DEVELOPMENT OF SENSITIVITY TERM IN CAR-FOLLOWING MODEL CONSIDERING PRACTICAL DRIVING BEHAVIOR OF PREVENTING REAR END COLLISION

Transcription

1 DEVELOPMENT OF SENSITIVITY TERM IN CAR-FOLLOWING MODEL CONSIDERING PRACTICAL DRIVING BEHAVIOR OF PREVENTING REAR END COLLISION CHUNG, Sung Bong Research Fellow Center or Transport Inrastructure Investment The Korea Transport Institute 311, Daehwa-dong, Ilsan-gu, Goyang-si Gyeonggi-do, , Korea Fax: HONG, Sang Yeon Ph. D. Course Student Dept. o Civil, Urban and Geo-systems Engineering Seoul National University San 56-1 Shillim-dong, Gwanak-gu Seoul, Republic o Korea Fax: zenadu@snu.ac.kr SONG, Ki Han Graduate Student Dept. o Civil, Urban and Geo-systems Engineering Seoul National University San 56-1 Shillim-dong, Gwanak-gu Seoul, Republic o Korea Fax: willkh@hanmail.net KHO, Seung Young Proessor Dept. o Civil, Urban and Geo-systems Engineering Seoul National University San 56-1 Shillim-dong, Gwanak-gu Seoul, Republic o Korea Fax: sykho@snu.ac.kr Abstract: In this study, the deceleration model and the sensitivity term were developed by considering actual driving behaviors to prevent rear end collisions in the car-ollowing situations, especially in deceleration situations. Test vehicles capable o measuring the speed, acceleration, and spacing between the vehicles were used to collect data in the ield survey. The deceleration model developed by linear regression has a high coeicient o determination (0.93) and or the goodness-o-itness test o the sensitivity term, the chi-square test was perormed under 95% conidence level. The results ( χ = < χ 0.05 = ) showed that the sensitivity term has practical signiicance. According to the results, the actor that determines the drivers characteristics in deceleration situations is the lapsed time until the driver starts to respond to the given stimuli. The results indicate that the lapsed time until driver reaction corresponds to the α value, which represents the drivers characteristics in the GM model. Key Words: deceleration model, sensitivity term, car-ollowing, rear end collision 1. INTRODUCTION The basic concept o car-ollowing theories is the relationship between Stimuli and Response. In the classic car-ollowing theory, the stimuli are represented by the relative speed o ollowing and leading vehicle, and the response is represented by the acceleration (or deceleration) rate o the ollowing vehicle. During the last 50 years, various car-ollowing theories have been studied in terms o rom the earlier deterministic relationships to the recent stochastic relationship. 1354

2 The most signiicant improvement in the theory was achieved when the reaction time and sensitivity parameter, which can show the level o responsiveness, were considered. The GM model integrated previous car-ollowing models and the research team developed ive generations o car-ollowing models, all o which took the orm o response = unc( sensitivity, stimuli). The dierence in the dierent levels o models was the representation o sensitivity. Many researches have been conducted to develop the reliable sensitivity term, but they just ocused on inding the values o the parameters o the sensitivity term. The GM model considered driving behaviors by introducing m and l exponents and α in the sensitivity term, but did not explain the actors that aect these parameters. In actuality, a driver does not directly respond to the relative velocity between the vehicles, but rather to the amount o risk o rear end collision he perceives rom the spacing, the velocity, and the probability o a collision. Accordingly, the stimuli to the response in the car ollowing situation is the risk o a rear end collision perceived by the drivers on the basis o their experiences. In this study, the deceleration model and the sensitivity term were developed and deduced based on these concepts.. LITERATURE REVIEW.1 Car-Following Theory Reushel (1950) and Pipes (1953) gained the public acceptance o their car-ollowing theory in the early 1950 s or the irst time, and since then, many have studied this theory. The classic car-ollowing theory is classiied into two conceptual models. The irst one was established by Chandler, Herman, and Montroll (1958), and they proposed a conceptual ramework like equation (1). Response = F(sensitivity,stimuli) (1) They also suggested that drivers exert acceleration or deceleration in proportion to Force meaning the relative speed. From this concept, they developed the linear model like equation () on the assumption that the driver o the ollowing vehicle controls the accelerator (or brake) to keep zero relative speed to the leading vehicle. where, M n a n( v( n 1) / n t t + T ) = λ ( ) () M n : Mass o nth vehicle an ( t + T ) : Acceleration o nth vehicle ater reaction v( n 1) / n( t) : Relative speed o ( n 1) to the th nth vehicle in time t T : Response delay o a driver λ : Sensitivity Term 1355

