CALIBRATION PROCEDURE FOR GIPPS CAR-FOLLOWING MODEL Hesham Rakha (Corresponding Author) Charles Via Department of Civil and Environmental Engineering Virginia Tech Transportation Institute 35 Transportation Research Plaza (536) Blacksurg, VA 2461 Phone: (54) 231-155 Fax: (54) 231-1555 E-mail: hrakha@vt.edu Caroline Cavagni Pecker Laoratório de Sistemas de Transportes Universidade Federal do Rio Grande do Sul Praça Argentina, no. 9 sala 42 Porto Alegre, Brazil 94-2 Phone: +55 51 3316-44 Fax: +55 51 3316-47 E-mail: carol@producao.ufrgs.r Helena Beatriz Bettella Cyis Laoratório de Sistemas de Transportes Universidade Federal do Rio Grande do Sul E-mail: helenac@producao.ufrgs.r Word count: 432 (text) + 325 (tales and figures) = 7,552 (total) Revised paper sumitted for pulication in the Transportation Research Record. March, 3 th 27
Rakha, Pecker, and Cyis 2 ABSTRACT This paper presents a methodology for calirating the car-following model proposed y Gipps. This caliration procedure is concerned with steady-state conditions. Steady-state occurs when the leader and follower vehicles travel at very similar and near constant speeds, maintaining similar space headways etween each other. The steady-state caliration is important ecause it determines the roadway capacity, the speed-at-capacity, and jam density (spatial extent of queues when fully stopped). The caliration process allows the identification of adequate values for deceleration rates, (maximum deceleration rate the driver is willing to use), (the maximum deceleration rate estimated for the leader), and driver reaction time (T). Two different ehavior assumptions were developed concerning the deceleration rates: (i) equal to, and (ii) different from. The first hypothesis assumes the leader will e as aggressive as the follower. The second assumes that the driver considers his/her leader to have a different maximum deceleration rate. The results are presented in tales and graphs that correlate the car-following parameters to the fundamental traffic stream variales for the different ehavior assumptions. The procedures are then tested on sample field data to demonstrate the adequacy of the caliration procedures.
Rakha, Pecker, and Cyis 3 INTRODUCTION Traffic stream motion can e formulated either as discrete entities or as continuous compressile fluid. Microscopic car-following models, which form the discrete entity approach, characterize the relationship etween a vehicle s desired speed and the distance headway (h) to its preceding vehicle in the same lane. Since the first car-following theory was presented half a century ago, many models have een developed. Among others, the most studied are the Gazis-Herman- Rothery (GM) model and the collision avoidance formulation (1). A detailed description of the various car-following model formulations is eyond the scope of this paper ut can e found elsewhere in the literature (1). Alternatively, macroscopic traffic models descrie the motion of the traffic stream y approximating the flow to a continuous compressile fluid. These models relate three traffic stream variales, flow rate (q), density (k), and space-mean-speed (ū). The early models assumed flow condition as a single regime phenomenon. Some of these models were developed y Greenshields, in 1935, Greenerg, in 1958, and Underwood, in 1961. Also in 1961, Edie proposed the first multiregime model using Underwood and Greenerg formulations for free-flow and congested conditions, respectively (2). Macroscopic and microscopic traffic models can e related to each other, ased on the relationship etween the respective approach macroscopic (flow, density, and space-mean-speed) and microscopic (headway, spacing, and individual vehicle speeds) parameters. In such case, the sum of successive time headways (s n ) within a long analysis period can e approximated to its duration (T ), as demonstrated in Equation [1]. Therefore, the flow rate (q) can e expressed as the inverse of the average vehicle time headway. In the same way, the traffic density (k) can e approximated for the inverse of the average distance headway (h n ) for all vehicles within a section of roadway of length (L), as demonstrated in Equation [2]. T N sn n= 1 N 1 T 1 1 n N N n= 1 q = s q s L N hn n= 1 N 1 L 1 1 n N N n= 1 k = h k h Microscopic traffic simulation uses car-following models to capture the interaction among vehicles traveling in the same lane. The car-following process can e modeled y an equation of motion for steady-state conditions and two additional constraints that govern the ehavior of vehicles moving from one steady-state to another (decelerating and accelerating). The first constraint governs the vehicle acceleration ehavior, which is typically a function of the vehicle dynamics. The second constraint ensures that vehicles maintain a safe position relative to the lead vehicle. The caliration of car-following ehavior within a microscopic simulation model can e viewed as a two-step process. The first step calirates the steady-state condition. The steady-state caliration has a crucial importance ecause it imposes the roadway capacity and determines the speed-at-capacity and the jam density (spatial extent of queues when fully stopped). [1] [2]
Rakha, Pecker, and Cyis 4 The second step is the non-steady-state caliration. Non-steady-state ehavior influences how vehicles move from one steady-state to another. It captures the capacity reduction that results from traffic reakdown and the loss of capacity during the first few seconds as vehicles depart at traffic lights, for instance, usually known as the start loss. Under certain circumstances, the nonsteady-state ehavior can influence steady-state ehavior. For example, vehicle dynamics may prevent a vehicle from attaining steady-state conditions (a desired free-flow speed). A typical example of such a case is the motion of a truck along a significant upgrade section when the truck is unale to attain its desired free-flow speed. This paper presents a methodology for calirating the car-following model proposed y Gipps (3). Gipps formulation is an important collision avoidance model which has een incorporated to several microscopic simulation models, such as AIMSUN (4), SISTM (5) and DRACULA (6). This caliration process is concerned with the steady-state condition. Steady-state occurs when the leader and follower vehicles travel at very similar and constant speeds, maintaining similar space headways etween each other (7). The process is performed for a single lane of unidirectional traffic, assuming that all vehicles have similar parameters and all drivers have similar characteristics. Section 2 reviews the details of Gipps model, descriing the constraints (acceleration/deceleration) and parameters. In section 3, an analysis of the Gipps car-following model for steady-state conditions is presented. The collision avoidance equation is descried as a macroscopic traffic stream model and the analysis of steady-state is performed. The caliration of two different hypotheses of steady-state ehavior is presented in Sections 4 and 5. In Section 6 the caliration procedures that were developed in Sections 4 and 5 are applied to field data. Finally, Section 7 presents the conclusions of the study and recommendations for further work. OVERVIEW OF GIPPS CAR FOLLOWING MODEL This section descries the Gipps car-following formulation. Prior to descriing the model formulation, the set of notations and their definitions are summarized in Figure 1 that illustrates two vehicles moving from left to right. Vehicle n-1 is the leader with a length of L n-1 while vehicle n is the follower. The vehicles positions, speeds, and acceleration rates at time t are denoted as x(t), u(t), and a(t), respectively. The distance headway etween vehicles n and n-1 is denoted as h n (t). According to Gipps, the speed of the follower vehicle is controlled y three conditions. The first condition ensures that the vehicle does not exceed its desired speed (U n ). The second condition ensures that the vehicle accelerates to its desired speed with an acceleration rate that initially increases with speed and then decreases to zero as the vehicle approaches its desired speed. The comination of these conditions results in Equation [3] which controls the vehicle acceleration while vehicles are distant from each other (free-flow ehavior). The equation coefficients were otained from fitting a curve to field data collected on a road of moderate traffic. un() t un() t un( t + T) = un( t) + 2.5 ant(1 ).25 U + U [3] n n where u n (t) is the speed of vehicle n at time t (m/s); a n is the maximum desired acceleration rate of vehicle n (m/s 2 ); T it the driver s reaction time (s); and U n is the desired speed of vehicle n (m/s). In a constrained traffic situation, when vehicles are traveling close to each other, the third condition ecomes dominant and controls the ehavior of the follower vehicle while decelerating. The speed
