, the frequency of approach vehicles. However, these variables were not statistically significant even at 10% significance level during the morning peak period. Mahmassani and Sheffi (7), Polus et al.(8) , Hamed et al. (9), Shiftan et al. (10) presented gap acceptance models with significant impact of waiting times on vehicles' critical gaps. Namely, their estimation results shows that drivers' average waiting times reduces the average critical gap. This result, however, is not observed in this present study. Teply et al. (12, 13) reported that complex gap acceptance models with several variables present a better descriptive and predictive power of capturing the drivers' behavior. They developed a binary logit model, which considers the nature of opposing traffic such as the type, time gap, space gap, speed of approaching vehicles, and minor traffic characteristics such as queue delay, front delay, and also driver age. They conclude that gap value only alone can yield reasonable approximations in modeling drivers' entry behavior for uncontrolled intersections. The probit models in this study are based on an exhaustive dataset as shown in Table 1. For the probit models developed for both circles, 12,923 gap data points were used. Note that this dataset size includes both rejected and accepted gaps. Also note that, for the probit estimation model only the logical gaps are taken into consideration. For example, it is obvious that the probability of accepting a gap of 0.1 second is 0.0. Likewise, the probability of accepting a gap of 15 seconds is 1.0. Therefore, only the gaps that can provide information on how drivers will behave were taken into consideration. This approach was also adopted by Gattis and Low (11). The dataset is far more extensive than most of the ones employed in the literature. For instance, Mahmassani and Sheffi (7) and Daganzo (14) used the same dataset of 406 data points for their probit models; Hamed et al.(9) employed 592 data points for developing a binary probit model; Shiftan et al. (10) used a dataset of 743 observations to develop a logit model of critical gaps at roundabouts. Teply et al. (12,13) used a dataset of 8,652 gap values with the above-mentioned variables.

PARAMICS API

PARAMICS API enables users to simulate the developed gap acceptance/rejection model. Basically, at each time step if the vehicle is within the link that has a yield (such as locations 2 or 3 in Figure 1 for both circles) or a stop sign (such as location 1 at the Collingwood circle Figure 1), the model checks the approach link associated with that sign. It then detects the leading vehicle on the approach link and calculates the approximate time, g, it would take the approaching vehicle to arrive at that location. Thus, for every approaching vehicle the model calculates the probability of accepting the gap g at each location at each simulation time step. Figure 3 demonstrates the flowchart of the implementation of the gap acceptance models in PARAMICS API. There is abundant number of studies in the literature that focus on gap acceptance models and critical gaps at uncontrolled intersections and at roundabouts. (See 4, 9, 10, 11, 12, 13 and 14). However, to the best of our knowledge, there are no guidelines on how to implement these models using microscopic simulation tools. For example, at location 2 at the Brooklawn circle, if the estimated gap is 3 seconds (300 milliseconds), then the Vn1 value is estimated as 0.171 using the model parameters given in Table 1 (i.e. the probability of accepting the gap with respect to rejecting). Using the standard normal table this value corresponds to an acceptance probability of 0.568. As mentioned above, there are no guidelines as to how to drive the simulation using this probability value. The analysis of vehicles' gap rejection times show that rejected gaps follow a negative exponential probability distribution and lognormal probability distribution at the selected locations of the Collingwood circle and the Brooklawn circle, respectively. Table 2 shows the chi-square goodness of fit test results of the rejected gaps at the selected locations. Using the inverse of the cumulative gap rejection distribution at each location, a critical value tc

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(seconds) is assigned using the probability value obtained from the binary probit model. If g < tc , the vehicle rejects the gap, accepts otherwise.