3 Gazis, Herman, and Rothery (1961) suggested that the earlier car ollowing model should be amended by relecting psychological actors represented by equation (3) Response = Sensitivity * Stimuli (3) In equation (3), the mass o vehicle was deleted and the response o a driver, the sensitivity term and the stimuli, are denoted as an ( t + T), λ and v( n 1) / n ( t ) respectively. The equation (3) is important because it introduces the psychological actor into the car-ollowing theory. Hence, the need or an appropriate unction that could express the sensitivity o a driver began to be recognized. Edie (1961), Newell (1960), Gazis, Herman, and Rothery (1960) studied nonlinear carollowing model considering the human actor, and especially Gazis et al (1960) stated that although it is diicult to ind a reliable non-linear model which can represent real carollowing conditions, only a nonlinear model can relect real car ollowing situations. The research team o General Motors integrated previous car-ollowing models, and arranged them into ive generations o car-ollowing models, all o which took the orm o Response = Sensitivity * Stimuli. With this conceptual base, they derived the generalized car-ollowing equation by introducing generalized exponents into the sensitivity term. where, a λ l, m[ ( )] ) v n t t + T = v ( t) (4) l ( n 1). n [ x ( n 1). / n( t) ] n ( / m m, l : Parameters or speed and distance headway λ l, m : Constants showing the characteristics o a driver Equation (4) is better known as the GHR model because Gazis, Herman, and Rothery (1960) introduced the parameters m and l into the sensitivity term o the car-ollowing model. For this equation, they also classiied the traic low conditions into congested and noncongested conditions in analyzing the relation o variables used in analyzing traic low.. Sensitivity Term Greenshields (1934) established the basis o the car-ollowing equation that could induce the equation o state or the steady low by presenting the sensitivity term as the square o the inverse o the distance headway. Edie (1961) proposed a modiied sensitivity term by adding a speed actor into Greenshields model, λ = c x& n+1 s, and this equation came to be basis o the sensitivity term o the GHR model shown in equation (5). He showed that with this sensitivity term, characteristics o traic situations such as stability o traic low and steady-state low characteristics could be explained. m λl, m x& n+ 1( t + T ) λ = (5) l [ t) ( t) ] x n( xn

