Comparison of Characteristics and Computational Performance: Car- Following Versus Cellular Automata Models

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1 Comparison of Characteristics and Computational Performance: Car- Following Versus Cellular Automata Models Ghulam H. Bham, M.S. * Graduate Student Fax (17) , Phone (17) bham@uiuc.edu 316 Newmark Civil Engineering Laboratory 05 N. Mathews Avenue Urbana, IL Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, IL, USA Paper prepared for presentation at the 8 nd Annual Meeting January, 003 Washington, D. C. Paper submitted to the TRB Committee on Artificial Intelligence (A5008C) November 15 th, 00 * corresponding author

2 Bham Comparison of Characteristics and Computational Performance: Car- Following Versus Cellular Automata Models Ghulam H. Bham, University of Illinois, Urbana-Champaign ABSTRACT The paper presents comparison of characteristics of car-following (CF) and cellular automata (CA) models. CELLSIM based on concepts of CF and CA models is also compared with these models. CF, CA and CELLSIM models have been compared in terms of their acceleration and deceleration models, use of continuous and discontinuous space, consideration of length of vehicles, use of integer versus floating point numbers, reaction times and model applications. The strength and weaknesses of these models are also described. The paper also compares the computational performance of CELLSIM and Stochastic Traffic Cellular Automata model. Keywords: microscopic simulation, car-following, cellular automata, acceleration model, deceleration model, computational performance INTRODUCTION The use of microscopic models has been limited in scope and scale of application in the past. Applications were restricted to arterial or small networks with limited number of vehicles. With the advent of faster computers, microscopic models are now used in simulating traffic on the level of cities and freeway networks. The INTEGRATION model has been used to simulate traffic for the Salt Lake Metropolitan Area (1). The MITSIM model () has been used to evaluate aspects of both the traffic control system and the ramp configurations of the Central Artery/Tunnel project in Boston. Similarly, AIMSUN have been used to simulate the Rings Roads of Barcelona (3). These models use various techniques to simulate high volume of traffic on large networks with shorter execution time. MITSIM supports distributed implementation. AIMSUN uses parallel computers to shorten the execution time. Similarly, the TRANSIMS project used CA models to simulate traffic for the city of Fortworth-Dallas using parallel computers, mostly coupled workstations (4). Traffic simulation using CA models have also been performed on vector supercomputers, to simulate traffic in shortest possible time (5). CELLSIM (6,7) on the other hand, is developed based on concepts of car-following (CF) and cellular automata (CA) models. CF concepts are used to simulate traffic realistically and CA concepts are used to shorten the execution time by minimizing the use of computational resources. CELLSIM thus represents a unique approach to the development of microscopic simulation models. CELLSIM is currently under development for simulation of traffic on a regional network of Illinois. The paper is organized as follows: first CA and CF are briefly described, CELLSIM is then described in the context of both CA and CF models. Characteristic differences between CF, CA and the CELLSIM model are then discussed in terms of their acceleration and deceleration models, use of continuous and discontinuous space, consideration of length of vehicles, use of integer versus floating point numbers, reaction times, model applications and computational performance.