Rakha, Pecker, and Cyis 5 of the follower vehicle (see Equation [4]) is affected y the driver reaction time, the speed and distance difference etween leader and follower, and the deceleration rates they are willing to employ. Gipps pointed out that a safety margin should e added to the driver s reaction time. The safety margin would assure the vehicle s aility to stop even when there is a delay to initiate its reaction for some reason. For the purpose of this study, the safety margin is constant and equivalent to T/2 (reaction time divided y two). This safety value is implicit in Equation [4]. 2 2 u un( t + T) = T + T + 2 [ xn 1( t) Ln 1 xn( t) ] un( t) T + 2 () t where and are deceleration parameters of vehicle n (m/s 2 ); is the actual most severe deceleration rate the vehicle is willing to employ in order to avoid a collision; and is the estimated most severe deceleration rate the leader vehicle is willing to employ. It is an estimated value ecause it is impossile for the follower to evaluate the real intention of his/her leader; L n-1 is the effective length of vehicle n-1 (the actual length plus a safety margin); x n (t) is the position of vehicle n at time t; x n-1 (t) is the position of the preceding vehicle at time t; and u n-1 (t) is the speed of the preceding vehicle (m/s). The parameters related to deceleration rates ( and ) are very important for the raking process modeling. These parameters influence the distance headway etween follower and leader vehicles and thus affect the lane capacity. Assuming the vehicles will travel as close to their desired speed as possile and considering the dynamics limitations, the speed of vehicle n at time t + T can e computed as un() t un() t un( t) + 2.5 ant(1 ).25 +, Un U n un ( t + T) = min [5] 2 2 2 un 1() t - T + T + {2 [ xn 1( t) Ln 1 xn( t) ] un( t) T + } According to the aove formulation, once the road is unconstrained and the space headways etween the vehicles are large enough to allow them to travel at their desired speed, the first argument of Equation [5] is applied. In this case, the following vehicle is ale to accelerate according to the empirical equation of vehicle dynamics. Alternatively, in congested conditions, where short headways are typical, the second argument of Equation [5] is applied. In such a case, the speed is limited y the leader vehicle performance. Each vehicle estalishes its speed in order to avoid a collision ased on the assumption that the leader deceleration rate will not exceed. The following section presents the proposed caliration procedures for the Gipps carfollowing model for steady-state conditions. GIPPS CAR-FOLLOWING MODEL IN STATIONARY STATE (STEADY-STATE) A detailed mathematical analysis of Gipps car-following model under steady-state conditions is presented in the literature (8). Consequently, the paper will only summarize the major findings of the Wilson study and present a caliration procedure of the model. In his study, Wilson (8) presented a mathematical analysis of simplified scenarios and identified parameter regimes that deserve further investigation. The paper also showed the derivation of uniform flow solutions (steady-state) and speed-headway functions under simplifying conditions concerning parameters,, and T, and an analysis of the linear staility of the uniform flow, identifying stale and nonstale flow regimes. n 1 [4]
Rakha, Pecker, and Cyis 6 However, in order to enhance the practice of traffic modeling it is desirale to relate the model parameters to some measurale traffic stream parameters. The present work aims to estalish mathematical relationships etween Gipps parameters (, and T) and the fundamental traffic variales (free-flow speed, speed-at-capacity, capacity, and jam density). The main assumptions considered to validate the mathematical relationships are (i) traffic flows on a single lane highway (multilane scenarios with lane-changing effects are not considered) and (ii) characteristics of all drivers and vehicles are similar. Once steady-state conditions are achieved (characterized y similar headways etween vehicles traveling at similar speeds), the following equity is true u ( t + T) = u () t = u () t = u, [6] n n n 1 where ū is defined as the space-mean-speed of the traffic stream of vehicles. This equation holds ecause all vehicles are traveling at near equal constant speeds, and thus the time lag T has no impact on the model. Wilson applied the aove statement into Equation [4] to yield the following quadratic equation: ( 1) u 2 3Tu + 2( xn 1() t Ln 1 xn()) t = [7] The microscopic headway etween vehicles n and n-1, h n (t) is no longer time dependent and thus ecomes constant, and denoted y h. Furthermore, Gipps formulation considers that a safety margin is included in the vehicle length (L n-1 ) in order to ensure that vehicles do not invade other vehicle space. Consequently, L n-1 can e defined as the minimum space headway etween two vehicles (h j ) and, according to Equation [2], is the inverse of the jam density. Equation [7] then reverts to ( 1) u 2 3Tu 2 ( h hj ) + = [8] Equation [8] can then e simplified to the following equation if we assume that the traffic stream speed u is in km/h ( 1 ) 1 1 2 2 h = hj + Tu + u = a1 + a2u + a3u 2.4 25.92. [9] 1 1 1 where: a = h = ; a = T; a = ( 1 k 2.4 25.92 ) 1 j 2 3 j Wilson demonstrated that using Equation [9] it is possile to develop two different hypotheses concerning and. Section 4 presents the solutions for equal to. This condition underlines the hypothesis that the driver assumes his leader will e identical in the level of aggressiveness. Section 5, on the other hand, shows the solutions when is different from. In this case, the driver considers his leader may have a different maximum deceleration rate. HYPOTHESIS OF EQUAL TO When is assumed to e equal to, the constant a3 reverts to zero and thus Equation [9] is simplified to 2.4 u = ( h h j ). [1] T
Rakha, Pecker, and Cyis 7 Under these conditions, the Gipps model presented in Equation [9] reverts to the Pipes model (9), which is a stimulus-response model. The vehicle speed is the result of a stimulus weighed y a sensitivity factor. An earlier study (7) demonstrated that the sensitivity parameter can e related to the traffic variales (capacity, jam density, and free-flow speed), as 1 1 u = ( h hj ), [11] 1 1 1 q k u c j f where q c is the roadway capacity (veh/h/lane), k j is the jam density (veh/km/lane), and u f is the traffic stream free-flow speed. The space headway acts as the stimulus and the fundamental traffic variales (q c, k j and u f ) are used to compute the sensitivity factor. Considering the analogy etween the simplified version of the Gipps model (Equation [1]) and Pipes (Equation [11]) the reaction time in units of seconds (T) can e computed as 1 1 T = 24 q k u. [12] c j f Equation [12] provides a unique formula that can e used to calirate the driver reaction time using desired macroscopic traffic stream parameters without the need to gather microscopic carfollowing data. This formula is the first contriution of the paper. For example, considering a freeway facility with a lane capacity of 24 veh/h/lane, a free-flow speed of 1 km/h, and a jam density of 1 veh/km/lane, this would correspond to a driver reaction time of.76 s. Alternatively, the modeler would need to code a driver reaction time of.76 s in order to model the aove identified macroscopic parameters. Tale 1 demonstrates the variation in roadway capacity (veh/h/lane) as a function of the driver reaction time parameter for different roadway free-flow speeds and jam densities using Equation [12]. The capacities that exceed 23 veh/h/lane are highlighted to demonstrate the various trends. First, as the driver reaction time increases, the roadway capacity decreases. Furthermore, as the roadway free-flow speed and jam density parameters increase, the roadway capacity increases. Figures 2 and 3 demonstrate the influence of the equation parameters on the model ehavior. Figure 2 presents the model results for the following set of parameters: driver reaction time (T) equal to 1. s, a free-flow speed (u f ) equal to 8 km/h, and a minimum vehicle spacing of 7.14 m. Alternatively, Figure 3 illustrates the speed-flow-density curves considering different reaction times: a) T =.8s and ) T = 1.2s. Figure 2 shows that the Gipps model ( = ) reverts to the Pipes model shape. In the uncongested regime, speed is insensitive to traffic flow and density (see Figure 2a and 2) as is the case with the model states. The speed-headway relationship is linear (see Figure 2c) and the flowdensity curve has an inverted V shape (see Figure 2d). HYPOTHESIS OF DIFFERENT FROM Another possile assumption is the assignment of different values to the vehicle deceleration parameters ( ). The driver may estimate that his/her leader is either more aggressive or less aggressive than him/herself. The equation that corresponds to this situation is the resultant speed y solving Equation [8] to derive