4 May and Keller (1967) calibrated the parameters o the sensitivity term in the GM 5 th Model by setting m = 1, l = 3, and they also showed that i the parameters did not have integer values, they would be approximately equaled to m = 0.8, l =. 8. Keiichi Sato and Hideo Igarashi (197) induced a macroscopic traic low equation rom the car-ollowing equation and classiied it into two models. The irst one is the Exponential nonlinear model, which can be derived rom equation (5) when m = 1, l > 1 and one o the representative models o exponential non-linear model is the Underwood s model. The second one is the N-power curve model, which was derived rom equation (5) when m = 0. In this model, i l = 1, the model becomes the Greenberg s model, and i l > 1, then it becomes the Greenshield s model and Drew s model. Heyes and Ashworth (197) introduced a new model to look into the relationship between stimuli and response. In their model, the stimuli and the sensitivity term were denoted as v x and t p, respectively, and the sensitivity term was estimated as 0.8 based on the survey ield data o the Mersey tunnel in UK. Treiterer and Myers (1974) calibrated the parameters o sensitivity in GM model with the data collected rom an aerial photograph, and they developed two distinct models or acceleration and deceleration, respectively, to consider the imperection and heterogeneity o human behaviors. Ceder and May (1976) classiied the traic low conditions into two groups, congested and non-congested condition, and or each group, they estimated the parameters, l, m rom a large data base o real data. Ceder (1976, 1978) modiied the GHR model by substituting the sensitivity term with equation (6), and S, A in equation (6) denote distance headway in a congested situation, constant that varies with traic low conditions respectively.. S x A (6) x Aron (1988), who tested drivers response in various traic conditions, classiied the drivers responses into 3 types: deceleration, constant speed, and acceleration responses. Ozaki (1993) investigated the sensitivity term with the data ilmed rom a 3 story building, and he analyzed the time series data under ten seconds due to the technical limitation o video ilm. From this analysis, he showed that the driver has a dierent sensitivity or acceleration and deceleration..3 Collision Avoidance Model Kometani and Sasaki (1959) tried to improve the traic dynamic theory o Pipes (1953) and ound the undamental equation o traic dynamics. They also showed that the saety o the ollowing vehicle with respect to rear end collision when the lead car is in sinusoidal motion can be quantiied by introducing a saety index. Through this study, they established the basic concept o the collision avoidance model in car-ollowing situations. 1357

5 Gipps (1981) made a signiicant improvement in collision avoidance study. To mimic similar car-ollowing behavior to real situations, he gave limitations to drivers abilities and the perormance o vehicles, and with this hypothesis, he calculated the speed o ollowing vehicle required or sae ollowing. Considering reaction time and sensitivity term, he developed a realistic simulation model with real data collected rom ield survey, but did not calibrate the parameters. Touran et al (1999) developed a crash model to evaluate the saety o the AICC (Autonomous Intelligent Cruise Control) system and examined the unction o this equipment to show how it controls a vehicle to prevent crash. He also tested the ield survey with our consecutive vehicles, and through this survey, he showed that under the given hypothesis, a vehicle with AICC could reduce the possibility o collision with a leading vehicle. One o the various collision avoidance models, which have been researched to ind an appropriate Simulation model, is CARSIM developed by Benekohal and Treiterer (1989). This model consists o two algorithms, which are the ollowing algorithm without speed variation and the acceleration (or deceleration) algorithm. In the ollowing algorithm, both the control algorithm o the distance headway and the collision restraint algorithm are considered and in the acceleration (deceleration) algorithm, various acceleration (deceleration) rates are applied to the model according to 5 types o traveling conditions. Especially, by assigning 1 types o reaction time to each driver randomly, they tried to simulate the variety o the drivers as well as the various acceleration and deceleration patterns in car-ollowing. 3. MODEL DEVELOPMENT In this chapter, based on the deceleration model developed by assuming that the risk o rear end collision is a stimulus to a driver, a new sensitivity term is proposed to compare with that o the earlier car-ollowing model. 3.1 Deceleration Model (1) Stimuli and Response The GM model says that a driver o the ollowing vehicle accelerates (or decelerates) in response to the stimulus o relative speed, but in real car-ollowing situations, especially in the deceleration condition, a driver responds not to the speed dierence but to the perceived risk o rear end collision caused by the speed dierence. In other words, a driver recognizes the level o risk rom the current traveling speed, distance headway, and the probability o collision. And then, he/she ollows the leading vehicle only within the region o risk he/she can bear. I the driver recognizes large risk, the driver would decelerate quickly; i the driver recognizes small risk, then decelerate slowly. I the leading car decelerates, the ollowing car driver would select a proper reaction time and deceleration rate to prevent rear end collision according to his/her experiences. Hence, all drivers have their own reaction area with respect to reaction time and deceleration rate, and they ollow the leading vehicle to a proper distance considering their reaction area. 1358