3 Bham 3 COMPARISON OF CAR-FOLLOWING AND CELLULAR AUTOMATA MODELS This section describes car following and cellular automata simulation models, their features and differences in approach as well as their weaknesses and strengths. CELLSIM is also described in context of these models. Car-Following (CF) Models Car-following models are detailed microscopic models, which consume more computing time and resources while providing greater resolution and potentially more accuracy. CF models use realistic driver behavior and detailed vehicle characteristics that require higher computational resources. Models like INTELSIM (8), INTRAS (9), CORSIM (NETSIM & FRESIM) (10), and CARSIM (11) use detailed acceleration models. In NETSIM, FRESIM, CARSIM and INTELSIM quadratic equations are solved to calculate acceleration rates. Since these programs require higher computational resources, the execution time can be longer than real time when large number of vehicles are simulated on a network. The INTEGRATION model required an execution time to 17 times the real time depending on the number of vehicles in the system on a 00 MHz Pentium PC (1). These times are clearly not feasible for practical purposes even with the faster computers available today. The MITSIM model reported time factors ranging from 0.48 to. on an SGI Indy R4400 workstation with 00 MHz of speed (,1) and was able to simulate,000 vehicles (1). Current car following models have detailed routines that require higher computational resources and longer execution time. Thus there was a need to simulate high number of vehicles with shorter execution time using efficient algorithms on personal computers. Cellular Automata (CA) Models CA models can be considered simple microscopic models that are straightforward with a logic that usually consist of a few integer operations. Because of their simplicity, they are able to perform several millions updates in a second (13,14). Thus, they may be used for simulating high volume of traffic over large networks. However, CA models do not have realistic driver and vehicular behavior models. Vehicles are modeled as particles having unrealistic acceleration and deceleration rates. Vehicles accelerate independently of their speed and have speed jumps of about 7 km/hr in one-second (15). Vehicles tend to change speed abruptly; they can come to a stop from a maximum speed of 135 km/hr in one-second (15). Hence they have very erratic acceleration and deceleration behavior. In STCA, space is discretized into cells of 7.5 meters (4.61 feet) (15). Thus vehicle movement is restricted to a multiple of 7.5 meters and this obviously is not suitable for studying real world problems since the model is too coarse. Because of these shortcomings they cannot be used for conducting detailed traffic flow studies at the high fidelity level. Since CA models allow unrealistic acceleration and deceleration the validation of CA model is only possible at the macroscopic level (16). When macroscopic comparison is made, the simulation results describe the quality of traffic flow in an acceptable manner but to describe the traffic flow quantitatively correct the CA model has to be modified (16). Research to improve the behavior of CA models by finer discretization of cells was carried out by Krauss (17,18) and more recently by Knospe (19). However, the results from Krauss' model, Figure 3 (17) clearly showed deceleration from the model that exceeds deceleration in reality. In

4 Bham 4 some cases of the Krauss' model, both acceleration and deceleration rates were set equal to constant values. Although Krauss' and Knospe's models use finer discretization, they do not provide realistic representation of a platoon of vehicles going through a disturbance. Additionally, no stability analyses of these models have been performed to observe their behavior in normal and emergency braking conditions. CA simulations have been used by several researchers for simulating different types of traffic conditions (0,1,). Networks on the scale of cities have been simulated using these models (4). These particle-based models have proven to be extremely useful in complex geometrics where they avoid the instabilities of the discretized partial differential equations (3). CA models are classified as either site oriented models or particle oriented models. In site oriented models the state of the system is specified by storing the state of each cell, which can either be empty or occupied by a vehicle with a certain velocity, where velocity equals integer values from 0 to a maximum velocity (in most cases to be 5). In car or particle oriented models the state of the system is stored as the velocity of each car and the distance to the next car ahead. In the following, the most researched CA model is explained. Stochastic Traffic Cellular Automaton (STCA) The update rules of the models are as follows. For all vehicles 'i' speed update is performed as (15): If (v i gap i ) [close following/braking] v f = {gap i - 1 with probability p noise (v < 0 not allowed) {gap i otherwise Else If (v i < v max ) [acceleration] v f = {v i with probability p noise {v i + 1 otherwise Else (v i = v max AND v i < gap i ) [free driving] v f = {v max - 1 with probability p noise {v max otherwise where p noise = random noise parameter, v i = v t = initial speed, v f = v t+1 = updated speed. Next, vehicles are updated as: x t+1 = x t + v t + 1 (1) p noise adds randomness in close following and braking (sometimes be slower than you can), in acceleration (sometimes do not accelerate), and in free driving (fluctuate around current speed) (0). The model is rescaled to real world units as follows:

5 Bham 5 Length of cells = 7.5 m. Time Step = 1 second. Maximum velocity = 5 boxes per time step = 5 * 7.5 m/sec = 135km/h 85 mph. Realistic maximum flow values are then reached with p noise = 0., indicating that 0 percent of the times drivers do not accelerate. STCA is extremely easy to implement as 3 conditions are considered in the model i.e. close following/braking, acceleration and free driving. Moreover, during the update of a vehicle's speed only variables are required i.e. speed and available gap between vehicles. Similarly, updating position of a vehicle is extremely simple, as it doesn't use rate of acceleration. This cuts down the number of calculations required to update a vehicle's speed and position at every update. CELLSIM Model The limitations in CF and CA models motivated the development of CELLSIM. CELLSIM presents a simplified approach to the development of a microscopic traffic simulation model. The approach has been adapted to minimize the amount of computational resources, allowing higher number of vehicles to be simulated realistically on a regional highway network. The model formulation uses concepts of CA for simplicity and speed. Like CA models, the cell based or spatial discretization approach has been adopted. Integer values instead of floating point numbers have been used as much as possible to shorten the model's execution time. The concepts of CF models are used for realistic modeling of driver and vehicular behavior. Realistic driver behavior is achieved using preferred time headway (TP). A dual-regime acceleration model is formulated which requires minimal calculation compared to detailed acceleration models used in CF models. A simple deceleration model is also used. Moreover, the model introduces a simplified car-following logic. Details of the CELLSIM model can be found in (6,7) and is briefly described in this paper. In CELLSIM the car following logic is implemented when a pair of vehicles; leader and follower, are at a space gap of less than or equal to 76m (50 feet). Space gap is measured from the rear bumper of the leader to the front bumper of the follower. Space gap of 76 m is used because in separation greater than 61m (00 feet), car following was observed to be negligible on driver's behavior (4). Therefore, when vehicles are separated by more than 76m (50 feet), they are considered to be free flowing. In free flow condition vehicles accelerate to reach their desired speed and then coast. Maximum speed of vehicles is calculated as the minimum of: current speed, desired speed of the driver and maximum speed attainable by the vehicle. In car following, a follower tries to maintain a space gap equal to his/her desired space gap behind the leader. Desired space gap is defined as the product of follower's speed and TP. TP is the time headway a follower prefers to maintain during steady-state car following. TP can be found from the field data by averaging the time headways of a follower at the same speeds with the leader (8). It can also be found from an appropriate distribution that represents reaction time of drivers. The car-following logic is based on comparison of desired space gap and available space gap and the relative speeds of the leader and follower (6).

6 Bham 6 CELLSIM has been validated comprehensively at the macroscopic and microscopic levels using two sets of field data. Validation at the macroscopic level has been performed for average speed, density and volume. Unlike CA models, whose validation is only possible at the macroscopic level (16), CELLSIM's result show very close agreement with the field data. Validation at the microscopic level has been conducted for trajectories and speeds of individual vehicles. The microscopic results from CELLSIM also showed close agreement to the field data. Additionally, error tests and techniques from the field of econometrics have been used to evaluate the results of simulation (7). CELLSIM performed well when tested for stability analyses using mild and severe disturbance conditions (6,7). Characteristic Comparison between CF and CA Models This section describes the characteristic differences in modeling between CF and CA models. Acceleration Model Detailed acceleration models are used in car-following models. CARSIM, NETSIM, FRESIM and INTELSIM require quadratic roots to be solved to calculate an acceleration rate. Acceleration models used in FRESIM/INTRAS, and CELLSIM are discussed here. In INTRAS and FRESIM which use the PITT's car-following model, the basic assumption is that the following vehicle tries to maintain a space gap equal to (9): L kv + bk (u t - v t ) () where L = vehicle length of the leader, k = driver sensitivity factor, u t = speed of the leader at time t, v t = speed of the follower at time t, and 0.1for ( ut vt ) 10 b = calibration constant = 0 for ( ut vt ) 10 In both models, the leader is first advanced to its news position at the end of time step and then the follower is brought a space gap behind the leader. An acceleration value to bring the following vehicle to this space gap in one time step is calculated as (9): where *[x t T - y t - L v t (k T) - bk(u t T - v t ) ] a f (3) (T kt) a f = acceleration of follower in the interval (t, t + T), x t+t = position of leader at time t + T, y t = position of follower at time t, T = time step, and u t+t = speed of leader at time t + T, The driver reaction time 'c' is introduced in the car-following equations after 'a f ' has been calculated, the new position and speed are then updated as (9):

7 Bham 7 where c < T. vt T vt a f ( T c) (4) a f ( T c) yt T yt vtt (5) The above equations present only part of several equations used in acceleration and deceleration of vehicles. Detailed account of INTRAS model used can be found in (9). CELLSIM on the other hand uses a simple equation to update its velocity. No equation is used to calculate the rate of acceleration. Speed is updated as follows (6): v u a * t, 0 v < 1.19 m/sec (40 ft/sec) (6) 1 and where v u a * t, v 1.19 m/sec (40 ft/sec) (7) v = updated speed, ft/sec u = initial speed, ft/sec a 1 = acceleration constant 1 = 1.1 m/sec (3.6 ft/sec ), a = acceleration constant = 0.37 m/sec (1. ft/sec ). These equations are basic kinematic equations. The two acceleration rates were found from the average speed profile of vehicles from the Ohio State Data (5). The acceleration rates used can be updated for different data sets using the procedure described in (6,7). The position of vehicles is then updated as (again a basic kinematic equation) (6): where a * t xt 1 xt v * t (8) a = acceleration constant (a 1 or a ), ft/sec. In STCA, vehicles accelerate with a probability of p noise, with a rate of 7km/hr/sec. The major shortcoming of CA models is their acceleration and deceleration behavior. Vehicles accelerate independent of their current speed (15) and without taking into account the speed of the leader. When a vehicle's speed is excessive it decelerates and tends to change its speed abruptly. The behavior thus tends to be erratic. To summarize acceleration models used in car-following models are long and detailed which involves evaluation of several terms whereas CA acceleration models are simple but unrealistic. CELLSIM's acceleration model is simpler as it involves fewer variables and doesn't require calculation of acceleration rate at every time step compared to acceleration models used in other car following models and the results are also realistic. Deceleration Model In CF models, acceleration/deceleration is based on rate of acceleration and if the rate of acceleration is negative, vehicles decelerate otherwise accelerate. In CELLSIM, first it is