Rakha, Pecker, and Cyis 8 u 8 1 3T 1 1 = ± + 2. [13] 2 9 ( 1 T ) ( h hj )( ) Considering different and, two separate cases are possile: (i) > or (ii) <. Based on the mathematical analysis from (8), the formulation for each case is derived. > When is greater than, the third parameter a 3 of the speed-headway (see Equation [9]) function is negative, which implies that the speed-headway curve is convex as opposed to concave. This ehavior is inconsistent with field data, as illustrated in Figure 4. Furthermore, Wilson (8) demonstrated that the speed-headway relationship may ecome unphysical and produce multiple solutions for some sets of parameters. In order to guarantee feasiility, the user must choose parameters that satisfy u f < 2.4 T ( 1). [14] In order to satisfy the condition of Equation [14] the ratio of to must e very close to 1. (e.g., 1.1), which means that the speed-headway function is near linear and thus would e similar to a Pipes model. Given the unrealistic ehavior of the model for such conditions and that these are not recommended within the commercial simulation software, this case is not discussed any further. < When is smaller than (which implies the leader willingness to decelerate is overestimated or that the vehicle characteristics are different), the model yields 8 1 3T u 1 1 = + + 2. [15] 2 9 ( 1 T ) ( h hj )( ) In this equation we have removed the negative solution generated y Equation [13] and only included the positive solution. Comining the uncongested and congested regimes the general formulation for the speed-headway relationship can e cast as 8 1 5.4T u min uf, 1 1 = + + 2 9 ( 1 T. [16] ) ( h hj )( ) Equation [16] is adjusted to generate the speed in units of km/h instead of m/s as is the case in Equation [15]. The final speed-headway is illustrated in Figure 5. The vehicle speed is zero for headways smaller than the minimum headway (h j ), and is constant after it reaches the free-flow speed (u f ). The curvature of the curve is convex as was discussed earlier in the paper.
Rakha, Pecker, and Cyis 9 Using the speed-headway relationship (Equation [16]), the speed-density relationship can e derived as ( ) 1 1 8 1 5.4T k kj u = min uf, 1 1 + + 2 9 ( 1 T. [17] ) Equation [17] considers the units conversions to ensure that estimated speed is in units of km/h. Alternatively, the density-speed relationship can e cast as 1 k =, [18] 2 a + a u + a u 1 2 3 where a 1, a 2, and a 3 are as defined earlier. The speed-flow relationship can then e cast as 1u q =. [19] 2 a + a u + a u 1 2 3 Recognizing the convex nature of the congested regime of the speed-flow relationship, if speed-atcapacity occurs prior to the desired speed it can e estimated y taking the derivative flow with respect to speed and setting it to zero as q 1 u( a2 a3u) + = 1 2 2 2 = u u c a1 a2u a3u + + ( a1 + a2u + a3u ). [2] c1 2hj uc =± = 3.6 c3 ( 1 ) Given that the model is a dual-regime model, the speed-at-capacity may exceed the free-flow speed and thus should e constrained y the uncongested regime desired speed as u c 2 = min uf, 3.6. [21] kj ( 1 ) Equation [21] demonstrates that the speed-at-capacity is not impacted y the driver reaction time (T), ut instead is impacted y the,, and k j parameters, as demonstrated in Tale 2. Tale 2 demonstrates an increase in the speed-at-capacity as the / ratio approaches 1. (Pipes model) as demonstrated y the highlighted cells. Furthermore, as the driver deceleration rate increases, the speed-at-capacity increases. Using Equation [21] the roadway capacity can e estimated as 1uc qc =. [22] 2 1 Tuc uc + + 1 ( ) k 2.4 25.92 j Unlike the speed-at-capacity, the roadway capacity is impacted y the driver reaction time. Specifically, as the driver reaction time increases, the roadway capacity decreases, as demonstrated y comparing Tale 3, Tale 4, and Tale 5. The roadway capacity was calculated considering the free-flow speed (u f ) ranged from 6 to 12 km/h, the jam density (k j ) ranged etween 1 and 16 veh/km/lane, for three reaction times (T) of.5,.6, and.7s, respectively. The deceleration parameter () ranged from 2 to 8 m/s 2, while was set y the / ratio which varied from.45 to
Rakha, Pecker, and Cyis 1.95. The tales demonstrate that the roadway capacity increases as the jam density increases, decreases as increases, and increases as the / ratio increases. SAMPLE APPLICATIONS OF PROPOSED CALIBRATION PROCEDURES This section presents some sample applications of the proposed caliration procedure using field data from two freeways and a single arterial roadway facility. The caliration tool was tested on data provided in the literature (1). These data included a freeway with a speed limit of 88 km/h (55 mph) (Figure 6), the Holland tunnel data, and data gathered from an arterial street that was monitored using the Split Cycle and Offset Optimization Tool (SCOOT) (Figure 7). Other data sources included a sample dataset of 5-min. data from Highway 41 in Toronto, Canada (Figure 4). Tale 6 summarizes the estimates of the four traffic stream parameters for different traffic stream models, as taulated in the literature (1), and as computed using the Gipps and Van Aerde (7, 1, and 11) functional forms. In addition, the literature provides independent estimates of the valid ranges of oserved values for each of the four parameters of interest. These oserved validity ranges serve as an independent measure of the quality of fit of the various models to the suject data. The results of Tale 6 demonstrate the efficiency of the Van Aerde functional form in fitting the data and highlight some of the deficiencies of the Gipps functional form in matching the field data. The caliration of the macroscopic traffic stream parameters (u f, u c, q c, and k j ) was achieved using a heuristic automated tool (SPD_CAL) for calirating steady-state traffic stream models (12,13). These parameters are summarized in Figure 6. Once the macroscopic parameters are derived, the next step is to use the caliration procedures developed in this paper to compute the Gipps car-following model input parameters. By considering the most severe deceleration rate to e employed to avoid a collision to e -5 m/s 2, the / ratio is computed to e.798 (i.e., = 6.27 m/s 2 ) y applying the Excel Solver to Equation [21] considering a desired speed-at-capacity of 74 km/h. The driver reaction time is then estimated to e.856 s y applying the Excel Solver to Equation [22] considering a desired capacity of 1699 veh/h. It should e noted that y changing the parameter from -5 m/s 2 to -4 m/s 2 another set of parameters that produce the same macroscopic ehavior could e achieved. The proposed caliration procedures were also applied to a sample arterial dataset that was extracted from the literature (1). The first step was to calirate the macroscopic traffic stream parameters, as illustrated in Figure 7 using the SPD_CAL heuristic. The figure illustrates a more paraolic functional form for the speed-flow relationship in comparison to the freeway fit that was presented earlier. The Gipps model is ale to predict a speed-at-capacity and capacity that is consistent with field data. In this case the speed-at-capacity is less than the desired speed (free-flow speed). The next step was to input the four macroscopic parameters into Equations [21] and [22] to compute the microscopic parameters,, and T. Using the Excel Solver the / ratio was set at.3 and the parameter was estimated to e 1.44 m/s 2 to produce a speed-at-capacity of 19 km/h. The driver reaction time (T) was computed to e 2.667 s to produce a capacity of 55 veh/h. It should e noted that there are an infinite numer of cominations of the microscopic input parameters that would produce the same macroscopic ehavior. For example, the following solutions produce the same macroscopic ehavior (/ =.4, = 1.21 m/s 2, = 3.3 m/s 2, T = 2.65 s and / =.5, = 1.2 m/s 2, = 2.3 m/s 2, T = 2.66 s).