6 For instance, when the leading vehicle decelerates at some rate in the car-ollowing situation, the driver o the ollowing vehicle will choose one o the response sets, ( a, t r) rom his/her experience, and then he/she will decelerate at that rate to minimize the risk o rear end collision as shown in Figure 1. I this empirical decision is wrong, that is to say, i he/she 1 1 selects the response set o c1( a, t r) the driver would recognize a larger risk than beore, so he/she will choose an even higher deceleration rate with lower reaction time like (, ) c a t r. Figure 1. Response Area The driver o the ollowing vehicle ollows the leading car and tries to retain proper ollowing situation by employing this continuous selection process. Thereore, to some stimuli, a driver has his/her own response area, which is learned rom his/her experience, and in general carollowing conditions, most drivers try to ollow the leading vehicle within their response areas. Consequently, the knowledge about the response area o a driver or various car-ollowing situations can help us to identiy the car ollowing behavior o the driver. () Deceleration Behaviors To understand the relation between stimuli and response in a deceleration situation, let us think o two vehicles which have the traveling trajectory shown in Figure. d Lead vehicle Collision Following vehicle a a dl d tl tr tr ( c) Figure. Relation between Response Time and Deceleration Rate t 1359

7 In Figure, tl, t, tc, h = d l d, t m, a and a denote time at which leading vehicle begins to decelerate, time at which a driver o ollowing vehicle begins to decelerate, time at which a driver can prevent rear end collision only i he/she decelerates at imum deceleration rate, distance headway at time 0, duration time o imum deceleration, deceleration rate at which the driver o the ollowing vehicle should choose at t to avoid rear end collision, and imum deceleration rate at which the driver o the ollowing vehicle should choose at t c to avoid rear end collision respectively. In Figure, as the leading vehicle decelerates at time t l, the distance headway between the two vehicles shortens, and hence, the driver o the ollowing vehicle comes to perceive the risk o rear end collision. In this situation, the deceleration behavior o the ollowing vehicle depends on t and accordingly, to ollow leading vehicle while minimizing the risk o rear end collision, the driver should increase the deceleration rate rom a to a as t t c. From this point o view, the level o the deceleration rate chosen by the driver o the ollowing vehicle is related to t, t, a and the critical reaction time is represented as ollows; t c ( c) = tl tc (7) In act, the distance headway and the speed o the ollowing vehicle also aect the deceleration rate, but the eects o these actors are considered in determining t and t c. Thus, the deceleration rate which the driver chooses can be expressed as a unction o such variables as t, t c and a. a = ( t, t ( c), a ) (8) The most important actor in equation (8) is t (c), which is calculated rom the relationship among the speed o both vehicles, spacing, and imum deceleration rate. In this study, t (c) is calculated by the EXCEL program at every second. (3) Data Collection To validate the relationship among actors related to the deceleration rate o the ollowing vehicle, a ield survey was perormed on the straight roadway at Tong-Il Hill in Gyeonggi-do. To minimize the error caused by imperection and heterogeneity o human behaviors, the ield survey was perormed or 0 test drivers o dierent driving skill levels with two test vehicles equipped with a tachometer and laptop computer system capable o measuring the speed, acceleration, and spacing between the vehicles. The data collected rom the tachometer depended on vehicle speciications such as the wheel size and driving gear o the transmission and the like, which were provided by ROTIS Inc. 1360