FIGURE 3. PARAMICS API Gap acceptance/ rejection flowchart

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TABLE 2. The results of Chi-Square Goodness of Fit Tests of rejected gaps Location Number of Observations 890 615 1134 281 914 Number of Observations 280 420 803 1091 320 Chi-Square* (Observed)

2 9,0.90 2 9,0.90 2 9,0.90 2 9,0.90 2 9,0.90

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c.d.f. parameters

F ( x) = 1 exp x

1 2 - Inner

Collingwood Circle

2- Outer 3- Inner 3- Outer

Location

= 9.60 = 5.60 = 4.2 = 5.34 = 7.29

= 3.00, = 1.75, = 2.00, = 1.10, = 1.10,

f ( x) = 1 x 2

2

= 1.336 = 0.697 = 0.737 = 0.899 = 1.027

exp (ln x µ )2 2 2

Chi-Square* (Observed)

p.d.f. parameters

1 2

Brooklawn Circle

3-Inner 3-Outer 4

= 0.504 5 386 Note: Inner lane of the approaching vehicle refers to the lane that is closer to the vehicle at the minor stream.

VALIDATION RESULTS

= 6.52 2 9,0.90 = 12.96 2 9,0.90 = 8.66 2 9,0.90 = 9.35 2 12,0.90 = 5.69 2 12,0.90 = 17.91

2 9,0.90

µ = 0.249, µ = 0.438, µ = 0.164, µ = 0.121, µ = 0.567, µ = 0.513,

= 0.552 = 0.598 = 0.626 = 0.659 = 0.495

Average wait times and average interarrival times at the selected locations are used as output variables for the simulation model validation. As a direct result of the central limit theorem, it is known that the sampling distributions of the output variables follow a Gaussian distribution with unknown variances regardless of their initial distributions. The model is simulated with independent replications until model outputs attain a confidence level of 95% with a relative error of 5% for all the selected locations. The obtained values of the selected output variables are then used to construct the 95% confidence interval for the population mean using a Student t-distribution. Table 3 presents the simulated and the observed system outputs for the morning peak period for both study circles. Table 3 shows that some of the output ranges do not cover the corresponding observed output values. However, it should be pointed out that the output collection of PARAMICS and the analyst's data extraction methods are never identical. Furthermore, it can be seen in Table 3 that none of the simulated output values are substantially out of range. Therefore, the outputs of the simulation model can be assumed valid.

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TABLE 3. Validation Results - PARAMICS Output Location Average Wait Time (Seconds) Paramics Observed Paramics API Default [7.15-7.91] 7.64 [3.05-3.34] [6.00-6.28] 6.20 [2.95-3.22] [2.52-2.63] 2.51 [1.83-1.96] [1.17-1.22] 1.27 [1.01-1.10] [1.79-1.98] 1.74 [0.66-0.71] [2.97-3.08] 3.08 [2.58-2.86] [2.23-2.49] 2.05 [0.91-1.00] [2.47-2.70] 2.55 [1.42-1.48] Average Interarrival Time (Seconds) Simulated Observed [3.15-3.26] [2.75-2.84] [3.26-3.38] [11.80-12.57] [4.58-4.75] [2.86-2.93] [6.03-6.25] [4.55-4.68] 3.28 2.60 3.42 12.26 4.94 2.79 6.13 4.56

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Collingwood Circle Brooklawn Circle

1 2 3 1 2 3 4 5

Table 3 also presents the average wait time variable obtained using the default PARAMICS gap acceptance/rejection model. It can be seen that the wait time as obtained by the default PARAMICS gap acceptance/rejection model, is considerably lower than the actual wait time observed at the field. It is seen in Table 3 that the difference in accuracy between the default PARAMICS and the enhanced model results vary considerably at locations 1 and 2 at the Collingwood circle locations 2 and 4 at the Brooklawn circle. Though the difference in wait time averages (1-4 seconds) seem insignificant, when the high traffic volumes are considered at these locations, even small differences in wait times per vehicle has a substantial effect on the network performance. It should also be underlined that, the effect of the enhanced model is not only the accuracy in the output, but also the realistic vehicle crossings at these uncontrolled junctions. Figure 4 shows the comparison of the cumulative gap acceptance frequencies obtained from the default PARAMICS output and the PARAMICS API output with respect to the observed cumulative gap acceptance frequency at location 1 of the Collingwood circle. Considerable difference between the observed average wait time and the default PARAMICS outputs result as shown in Table 3 is due to the high frequency of gap acceptance values by the default PARAMICS gap acceptance model within the range of 2-4 seconds.