Rakha, Pecker, and Cyis 11 CONCLUSIONS The caliration of micro-simulation models is one of the major challenges that traffic modelers have to deal with. Furthermore, the model s ehavior is extremely sensitive to the parameters values input to the model. To make matters worse, it is extremely data intensive and difficult to gather microscopic driver ehavior. This paper presented a methodology for calirating the car-following model proposed y Gipps. The caliration procedure focuses on steady-state conditions, assuming that all drivers have similar ehavior and characteristics. The caliration procedure converts the car-following model to its associated macroscopic traffic stream model, and key macroscopic traffic stream parameters (free-flow speed, speed-at-capacity, capacity, and jam density) are calirated using loop detector data. The paper then develops procedures to estimate the model s microscopic parameters from the four macroscopic parameters. The procedure application was demonstrated through some example illustrations using data from two North American freeways and an arterial roadway. The Gipps model does have its limitations in representing traffic stream ehavior; however, the proposed caliration approach should assist modelers in identifying the optimum values of the microscopic car-following parameters to achieve desired macroscopic traffic stream ehavior.
REFERENCES 1. Brackstone, M., and M. McDonald. Car-following: A Historical Review. Transportation Research, 2F, 1999, pp. 181-196. 2. Gartner, Nathan H., Carrol J. Messer, and Ajay Rathi. Traffic Flow Theory - A State-of-the-Art Report: Revised Monograph on Traffic Flow Theory. FHWA, 1994 http://www.tfhrc.gov/its/tft/tft.htm. Accessed July, 26. 3. Gipps, P.G. A Behavioral Car-following Model for Computer Simulation. Transportation Research, 15B, 1981, pp. 15-111. 4. Barceló, J. Microscopic Traffic Simulation: a Tool for the Analysis and Assessment of ITS Systems. http://www.tss- cn.com. Accessed July, 26. 5. Hardman, E. J. Motorway Speed Control Strategies Using SISTM. In Proceedings of the 8 th International Conference on Road Traffic Monitoring and Control. London, UK, 1996, pp. 169-72. 6. Liu, R., D. Van Vliet, and D.P. Watling. DRACULA: Dynamic route assignment comining user learning and microsimulation. Paper presented at PTRC, Vol E, 1995, pp 143-152. 7. Rakha, H., and B. Crowther. Comparison and Caliration of FRESIM and INTEGRATION steadystate Car-following Behavior, Transportation Research, 37A, 23, pp. 1-27. 8. Wilson, R.E. An analysis of Gipps' car-following model of highway traffic. In IMA Journal of Applied Mathematics 66, 21, pp.59-537. 9. Pipes, L.A. Car-following Models and Fundamental Diagram of Road Traffic. Transportation Research, Vol 1, 1967, pp 21-29. 1. May, A.D. Traffic flow fundamentals. 199: Prentice Hall. 11. Van Aerde, M., Single regime speed-flow-density relationship for congested and uncongested highways. Presented at the 74th TRB Annual Conference, Washington DC, Paper No. 9582, 1995. 12. Van Aerde, M. and H. Rakha. Multivariate caliration of single regime speed-flow-density relationships. in Proceedings of the 6th 1995 Vehicle Navigation and Information Systems Conference. 1995. Seattle, WA, USA: Vehicle Navigation and Information Systems Conference (VNIS) 1995. IEEE, Piscataway, NJ, USA, 95CH35776. 13. Rakha, H. and M. Arafeh. Tool for Calirating Steady-State Traffic Stream and Car-Following Models. Accepted for presentation at the 86 th Transportation Research Board Annual Meeting, Washington DC, 27. ACKNOWLEDGEMENTS The authors acknowledge the financial support of the Mid-Atlantic University Transportation Center (MAUTC) and the editorial help of Vikki Fitchett.