8 To obtain a reliable model, ield survey should be perormed under various traic conditions, but in this study, it was perormed in a speciic driving condition because o the restraints o the survey environment, test equipment, and saety. The survey was perormed in three dierent situations, namely when the ollowing vehicle ollows the leading vehicle at 30 km/h, the drivers o the ollowing vehicles were instructed to respond to the stimuli rom the leading vehicle with three responses such as immediate reaction, general reaction, and slightly delayed reaction. To ilter the noises caused by data conversion rom the tachometer, the non-latness o the surace and the dierence o driving distance, two correction steps were chosen. At irst, the dierence between the traveling distance by a vehicle and the distance in the map was corrected by the average dividing method. The noise produced by data conversion was iltered by eliminating abnormal data, and then the noise o the data which has no abnormal point was iltered by using the wavelet tool in Matlab. (4) Data Collection and Analysis With the ield data, the relationships between deceleration rate and related actors were analyzed. The results showed that the deceleration rate was linearly related to the reaction time/critical reaction time, as shown in Figure Deceleration Rate Reaction time/critical Reaction time Figure 3. Scatter Diagram o Reaction time In Figure 3, the ield data has three dierent areas o distribution because o the three dierent test conditions. The distance among each distribution area is not uniorm because the test drivers did not respond exactly to the three dierent requests during the test. Figure 3 shows the irst order linear relationship between deceleration rate and reaction time/critical reaction time and the model could be ormulated by regression analysis based on equation (9). t a = β 0 + β 1 (9) t ( c) 1361

9 Table 1. Results o Regression Analysis Coeicient SD t-statistics P-value Intercept E-14 X variable E-88 Table. Results o Regression Analysis (ANOVA) Degree o Mean o Sum o squares Freedom squares F-statistics Regression Residual Sum To check the statistical signiicance o the model, -test and t-test were perormed. The F- statistics was ( > F ) and the t-statistics o intercept and x variable were 8.40, ( > t ), respectively. Moreover, the Coeicient o Determination was 0.93 and the adjusted one was so the estimated model has statistical signiicance. As a result, the estimated model is as ollows; t a = (10) t ( c) As mentioned in section 3., i a driver responds to the stimuli with the critical reaction time, he/she should exert deceleration at a imum rate to avoid rear end collision. According to equation (10), the deceleration rate that is required when a driver responds with the critical reaction time is 7.94, which is similar to the imum deceleration rate o the test vehicle in slow traic conditions. Especially, the drivers should decelerate considering a buer distance to ollow the leading vehicle saely, so the intercept relects this buer distance considered by the drivers. Function (8) shows that the deceleration rate is aected by the imum deceleration rate considered by driver but in developing the model, the data on the imum deceleration rate was not included. However, the regression results show that the coeicient o the x variable is almost equal to the imum allowable deceleration rate in a slow traic condition. Thereore, this result shows that the relationship between the independent variable and dependent variable is consistent with the theoretical hypothesis o the model. 3. Deducement o Sensitivity Term As known in Figure, the deceleration rate that the driver selects to prevent rear end collision with the leading car is related to t (c), t, a. And this relationship was statistically proven in section 3.1. In equation (10), the coeicient o the x variable is almost imum deceleration rate or the vehicle in a slow traic condition, and the intercept can be deleted because it is negligible when trying to explain the trend o the deceleration by a driver. 136

10 a t = a (11) c t To deduce the sensitivity term, the next two assumptions are required. First, deceleration is a response which is executed by a driver to avoid rear end collision with the leading vehicle. Second, the speed o both vehicles will be equal at the end o the deceleraton o the ollowing vehicle. Especially, when the ollowing vehicle decelerates at its imum rate, the spacing between the two vehicles is close to 0. In the right side o equation (11), a is determined by the perormance and traveling speed o the ollowing vehicle, t is related to the characteristics o driver and t (c) is calculated rom the actors related to traveling condition such as the speeds o both vehicles, spacing etc. To deduce the sensitivity term rom equation (11) t c should be expanded to the variables related to the traveling condition actors. Thus, i a driver begins to decelerate at the time o t c, he/she should decelerate at the imum rate to avoid rear end collision. In this case, the ollowing equation should be satisied. where, where, vl = v + amtm (1) vl : Speed o leading vehicle ( m / s ) v : Speed o ollowing vehicle ( m / s ) am : Maximum deceleration rate at the traveling condition ( m s ) tm : Lapse time to the point when the speed o the ollowing vehicle becomes equal to the speed o leading vehicle at the imum deceleration rate (sec) * * 1 va ( t + tm) + h vb t + vb tm + am tm (13) h : spacing between two vehicles beore deceleration Substitute t with c t + tm and the equation (13) will be as ollows; v t + h = v 1 t + a m t m (14) Expanding equation (14) or t c, equation (14) will be as ollows ; t c 1 am tm h ( va vb) tm h ( va vb) tm = = (15) va vb ( va vb) Substituting equation (15) or equation (11) and expanding equation (11) or a 1363