8 Bham 8 determined if the vehicle needs to accelerate or decelerate. If it needs to decelerate, rate of deceleration is calculated and is a function of the speed of the leader and the follower. The speed of the leader is used so that the follower reduces its speed closer to the speed of the leader. In case of coming to a stop, the deceleration rate allows the follower to stop behind the leader and maintain a buffer space. The deceleration rate in slowing down is calculated as (6): where d F u L uf (9) * sp d F = desired deceleration rate of follower, u L = speed of leader, u F = speed of follower, sp = (space gap - buffer space), buffer space = distance between front bumper of follower and rear bumper of leader during stopped condition. The above equation is basically a derived form of the basic kinematic equation. The deceleration rate from the above equation is calculated as an integer number. CF models also use a collision avoidance sequence to bring the follower to a stop during emergency conditions. The deceleration rate from the acceleration model is overwritten when the collision constraint is not meet and another deceleration rate is calculated to bring the follower to a stop behind the leader. In CELLSIM for the collision avoidance condition to govern, it is first checked if the follower will be able to stop a distance equal to the buffer space behind the braking leader if it decelerates at a deceleration rate of 16 ft/sec. This condition can be expressed as (7): where vl space gap * d L vf * d F bs d F, d L = assumed deceleration rate of leader and follower = 16 ft/sec. (10) Secondly, the collision condition is used when a follower approaches a stopped vehicle. For this condition to govern the speed of the follower should be greater or equal to the buffer space between the follower and the stopped vehicle. If any of the conditions govern, collision avoidance sequence is used to reduce the speed of the follower. The following equation is used to calculate the deceleration rate (6): d F uf (11) * sp Again the deceleration rate from the above equation is calculated as an integer number. CA models follow simple rules for deceleration, as speed is a function of available gap (space gap). Vehicles decelerate if speed of follower is greater or equal to gap (v gap). They maintain

9 Bham 9 a distance equal to gap or gap - 1 (with probability p noise ), else speed is reduced by gap to zero. The model doesn't use a collision avoidance sequence and speed can reduce from a maximum of 135 km/hr to zero in one second. Length of Vehicle, Spatial Discretization and Continuous versus Discontinuous (Discretized) Space Models CA models are discontinuous or discretized space models as space is divided in cells. In STCA the length of each vehicle is fixed as 7.5 meters (4.61 feet) and represents vehicle in a traffic jam. Each vehicle thus moves a distance in multiples of 7.5 meters. Thus its movement is restricted to a multiple of 7.5 meters. This proves to be problematic for studying real world problems since the model is too coarse to use. CF models are continuous space models in which the model mainly keeps track of vehicle's starting and ending positions. However, division of space in cells provides advantage compared to continuous space used in CF models. The discreteness of the space can bear significant computational advantages, because operation on integer numbers can in general be performed much more efficiently than floating point numbers (14). The concept of spatial discretization is also used in CELLSIM, space is discretized as one unit (1 foot). Length of vehicle is a variable in the model. 1 foot also provides accurate results and does not prove to be problematic in validating the model using field data. Discretization in space proves useful as the occupancy of cells can be used for a variety of purposes. Keeping track of spaces works similar to loop detectors on a highway. They can be used for collecting traffic flow data, which in turn can be used to predict traffic flow parameters. Application of space discretization in CELLSIM is published elsewhere (8). Integer versus Floating Point Numbers CF models use floating point numbers. In CA models, speed and position of vehicles are integer values, however, for very vehicle at every time step, updating speed requires random numbers, which are real values. Floating point calculation is not computationally efficient to use when high number of vehicles is simulated on a large traffic network. Integer values have proved to be 10-15% faster than floating point numbers (9). CELLSIM like CA models, uses integer values for speed and position of vehicles. Using integer numbers makes the model computationally efficient and shortens the simulation run time. Moreover, in CELLSIM acceleration constants and preferred time headways are real numbers, but every effort is made to keep calculations integer based as much as possible. Reaction Time In CF models reactions time of drivers is taken into consideration in different ways. INTELSIM incorporates reaction time of drives in the most detailed manner. It incorporates reaction time by using brake reaction times and preferred time headway. Preferred time headway represents the maximum reaction time of the follower in case of deceleration by the lead car (30). CA models do not take into account reaction time of drivers. In CELLSIM, the reaction time of drivers in not incorporated directly. However, followers react to the leader's stimulus in the next time step. Therefore a reaction time equal to the time step is used for all drivers. Time step used in the model is 1 second. Driver behavior in CELLSIM is incorporated using preferred time headway.