Rakha, Pecker, and Cyis 13 LIST OF TABLES TABLE 1 Variation in Roadway Capacity (veh/h/lane) for =...14 TABLE 2 Variation in Speed-at-Capacity (< )...15 TABLE 3 Variation in Capacity (< ) (T=.6s)...16 TABLE 4 Variation in Capacity (< ) (T=.7 s)...17 TABLE 5 Variation in Capacity (< ) (T=.8 s)...18 TABLE 6 Comparison of Flow Parameters for Single-Regime, Multiple-Regime Models, and Gipps Model...19 LIST OF FIGURES FIGURE 1 Car-following notations...2 FIGURE 2 Steady-state Gipps traffic stream models ( = )...21 FIGURE 3 Sensitivity of traffic stream models to driver reaction times, a) T =.8s e ) T = 1.2s...22 FIGURE 4 Sample fit to North American freeway data....23 FIGURE 5 Sample speed-headway relationship ( < )...24 FIGURE 6 Calirated speed-flow-density fit to freeway data....25 FIGURE 7 Calirated speed-flow-density fit to arterial data....26
Rakha, Pecker, and Cyis 14 TABLE 1 Variation in Roadway Capacity (veh/h/lane) for = uf (km/h) 6 8 1 12 kj T (s) (veh/km).5.6.7.8.9 1. 1.1 1.2 1.3 1.4 1.5 1 2667 24 2182 2 1846 1714 16 15 1412 1333 1263 11 2779 2491 2256 263 1899 176 164 1535 1443 1361 1288 12 288 2571 2323 2118 1946 18 1674 1565 1469 1385 139 13 2971 2644 2382 2167 1987 1835 175 1592 1493 145 1328 14 355 271 2435 2211 224 1867 1732 1615 1514 1424 1344 15 313 2769 2483 225 257 1895 1756 1636 1532 144 1358 16 32 2824 2526 2286 287 192 1778 1655 1548 1455 1371 1 3 2667 24 2182 2 1846 1714 16 15 1412 1333 11 316 275 2467 2237 247 1886 1748 163 1526 1435 1354 12 32 2824 2526 2286 287 192 1778 1655 1548 1455 1371 13 3284 2889 2579 2328 2122 195 183 1677 1568 1472 1387 14 336 2947 2625 2366 2154 1976 1826 1697 1585 1487 14 15 3429 3 2667 24 2182 2 1846 1714 16 15 1412 16 3491 348 274 243 227 221 1864 173 1613 1512 1422 1 3243 2857 2553 238 215 1935 1791 1667 1558 1463 1379 11 3342 2933 2614 2357 2146 197 1821 1692 1581 1483 1397 12 3429 3 2667 24 2182 2 1846 1714 16 15 1412 13 356 359 2713 2438 2213 226 1868 1733 1617 1515 1425 14 3574 3111 2754 2471 224 249 1888 175 1631 1527 1436 15 3636 3158 2791 25 2264 269 195 1765 1644 1538 1446 16 3692 32 2824 2526 2286 287 192 1778 1655 1548 1455 1 3429 3 2667 24 2182 2 1846 1714 16 15 1412 11 352 37 2722 2444 2218 231 1872 1737 162 1517 1427 12 36 313 2769 2483 225 257 1895 1756 1636 1532 144 13 3671 3184 2811 2516 2277 28 1914 1773 1651 1545 1451 14 3733 3231 2847 2545 231 21 1931 1787 1663 1556 1461 15 3789 3273 288 2571 2323 2118 1946 18 1674 1565 1469 16 384 331 299 2595 2341 2133 1959 1811 1684 1574 1477
Rakha, Pecker, and Cyis 15 TABLE 2 Variation in Speed-at-Capacity (< ) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8.45 31 38 43 49 53 57 6 31 38 43 49 53 57 61 31 38 43 49 53 57 61.55 34 42 48 54 59 6 6 34 42 48 54 59 63 68 34 42 48 54 59 63 68.65 38 47 54 6 6 6 6 38 47 54 61 67 72 77 38 47 54 61 67 72 77.75 46 56 6 6 6 6 6 46 56 64 72 79 85 9 46 56 64 72 79 85 91.85 59 6 6 6 6 6 6 59 72 83 9 9 9 9 59 72 83 93 12 11 118.95 6 6 6 6 6 6 6 9 9 9 9 9 9 9 12 12 12 12 12 12 12.45 29 36 41 46 51 55 59 29 36 41 46 51 55 59 29 36 41 46 51 55 59.55 32 4 46 51 56 6 6 32 4 46 51 56 61 65 32 4 46 51 56 61 65.65 37 45 52 58 6 6 6 37 45 52 58 64 69 73 37 45 52 58 64 69 73.75 43 53 6 6 6 6 6 43 53 61 69 75 81 87 43 53 61 69 75 81 87.85 56 6 6 6 6 6 6 56 69 79 89 9 9 9 56 69 79 89 97 15 112.95 6 6 6 6 6 6 6 9 9 9 9 9 9 9 97 119 12 12 12 12 12.45 28 34 4 44 49 52 56 28 34 4 44 49 52 56 28 34 4 44 49 52 56.55 31 38 44 49 54 58 6 31 38 44 49 54 58 62 31 38 44 49 54 58 62.65 35 43 5 56 6 6 6 35 43 5 56 61 66 7 35 43 5 56 61 66 7.75 42 51 59 6 6 6 6 42 51 59 66 72 78 83 42 51 59 66 72 78 83.85 54 6 6 6 6 6 6 54 66 76 85 9 9 9 54 66 76 85 93 1 17.95 6 6 6 6 6 6 6 9 9 9 9 9 9 9 93 114 12 12 12 12 12.45 27 33 38 43 47 5 54 27 33 38 43 47 5 54 27 33 38 43 47 5 54.55 3 36 42 47 52 56 6 3 36 42 47 52 56 6 3 36 42 47 52 56 6.65 34 41 48 53 58 6 6 34 41 48 53 58 63 68 34 41 48 53 58 63 68.75 4 49 56 6 6 6 6 4 49 56 63 69 75 8 4 49 56 63 69 75 8.85 52 6 6 6 6 6 6 52 63 73 82 89 9 9 52 63 73 82 89 96 13.95 6 6 6 6 6 6 6 89 9 9 9 9 9 9 89 19 12 12 12 12 12.45 26 32 37 41 45 49 52 26 32 37 41 45 49 52 26 32 37 41 45 49 52.55 29 35 41 45 5 54 57 29 35 41 45 5 54 57 29 35 41 45 5 54 57.65 33 4 46 51 56 6 6 33 4 46 51 56 61 65 33 4 46 51 56 61 65.75 38 47 54 6 6 6 6 38 47 54 61 67 72 77 38 47 54 61 67 72 77.85 5 6 6 6 6 6 6 5 61 7 79 86 9 9 5 61 7 79 86 93 99.95 6 6 6 6 6 6 6 86 9 9 9 9 9 9 86 15 12 12 12 12 12.45 25 31 35 4 43 47 5 25 31 35 4 43 47 5 25 31 35 4 43 47 5.55 28 34 39 44 48 52 55 28 34 39 44 48 52 55 28 34 39 44 48 52 55.65 31 38 44 5 54 59 6 31 38 44 5 54 59 63 31 38 44 5 54 59 63.75 37 46 53 59 6 6 6 37 46 53 59 64 7 74 37 46 53 59 64 7 74.85 48 59 6 6 6 6 6 48 59 68 76 83 9 9 48 59 68 76 83 9 96.95 6 6 6 6 6 6 6 83 9 9 9 9 9 9 83 12 118 12 12 12 12.45 24 3 34 38 42 45 49 24 3 34 38 42 45 49 24 3 34 38 42 45 49.55 27 33 38 42 46 5 54 27 33 38 42 46 5 54 27 33 38 42 46 5 54.65 3 37 43 48 53 57 6 3 37 43 48 53 57 61 3 37 43 48 53 57 61.75 36 44 51 57 6 6 6 36 44 51 57 62 67 72 36 44 51 57 62 67 72.85 46 57 6 6 6 6 6 46 57 66 73 8 87 9 46 57 66 73 8 87 93.95 6 6 6 6 6 6 6 8 9 9 9 9 9 9 8 99 114 12 12 12 12