11 a ( vl v ) t = am h ( vl v ) (16) And expanding the equation (16) to the orm o GM model yields; a t am = ( vl v ) (17) ( vl v ) tm + h Comparing equation (17) with the GM model, the sensitivity term ( λ ) is expressed as; t am λ = (18) ( vl v ) tm + h In equation (18), the act that the sensitivity term deduced rom the deceleration model here increases as a, t, tm, and ( vl v ) increase and h decreases is consistent with general deceleration behaviors. 3.3 Goodness o itness Test To validate the sensitivity term, ield survey was perormed or three drivers with three dierent reaction times in the same test environment o the ield survey or deceleration model. According to the data collected rom the survey, the average reaction time, standard deviation and number o data o each test driver or each situation are summarized in Table 3 and the average deceleration rate o each test driver is also summarized in Table 4. Table 3. Average Reaction Time o Test Drivers Classiication Ave. time o Immediate reaction (SD, Num o data) Ave. time o Usual Reaction (SD, Num o data) Ave. time o Delayed Reaction (SD, Num o data) Driver (0.06, 15) 1.01(0.06, 15) 1.91(0.13, 0) Driver 0.70(0.07, 15) 1.04(0.08, 15) 1.98(0.15, 0) Driver (0.05, 15) 1.0(0.10, 15) 1.98(0.15, 0) Table 4. Average Deceleration Rate o Test Drivers Classiication Ave. Deceleration Rate or Immediate Reaction (SD) Ave. Deceleration Rate or Usual Reaction (SD) Ave. Deceleration Rate or Delayed Reaction (SD) Driver 1.17(0.17) 3.33(0.) 5.71(0.4) Driver.31(0.11) 3.35(0.17) 5.89(0.19) Driver 3.5(0.18) 3.36(0.3) 5.84(0.38) 1364

12 The results showed that the deceleration rates o drivers 1,, 3 became higher as the reaction time was delayed. Moreover, the model indicated no rear end collision as long as the spacing between two vehicles is over 0, but generally the drivers consider buer spacing or minimum saety so drivers are likely to decelerate at a slightly higher rate than that predicted by the model. To test goodness-o-itness o the model under the 95% conidence level, the chi-square test was perormed and the result ( χ = < χ 0.05 = ) showed that the model deduced in this study provides results that are consistent with the behaviors o the driver o ollowing vehicle. 4. CONCLUSIONS Deceleration in the classic car-ollowing model can be represented as the response to the stimulus o the relative speed between two vehicles. In this study, however, the deceleration model and the sensitivity term were developed based on the dierent concept o the stimuli; that is, the risk o rear end collision perceived by the driver o ollowing vehicle is considered as the stimulus. Stability Analysis which explains the rear end collision caused by unstable ollowing vehicle also states a close relationship between reaction time and the response o acceleration/deceleration. Namely, the drivers who respond slowly have high acceleration or deceleration rates. On the other hand the, the drivers who are attentive to the car-ollowing process are not likely to resort to sudden accelerations or decelerations except in emergencies. Accordingly, the car-ollowing behaviors o a driver in deceleration situation can be expressed by the relationship between reaction time and deceleration. The data rom ield survey showed that the deceleration rates chosen by the drivers are closely related to the ratio o the critical reaction time and the reaction time o the driver o the ollowing vehicle. The results o regression analysis showed that the coeicient o determination (0.93) is very high and F-statistics and t-statistics o intercept and x variable are ( > F ), 8.40, ( > t ), respectively. Accordingly, the estimated model has statistical signiicance. The sensitivity term was deduced theoretically rom the deceleration model, and goodness-oitness test o the model in the 95% conidence level showed that the sensitivity term is consistent with the behaviors o the drivers o ollowing vehicles. O the actors which constitute the sensitivity term, no one actor can explain the characteristics o driver theoretically. However, the model developed in this study ound that the characteristics o the driver in the sensitivity term can be represented by the reaction time o the driver in response to the risk o rear end collision. To develop a more reliable model, the data on reaction time o less than 1 second was required but in this study, due to the limitations o the equipment, the time unit o the data collected rom ield survey was 1 second rom slow traic conditions. 1365