10 Bham 10 Model Applications CA models are designed especially for transportation planning application whereas carfollowing models are mostly used for traffic engineering applications. CA models are designed for use with larger network on the scale of cities. Car-following models are designed for a particular section of highway/street or a small network of streets. CELLSIM is planned to be used for simulating traffic on the level of regional high networks. It is planned to be linked with a geographic information system to animate traffic in real time. This will provide a graphical tool to observe traffic on a regional highway network. Congested routes can be zoomed in and traffic conditions observed. The discretized space used in the model can be used for a variety of applications as they work like loop detectors. Application of discretized space in calculation of Space Occupancy to calculate a better state of traffic can be found in (8). CELLSIM has potential for studying Intelligent Transportation Systems such as Advanced Traffic Management Systems and real time optimization of traffic flow. Traffic jams can also be studied in detail since high volume of traffic can be simulated microscopically. Traffic phenomenon can be studied with much more detail. The model can also be used for studying acceleration characteristics of vehicles as well as driver behavior for various traffic conditions. COMPUTATIONAL PERFORMANCE This section compares the computational performance of CELLSIM, a simple car-following model and STCA. Comparison of computational performance of a CF and a CA model has not been made before and it is not known how fast are CA models compared to CF models. CELLSIM is implemented in Mathematica (Wolfram Research, Inc.) and being a powerful and an interpreted language is slower to implement compared to compiled languages such as C/C++ or Fortran (31). However, the simulation code has been kept as efficient as possible to rival the speed of other programming languages. The implementation of CELLSIM does not impose restriction on the length of the highway nor the number of vehicles simulated. However, the execution time depends heavily on the number of vehicles present in the system and it increases with increase in their number. CELLSIM can be used on different types of computers such as IBM compatible personal computers, Macintosh and workstations without making any change to the program. A simple test was performed to compare the computational performance of a single lane circular highway using CELLSIM and STCA. The tests were performed on a DELL Optiplex GX60, Pentium 4, 1.8 GHz personal computer with 56 MB of RAM. No parallel or super computers were used in the analysis. Detailed analysis of computational performance of CA models on different types of computers can be found in (13). Vehicles ranging from 500 to 3500 with increments of 1000 vehicles were simulated for 1800 seconds. Initially the vehicles were spaced out to have a density equal to LOS of C i.e. 4 vehicles per mile (3). The length of the highway was thus a function of the number of vehicles at the start of simulation. Vehicles had an initial speed of 7 mph and a maximum speed of 75 mph was allowed in the system. Extensive data e.g. speed, space gap, and position of every vehicle was available for every second of simulation time. Results of the test; number of vehicles in the system, simulation time, execution time, and their ratio are presented in Table 1. The simulation time denotes the number of time steps used in simulating the vehicles and the execution time denotes the time taken by CELLSIM to complete the task of simulating the vehicles. Time step of one-second was used in simulation.