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FIGURE 4. Comparison of cumulative gap acceptance frequencies at the Collingwood circle location 1 Sensitivity Analysis In experimental design, the input parameters are called factors, and the output performance measures are called responses (15). A 2k factorial design test is used in this paper to assess the stability of the developed model. Sensitivity of the gap rejection parameters at the selected locations are tested based on the average network travel time (response). Average network travel time is simply the sum of travel times of all vehicles in the network during the peak period divided by the total number of vehicles. This output measure is employed because it provides a better overall mobility measure than other output measures. Here, it is observed whether the model is sensitive to the (a) means of the gap rejection probability distributions of location 2 and 3 of the Collingwood circle and location 1, 3 of the Brooklawn circle as given in Table 2 and (b) target mean reaction time of vehicles2. The choice of these factors is due to their major impact on the efficiency of these circles.

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Target mean reaction time is the mean reaction time of all vehicles in the system throughout the simulation duration. PARAMICS generate vehicles with different reaction times to meet this criterion

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TABLE 4. Design Matrix for a 23 factorial design for the morning peak period Factor Combination 1 2 3 4 5 6 7 8 COLLINGWOOD CIRCLE Location 4 Location 5 Reaction Time + + + + + + + + + + + + BROOKLAWN CIRCLE Location 1 Location 3 Reaction Time Average Network Travel Time 177.52 seconds 176.32 174.78 171.75 176.05 173.73 173.75 171.28

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Factor Average Network Combination Travel Time 1 + + + 154.0 seconds 2 + + 151.4 3 + + 150.5 4 + 144.4 5 + + 155.0 6 + 146.8 7 + 152.5 8 145.6 Note: + and signs stand for +0.2 and 0.2 difference in the mean value ( ) of the negative exponential distribution, +0.70 and 0.70 difference in the mean value ( µ ) of the lognormal distribution, and +0.5 and 0.3 seconds difference in the mean reaction time of vehicles. Table 4 shows the levels of each input factor and the response values for the Collingwood and Brooklawn circles during the morning peak period. From these results, a 95% confidence interval with a relative error of 1% can be calculated as [172.3 176.2 seconds]. Similarly, a 95% confidence interval with a relative error of 2% can be calculated as [146.7 153.3 seconds]. It can therefore be concluded that the response does not vary drastically with minor changes in the gap rejection model. Thus, the simulation model of these circles can be assumed stable to minor changes in the input.

SUMMARY AND CONCLUSIONS In this study, the steps of model development and validation of Collingwood and Brooklawn traffic circles for the morning peak hours are described in detail. These modeled facilities shown in Figure 1 are not roundabouts, but traffic circles with unusual geometric and operational designs. Therefore, commonly used deterministic roundabout analysis models such as RODEL, aaSIDRA, HCS are not applicable for modeling of these traffic facilities, where the priority movements and geometric characteristics are different than those of regular roundabouts. PARAMICS is one of the few simulation software packages that can model and analyze such unconventional traffic circles. The use of a microscopic simulation tool offers various analysis capabilities. Instead of analyzing the system as a series of approaches, the system can be evaluated as a complete network. For instance, the signalized and unsignalized intersections located in the vicinity of the circle can be included in the system, and the arrival patterns of the vehicles into the circles can be modeled realistically. The impact of various changes to the operational and