13 In the uture, i more accurate equipment can collect more precise data than the present data, and saety related problem can be solved, the general deceleration model in various carollowing situations can be developed. And based on the general model, the sensitivity term which can consider the general characteristics o drivers can be also deduced. The sensitivity term developed in this study can help build the theoretical rame or analyzing the characteristics o traic low and the characteristics o the control algorithm used in advanced vehicles. And based on the sensitivity term, i the situations o car-ollowing can be better understood, which can provide higher reliability or analyzing capacity, the traic low saety and so on. Moreover, this study proposed a signiicant new approach or developing the sensitivity term which has not changed or about a hal century. REFERENCE A. Reuschel (1950), Vehicle Movements in a Platoon with Uniorm Acceleration or Deceleration o the Lead Vehicle, Zeitschrit des Oesterreichischen Ingenieur-und Architekten-Vereines, No.95, 59-6 and Brackstone, M. and Mike McDonald (1999), Car-ollowing: a historical review, Transportation Research Part F, Ceder, A., and A.D. May. (1976), Further Evaluation o Single- and Two-Regime Traic Flow Models, Transportation Research Record 567, Transportation Research Board, NRC, Washington, D.C.: Del Castillo, J. M. (1996), A Car Following Model Based On The Lighthill-Witham Theory, Proceedings o the 13th International Symposium on Transportation and Traic Theory, Edie, L. C. (1961), Car-Following and Steady-State Theory or Non-Congested Traic, Operations Research, Vol. 9, No. 1, Gazis, D. C., R. Herman and Richard W. Rothery (1961), Nonlinear Follow-The-Leader Models O Traic Flow, O. R. Vol. 9, Greenshields B. D. (1934), A study o traic capacity, Highway Research Proceeding, Vol. 14, 1934, Ozaki, H. (1993), Reaction and Anticipation in the Car-Following Behavior, Proceedings o the 1th International Symposium on Transportation and Traic Theory, Herman, R., Elliott W. Montroll, Renrey B. Potts, and Richard W. Rothery (1959), Traic Dynamics: Analysis o Stability In Car Following, Operations Research 7, Herman, R. and Richard W. Rothery (1963), Car Following and Steady State Flow, Proceedings o the nd International Symposium on The Theory o Road Traic Flow, London, Herman, R., and Richard W. Rothery (1967), Propagation o Disturbances in Vehicular Platoons, Proceedings o the 3rd international symposium on the theory o traic low,

14 Herman, R. and Tenny-Lam (1974), Stability o Vehicle Platoons, Proceedings o the 6th international symposium on the theory o traic low, Herman, R., Richard W. Rothery and Ronald G. Rule (1977), Analysis o car-ollowing experiments employing spectral analysis, Proceedings o the 7th International Symposium on Transportation and Traic Theory, L. A. Pipes, An Operational Analysis o Traic Dynamics (1953), Journal o Applied Physics, Vol. 4, No.3, Wilhelm, W. E. and J. W. Schmidt (1973), Review O Car-Following Theory, Transportation Engineering Journal,