Rakha, Pecker, and Cyis 16 TABLE 3 Variation in Capacity (< ) (T=.6s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8.45 119 191 165 127 999 971 949 119 191 165 127 999 971 942 119 191 165 127 999 971 942.55 1192 1173 1143 116 173 167 167 1192 1173 1143 116 173 146 113 1192 1173 1143 116 173 146 113.65 1299 1282 125 1217 1217 1217 1217 1299 1282 125 1211 1174 1143 1111 1299 1282 125 1211 1174 1143 1111.75 1451 1431 1416 1416 1416 1416 1416 1451 1431 1399 1358 132 1286 1258 1451 1431 1399 1358 132 1286 1252.85 1694 1694 1694 1694 1694 1694 1694 1694 1674 1638 169 169 169 169 1694 1674 1638 1596 1555 1517 1479.95 217 217 217 217 217 217 217 2233 2233 2233 2233 2233 2233 2233 224 2227 2227 2227 2227 2227 2227.45 1148 1131 113 169 132 11 971 1148 1131 113 169 132 11 971 1148 1131 113 169 132 11 971.55 1232 1213 1181 1148 1113 184 184 1232 1213 1181 1148 1113 177 148 1232 1213 1181 1148 1113 177 148.65 1341 1323 1289 1253 124 124 124 1341 1323 1289 1253 1213 118 1153 1341 1323 1289 1253 1213 118 1153.75 1495 1477 1447 1447 1447 1447 1447 1495 1477 1443 14 1364 1328 1292 1495 1477 1443 14 1364 1328 1292.85 1741 1739 1739 1739 1739 1739 1739 1741 172 1684 164 1636 1636 1636 1741 172 1684 164 162 1562 1526.95 2177 2177 2177 2177 2177 2177 2177 2284 2284 2284 2284 2284 2284 2284 2287 2267 2265 2265 2265 2265 2265.45 1184 1168 1133 113 164 14 17 1184 1168 1133 113 164 14 17 1184 1168 1133 113 164 14 17.55 1269 1251 1218 1183 1145 1115 199 1269 1251 1218 1183 1145 1115 184 1269 1251 1218 1183 1145 1115 184.65 138 1362 1326 1287 1259 1259 1259 138 1362 1326 1287 1252 1217 1188 138 1362 1326 1287 1252 1217 1188.75 1536 1517 148 1474 1474 1474 1474 1536 1517 148 144 143 1364 1332 1536 1517 148 144 143 1364 1332.85 1784 1778 1778 1778 1778 1778 1778 1784 1763 1725 1684 1658 1658 1658 1784 1763 1725 1684 1643 166 1569.95 2238 2238 2238 2238 2238 2238 2238 2329 2329 2329 2329 2329 2329 2329 2329 239 2298 2298 2298 2298 2298.45 1217 12 1169 113 196 17 136 1217 12 1169 113 196 17 136 1217 12 1169 113 196 17 136.55 134 1288 1254 1217 1177 1145 1112 134 1288 1254 1217 1177 1145 1112 134 1288 1254 1217 1177 1145 1112.65 1417 14 1362 1328 1292 1277 1277 1417 14 1362 1328 1292 1254 1216 1417 14 1362 1328 1292 1254 1216.75 1574 1555 1521 1498 1498 1498 1498 1574 1555 1521 1479 144 14 1366 1574 1555 1521 1479 144 14 1366.85 1824 1812 1812 1812 1812 1812 1812 1824 184 1765 1721 1684 1678 1678 1824 184 1765 1721 1684 1645 166.95 2293 2293 2293 2293 2293 2293 2293 2368 2368 2368 2368 2368 2368 2368 2368 2349 2326 2326 2326 2326 2326.45 1249 123 1197 1164 1128 192 165 1249 123 1197 1164 1128 192 165 1249 123 1197 1164 1128 192 165.55 1337 1319 1282 1251 129 1175 1149 1337 1319 1282 1251 129 1175 1149 1337 1319 1282 1251 129 1175 1149.65 1451 1431 1397 1362 1323 1292 1292 1451 1431 1397 1362 1323 1284 1252 1451 1431 1397 1362 1323 1284 1252.75 161 1591 1556 1519 1519 1519 1519 161 1591 1556 1512 1471 1436 14 161 1591 1556 1512 1471 1436 14.85 186 1843 1843 1843 1843 1843 1843 186 184 183 1757 1719 1696 1696 186 184 183 1757 1719 1678 1643.95 2342 2342 2342 2342 2342 2342 2342 244 243 243 243 243 243 243 244 2385 2351 2351 2351 2351 2351.45 1279 126 1232 1189 1161 1123 194 1279 126 1232 1189 1161 1123 194 1279 126 1232 1189 1161 1123 194.55 1368 1349 1317 1276 1241 125 1178 1368 1349 1317 1276 1241 125 1178 1368 1349 1317 1276 1241 125 1178.65 1483 1466 1431 1387 1355 1313 135 1483 1466 1431 1387 1355 1313 128 1483 1466 1431 1387 1355 1313 128.75 1643 1621 1584 1544 1537 1537 1537 1643 1621 1584 1544 159 1465 1435 1643 1621 1584 1544 159 1465 1435.85 1895 1874 187 187 187 187 187 1895 1874 1836 1793 1753 1711 1711 1895 1874 1836 1793 1753 1711 1674.95 2387 2387 2387 2387 2387 2387 2387 2437 2434 2434 2434 2434 2434 2434 2437 2417 2379 2374 2374 2374 2374.45 137 1288 1259 1223 1184 1154 1114 137 1288 1259 1223 1184 1154 1114 137 1288 1259 1223 1184 1154 1114.55 1397 1378 1343 131 1274 1236 1198 1397 1378 1343 131 1274 1236 1198 1397 1378 1343 131 1274 1236 1198.65 1513 1495 1458 142 1378 1343 1317 1513 1495 1458 142 1378 1343 138 1513 1495 1458 142 1378 1343 138.75 1674 1655 1617 1576 1554 1554 1554 1674 1655 1617 1576 1539 151 1462 1674 1655 1617 1576 1539 151 1462.85 1927 196 1895 1895 1895 1895 1895 1927 196 1867 1829 1787 1743 1725 1927 196 1867 1829 1787 1743 176.95 2427 2427 2427 2427 2427 2427 2427 2467 2462 2462 2462 2462 2462 2462 2467 2447 2411 2393 2393 2393 2393