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geometric designs can be evaluated using vehicle-by-vehicle data not only at the traffic circle, but also at the network level. Although microscopic simulation tools offer a variety of input-output analysis options, the analyst should still assess and ensure the validity of the model input parameters. Every traffic facility has distinct characteristics whether it is a traffic circle, roundabout, an intersection or a freeway corridor. Extensive validation/calibration efforts might be required when the system is evaluated by using either deterministic or stochastic tools. The extent of these efforts depends on the characteristics of the facility and the scope of the analysis. Gap acceptance/rejection behavior of vehicles in the study traffic circles is found to have a dominating impact on the performance of the developed simulation models. It is observed that the default gap acceptance/rejection behavior model of PARAMICS fails to simulate the study circles accurately. Extensive field data are required to represent site-specific gap acceptance/rejection behavior of drivers. Binary probit models of gap acceptance/rejection decisions based on field data are developed, and implemented in PARAMICS API (See Table 1). The differences between the system output values of the API enhanced gap acceptance/rejection model and the default PARAMICS model are presented in Table 3 and Figure 4. It is shown that the default PARAMICS underestimates the allowable gap time of vehicle at the selected minor streams. The same approach of improving the model input variables based on field data can be applied to car following (see (16), (17)) or lane changing, merging behavior of vehicles for a given study network depending on the specific objectives of why the system is simulated. Also the stability of the simulation models are tested using 2k factorial design, where the change in the system performance is tested based on minor changes to the calibrated input parameters. The developed simulation models can be further improved by incorporating the follow-up times at the yield or stop controlled junctions. The follow-up time is defined as the time gap between the two cars of the minor stream being queued and entering the same major stream gap one behind the other. The extraction of this data is possible using the already available collected field data. Location specific followup time models can thus be estimated using this data. As in the gap acceptance/rejection models, PARAMICS API can be used to simulate follow-up time models at each selected locations. Finally, the future validation/calibration works will follow non-parametric statistical methods, where not only the mean of the output data, but also the output distribution as a whole can be compared. REFERENCES 1. National Cooperative Highway Research Program Report, "NCHRP 3-65: Applying Roundabouts in the United States," 2004. 2. Delaware Valley Regional Planning Commission (2002). "US 130 Brooklawn Circles Concept Development Report." 3. Abrahamsson, T. (1998). "Estimation of origin-destination matrices using traffic counts A literature survey". International Institute for Applied Systems Analysis, Interim Report. 4. Train K. "Discrete Choice Methods with Simulation", Cambridge University Press, 2003. 5. Ben-Akiva, M., Lerman S., "Discrete Choice Analysis: Theory and Application to Travel Demand", MIT Press, Cambridge Mass., 1985. 6. STATA Software Website (2005) http://www.stata.com/products/overview.html 7. Mahmassani, H. and Sheffi, Y. (1981). "Using gap accepance sequences to estimate gap acceptance functions," Transportation Research Part B, Vol 15B, pp. 143-148. 8. Polus, A., Lazar, S. S. and Livneh, M. (2003), "Critical gap as a function of waiting time in determining roundabout capacity." Journal of Transportation Engineering. September/October 9. Hamed, M.M., Easa, S. M. and Batayneh, R.R. (1997). "Disaggregate gap-acceptance model for unsignalized T-intersections" Journal of Transportation Engineering, January/February.

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10. Shiftan, Y., Polus, A. and Lazar-Shmueli, S. (2003). "Evaluation of the waiting-time effect on critical gaps at roundabouts by a logit model," Presented at the 82nd Annual Transportation Research Meeting, Washington D.C. 11. Gattis, J. L. and Low, S., (1999). "Gap acceptance at atypical stop-controlled intersections." Journal of Transportation Engineering. May/June. 12. Teply, S., Abou-Henaidy, M. and Hunt, J. D. (1997) "Gap acceptance behavior aggregate and logit perspectives: Part 1." Traffic Engineering and Control. Vol. 37. No.9, pp. 474-482. 13. Teply, S., Abou-Henaidy, M. and Hunt, J. D. (1997) "Gap acceptance behavior aggregate and logit perspectives: Part 2." Traffic Engineering and Control. Vol. 37. No.9, pp. 544-544. 14. Daganzo, C. (1981). "Estimation of gap acceptance parameters within and across the population from direct roadside observation" Transportation Research Part B, Vol 15B, pp. 1-15. 15. Law, A. M. and Kelton, W. D., (1991). Simulation Modeling and Analysis. 2nd Edition, McGrawHill, Inc. 16. Benekohal, R. F. and Treiterer, J. (1989). "CARSIM: Car-Following Model for Simulation of Traffic in Normal and Stop-and-Go Conditions," Transportation Research Record 1194, pp. 99111. 17. Wu, J., Brackstone, M. and McDonald, M. (2003). "The validation of a microscopic simulation model: a methodological case study," Transportation Research Part C, vol. 11, pp. 463-479.

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##### Modeling and Simulation of Unconventional Traffic Circles

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