Rakha, Pecker, and Cyis 17 TABLE 4 Variation in Capacity (< ) (T=.7 s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8.45 16 144 12 985 959 933 913 16 144 12 985 959 933 97 16 144 12 985 959 933 97.55 1135 1118 191 158 127 121 121 1135 1118 191 158 127 13 972 1135 1118 191 158 127 13 972.65 1232 1217 1188 1158 1158 1158 1158 1232 1217 1188 1153 1119 191 162 1232 1217 1188 1153 1119 191 162.75 1368 1351 1337 1337 1337 1337 1337 1368 1351 1322 1286 1251 1221 1195 1368 1351 1322 1286 1251 1221 119.85 1583 1582 1582 1582 1582 1582 1582 1583 1565 1533 158 158 158 158 1583 1565 1533 1496 146 1427 1393.95 1937 1937 1937 1937 1937 1937 1937 243 243 243 243 243 243 243 249 238 238 238 238 238 238.45 196 18 154 123 989 961 933 196 18 154 123 989 961 933 196 18 154 123 989 961 933.55 1172 1155 1125 196 164 137 137 1172 1155 1125 196 164 131 14 1172 1155 1125 196 164 131 14.65 127 1254 1223 119 1179 1179 1179 127 1254 1223 119 1155 1125 11 127 1254 1223 119 1155 1125 11.75 148 1391 1365 1365 1365 1365 1365 148 1391 1361 1322 1291 1259 1226 148 1391 1361 1322 1291 1259 1226.85 1623 1621 1621 1621 1621 1621 1621 1623 165 1574 1535 1531 1531 1531 1623 165 1574 1535 151 1466 1435.95 1996 1996 1996 1996 1996 1996 1996 286 286 286 286 286 286 286 288 271 27 27 27 27 27.45 1128 1114 182 155 119 996 967 1128 1114 182 155 119 996 967 1128 1114 182 155 119 996 967.55 125 1189 1159 1127 193 165 151 125 1189 1159 1127 193 165 137 125 1189 1159 1127 193 165 137.65 135 1289 1256 1222 1197 1197 1197 135 1289 1256 1222 119 1158 1132 135 1289 1256 1222 119 1158 1132.75 1444 1427 1394 1389 1389 1389 1389 1444 1427 1394 1358 1325 1291 1262 1444 1427 1394 1358 1325 1291 1262.85 166 1655 1655 1655 1655 1655 1655 166 1642 161 1573 1551 1551 1551 166 1642 161 1573 1538 155 1472.95 247 247 247 247 247 247 247 2123 2123 2123 2123 2123 2123 2123 2123 217 297 297 297 297 297.45 1159 1143 1115 179 148 125 993 1159 1143 1115 179 148 125 993 1159 1143 1115 179 148 125 993.55 1237 1223 1191 1158 1122 193 163 1237 1223 1191 1158 1122 193 163 1237 1223 1191 1158 1122 193 163.65 1338 1323 1289 1258 1226 1212 1212 1338 1323 1289 1258 1226 1192 1158 1338 1323 1289 1258 1226 1192 1158.75 1477 146 143 141 141 141 141 1477 146 143 1394 1359 1323 1293 1477 146 143 1394 1359 1323 1293.85 1695 1685 1685 1685 1685 1685 1685 1695 1678 1644 166 1573 1568 1568 1695 1678 1644 166 1573 1539 155.95 293 293 293 293 293 293 293 2155 2155 2155 2155 2155 2155 2155 2155 214 2121 2121 2121 2121 2121.45 1187 117 114 111 178 145 12 1187 117 114 111 178 145 12 1187 117 114 111 178 145 12.55 1266 1251 1217 1189 1151 112 197 1266 1251 1217 1189 1151 112 197 1266 1251 1217 1189 1151 112 197.65 1368 1351 132 1289 1254 1226 1226 1368 1351 132 1289 1254 1219 119 1368 1351 132 1289 1254 1219 119.75 159 1492 1461 1428 1428 1428 1428 159 1492 1461 1423 1386 1355 1323 159 1492 1461 1423 1386 1355 1323.85 1727 1711 1711 1711 1711 1711 1711 1727 179 1677 1638 164 1584 1584 1727 179 1677 1638 164 1568 1538.95 2134 2134 2134 2134 2134 2134 2134 2185 2184 2184 2184 2184 2184 2184 2185 2169 2142 2142 2142 2142 2142.45 1214 1197 1172 1133 118 173 147 1214 1197 1172 1133 118 173 147 1214 1197 1172 1133 118 173 147.55 1294 1277 1248 1212 118 1148 1123 1294 1277 1248 1212 118 1148 1123 1294 1277 1248 1212 118 1148 1123.65 1397 1382 1351 1311 1283 1245 1238 1397 1382 1351 1311 1283 1245 1215 1397 1382 1351 1311 1283 1245 1215.75 1538 1519 1486 1451 1445 1445 1445 1538 1519 1486 1451 1419 138 1354 1538 1519 1486 1451 1419 138 1354.85 1756 1738 1735 1735 1735 1735 1735 1756 1738 175 1669 1634 1597 1597 1756 1738 175 1669 1634 1597 1565.95 2171 2171 2171 2171 2171 2171 2171 2212 221 221 221 221 221 221 2212 2196 2165 216 216 216 216.45 124 1222 1196 1164 1129 111 165 124 1222 1196 1164 1129 111 165 124 1222 1196 1164 1129 111 165.55 132 133 1272 1242 129 1176 1141 132 133 1272 1242 129 1176 1141 132 133 1272 1242 129 1176 1141.65 1423 148 1375 1341 133 1272 1249 1423 148 1375 1341 133 1272 1241 1423 148 1375 1341 133 1272 1241.75 1565 1548 1515 1479 1459 1459 1459 1565 1548 1515 1479 1446 1412 1378 1565 1548 1515 1479 1446 1412 1378.85 1784 1766 1756 1756 1756 1756 1756 1784 1766 1732 1699 1663 1625 169 1784 1766 1732 1699 1663 1625 1592.95 224 224 224 224 224 224 224 2237 2233 2233 2233 2233 2233 2233 2237 2221 2191 2176 2176 2176 2176
Rakha, Pecker, and Cyis 18 TABLE 5 Variation in Capacity (< ) (T=.8 s) kj (veh/km/ln) and /' k j = 14 k j = 13 k j = 12 k j = 11 k j = 1 k j = 15 k j = 16 u f (km/h) and (m/s 2 ) u f = 6 km/h u f = 9 km/h u f = 12 km/h 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8.45 115 1 978 946 922 898 88 115 1 978 946 922 898 874 115 1 978 946 922 898 874.55 184 168 143 113 985 98 98 184 168 143 113 985 963 934 184 168 143 113 985 963 934.65 1172 1158 1132 115 115 115 115 1172 1158 1132 11 17 143 117 1172 1158 1132 11 17 143 117.75 1294 1279 1267 1267 1267 1267 1267 1294 1279 1253 122 1189 1162 1138 1294 1279 1253 122 1189 1162 1134.85 1485 1485 1485 1485 1485 1485 1485 1485 1469 1441 1419 1419 1419 1419 1485 1469 1441 149 1376 1347 1317.95 1793 1793 1793 1793 1793 1793 1793 1882 1882 1882 1882 1882 1882 1882 1888 1878 1878 1878 1878 1878 1878.45 148 133 11 981 95 924 898 148 133 11 981 95 924 898 148 133 11 981 95 924 898.55 1117 112 175 148 119 994 994 1117 112 175 148 119 988 964 1117 112 175 148 119 988 964.65 127 1192 1164 1134 1124 1124 1124 127 1192 1164 1134 112 174 152 127 1192 1164 1134 112 174 152.75 133 1315 1292 1292 1292 1292 1292 133 1315 1288 1253 1225 1196 1166 133 1315 1288 1253 1225 1196 1166.85 152 1519 1519 1519 1519 1519 1519 152 154 1477 1443 1439 1439 1439 152 154 1477 1443 1413 1382 1354.95 1843 1843 1843 1843 1843 1843 1843 1919 1919 1919 1919 1919 1919 1919 1921 197 195 195 195 195 195.45 178 165 135 111 977 957 929 178 165 135 111 977 957 929 178 165 135 111 977 957 929.55 1148 1133 115 177 146 12 17 1148 1133 115 177 146 12 994 1148 1133 115 177 146 12 994.65 1238 1223 1194 1162 114 114 114 1238 1223 1194 1162 1134 115 181 1238 1223 1194 1162 1134 115 181.75 1362 1346 1317 1313 1313 1313 1313 1362 1346 1317 1286 1256 1225 1199 1362 1346 1317 1286 1256 1225 1199.85 1553 1548 1548 1548 1548 1548 1548 1553 1537 159 1476 1457 1457 1457 1553 1537 159 1476 1445 1416 1387.95 1886 1886 1886 1886 1886 1886 1886 195 195 195 195 195 195 195 1951 1937 1929 1929 1929 1929 1929.45 115 191 165 133 14 983 954 115 191 165 133 14 983 954 115 191 165 133 14 983 954.55 1176 1163 1135 115 172 145 118 1176 1163 1135 115 172 145 118 1176 1163 1135 115 172 145 118.65 1267 1254 1223 1196 1166 1154 1154 1267 1254 1223 1196 1166 1135 114 1267 1254 1223 1196 1166 1135 114.75 1392 1376 135 1332 1332 1332 1332 1392 1376 135 1317 1286 1254 1227 1392 1376 135 1317 1286 1254 1227.85 1583 1574 1574 1574 1574 1574 1574 1583 1568 1539 155 1476 1472 1472 1583 1568 1539 155 1476 1447 1416.95 1925 1925 1925 1925 1925 1925 1925 1978 1978 1978 1978 1978 1978 1978 1978 1964 1949 1949 1949 1949 1949.45 1131 1116 188 161 131 11 978 1131 1116 188 161 131 11 978 1131 1116 188 161 131 11 978.55 123 1189 1158 1133 199 17 149 123 1189 1158 1133 199 17 149 123 1189 1158 1133 199 17 149.65 1294 1279 1251 1223 1192 1166 1166 1294 1279 1251 1223 1192 116 1134 1294 1279 1251 1223 1192 116 1134.75 1419 145 1377 1348 1348 1348 1348 1419 145 1377 1343 1311 1282 1254 1419 145 1377 1343 1311 1282 1254.85 1611 1597 1597 1597 1597 1597 1597 1611 1595 1568 1533 153 1486 1486 1611 1595 1568 1533 153 1472 1445.95 196 196 196 196 196 196 196 23 22 22 22 22 22 22 23 199 1966 1966 1966 1966 1966.45 1156 114 1117 182 159 127 13 1156 114 1117 182 159 127 13 1156 114 1117 182 159 127 13.55 1228 1213 1186 1154 1125 195 173 1228 1213 1186 1154 1125 195 173 1228 1213 1186 1154 1125 195 173.65 132 137 1279 1243 1217 1184 1177 132 137 1279 1243 1217 1184 1157 132 137 1279 1243 1217 1184 1157.75 1445 1428 1399 1368 1363 1363 1363 1445 1428 1399 1368 134 135 1282 1445 1428 1399 1368 134 135 1282.85 1636 1621 1618 1618 1618 1618 1618 1636 1621 1592 156 1529 1497 1497 1636 1621 1592 156 1529 1497 1469.95 1991 1991 1991 1991 1991 1991 1991 225 223 223 223 223 223 223 225 212 1986 1982 1982 1982 1982.45 1179 1163 1139 111 178 153 12 1179 1163 1139 111 178 153 12 1179 1163 1139 111 178 153 12.55 1251 1236 128 1181 1151 1121 19 1251 1236 128 1181 1151 1121 19 1251 1236 128 1181 1151 1121 19.65 1344 133 13 127 1236 128 1187 1344 133 13 127 1236 128 118 1344 133 13 127 1236 128 118.75 1469 1454 1425 1393 1376 1376 1376 1469 1454 1425 1393 1364 1334 133 1469 1454 1425 1393 1364 1334 133.85 166 1645 1636 1636 1636 1636 1636 166 1645 1616 1587 1555 1522 158 166 1645 1616 1587 1555 1522 1493.95 219 219 219 219 219 219 219 247 243 243 243 243 243 243 247 233 28 1995 1995 1995 1995
Rakha, Pecker, and Cyis 19 TABLE 6 Comparison of Flow Parameters for Single-Regime, Multiple-Regime Models, and Gipps Model Type of Model Model Free-speed (km/h) Speed-at- Cap. (km/h) Capacity (veh/h/lane) Jam density (veh/km/lane) Valid Data Range (1) 8-88 45-61 18-2 116-156 Single- Regime Multi-Regime Greenshields 91 46 18 78 Greenerg 37 1565 116 Underwood 12 45 159 Northwestern 77 48 181 Edie 88 64 225 11 2-Regime 98 48 18 94 Modified 77 53 176 91 Greenerg 3-Regime 8 66 1815 94 Gipps 8 74 1699 116 Van Aerde 8 6 1827 116 Source: May, (1) pp. 3 and 33. Highlighted cells: Outside the valid data range for specified parameter.
Rakha, Pecker, and Cyis 2 L n-1 n n- 1 x n (t) x n-1 (t) h n (t) = x n-1 (t) - x n (t) FIGURE 1 Car-following notations.
Rakha, Pecker, and Cyis 21 1 1 8 6 4 2 8 6 4 2 5 1 15 2 25 2 4 6 8 1 12 14 16 Flow (veh/h) Density (veh/km) a) ) 1 8 6 4 2 2 4 6 8 1 12 14 16 Headway (m) Flow (veh/h) 25 2 15 1 5 2 4 6 8 1 12 14 16 Density (veh/km) c) d) FIGURE 2 Steady-state Gipps traffic stream models ( = ).
Rakha, Pecker, and Cyis 22 1 1 8 6 4 2 8 6 4 2 5 1 15 2 25 5 1 15 2 25 Flow (veh/h) Flow (veh/h) a) ) FIGURE 3 Sensitivity of traffic stream models to driver reaction times, a) T =.8s and ) T = 1.2s.
Rakha, Pecker, and Cyis 23 12 1 12 1 Field Data Gipps Model Van Aerde Model 8 6 4 8 6 4 2 2 5 1 15 2 25 Flow (veh/h/lane) 2 4 6 8 1 12 Density (veh/km) 12 25 1 2 8 6 4 Flow (veh/h) 15 1 2 5 1 2 3 4 5 Headway (m) 2 4 6 8 1 12 Density (veh/km) Gipps Model Van Aerde Model Free-Speed: 16 km/h Free-Speed: 16 km/h Capacity: 1892 veh/h/lane Capacity: 1888 veh/h/lane Speed-at-Capacity: 16 km/h Speed-at-Capacity: 85 km/h Jam Density: 1 veh/km/lane Jam Density: 1 veh/km/lane FIGURE 4 Sample fit to North American freeway data.