11 Bham 11 The result of tests was used in extrapolating the number of vehicles that can be simulated using CELLSIM in real time. Around vehicles can be simulated in real time for 1800 seconds of simulation. This was calculated using the ratio of number of vehicles in the system and the execution time. The execution time was found to be a function of the number of vehicles in the system and it increases linearly with the increase in number of vehicles simulated. STCA was also programmed in Mathematica and its execution times are also presented in Table 1. The simulation conditions remained the same as described for CELLSIM. Generally CA models are faster than CF simulation models, because of their coarseness and simplicity. This is also evident from the results in Table 1. The execution times for STCA are on average 1.88 times faster compared to CELLSIM. Figure 1 shows the ratio of simulation time and execution time versus number of vehicles in simulation. The figure shows that as the number of vehicles increase the ratio decreases sharply and when extrapolated it was found that the ratio would be around 1 when 6000 vehicles are simulated using STCA vehicles are around 1.8 times compared to vehicles for CELLSIM. Although the computational comparison of models shows that CA models are faster than CELLSIM model, it was felt that models should not be judged based on comparison of execution times only. STCA is much coarser than CELLSIM and the results from CELLSIM are realistic and can be validated using field data. This is not the case with STCA and CA models can only be validated at the macroscopic level (16). A particle in STCA moves in multiples of 7.5 meters whereas vehicles in CELLSIM move in multiples of 0.3m. STCA has speed jumps of 7 km/hr (17 mph) in one second, and it can come to a stop from a maximum speed of 135 km/hr, a deceleration rate of 37.5 m/sec (13 ft/sec ). This is not the case in CELLSIM and vehicles accelerate and decelerate realistically. The maximum deceleration rate in emergency conditions is 6.4 m/sec (1 ft/sec ) and the maximum acceleration is also limited to the acceleration constants. Thus a comparison of execution times of different types of models with varying levels of fidelity shouldn't be made. However, it is anticipated that the execution time of CELLSIM is shorter than most CF models found in the literature, if these programs are coded in the same programming language, run on the same computer with exactly the same level of detail. However, comparison of execution times between CELLSIM and other models would be misleading, since each model has its own characteristics. The level of detail in each model is also different. Furthermore each model is developed for a specific purpose. Therefore comparison of execution time of CELLSIM with other CF models was not performed. CONCLUSIONS The paper presents comparison of characteristics of car following (CF) and cellular automata (CA) models. CELLSIM based on concepts of CF and CA models is also compared with these models. CF models are detailed, require extensive computational resources whereas CA models are simple and faster to run. CA models have erratic acceleration and deceleration models and may be used for planning purposes for simulation of network traffic. Use of CA models for traffic engineering applications is not recommended. CELLSIM although simple, provides accurate and realistic results and is validated comprehensively at the macroscopic and microscopic levels. It uses integer values as much as possible to efficiently use computational resources. Driver behavior is also incorporated using preferred time headway. Straightforward

12 Bham 1 algorithms and efficient use of computational resources makes the model suitable for real time traffic simulation at the network level. It is anticipated that the simulation runtime of CELLSIM is shorter than other CF models, however no comparison between CELLSIM and other carfollowing models was conducted. Comparison between CELLSIM and a CA model showed that STCA is on average 1.88 times faster than CELLSIM. CELLSIM can simulate around vehicles and STCA around 6000 vehicles on a single lane in real time on a 1.8 GHz pentium 4 personal computer. However, it was felt that models should not be judged by their execution times only, since models have different levels of fidelity and application for which they are developed. REFERENCES 1. Rakha, H., M. Van Aerde, L. Bloomberg, and X. Huang. Construction and Calibration of a Large-Scale Microsimulation Model of the Salt Lake Area. Transportation Research Record 1644, TRB, National Research Council, Washington, D.C., pp , Yang, Q. A Simulation Laboratory for Evaluation of Dynamic Traffic Management Systems. Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts, Barcelo, J., J. F. Ferrer, D. Garcia, M. Florian, and E. Le Saux. The Parallelization of AIMSUN Microscopic Simulator for ITS Applications, In Proc., 3 rd World Congress on Intelligent Transportation Systems, Nagel, K., and C. L. Barrett. Using micro-simulation feedback for trip adaptation for realistic traffic in Dallas. International Journal of Modern Physics C-Physics & Computers, vol.8, no.3, pp , June Nagel, K., and A. Schleicher. Microscopic traffic modeling on parallel high performance computers, Parallel Computing, 0, pp , Bham, G. H. and R. F. Benekohal. ILLISIM: A Fast High Fidelity Traffic Simulation Model based on Car-Following and Cellular Automata Concepts, Paper No , 80 th Annual Conference of the Transportation Research Board, Washington, D.C Bham, G. H. and R. F. Benekohal. A High Fidelity Traffic Simulation Model based on Car- Following and Cellular Automata Concepts, Forthcoming in Transportation Research, Part C. 8. Aycin, M. F., and R. F. Benekohal. Linear Acceleration Car-Following Model Development and Validation, Transportation Research Record 1644, TRB, National Research Council, Washington, D.C., pp , FHWA. Development and Testing of INTRAS, A Microscopic Freeway Simulation Model, Vol. I: Program Design, Parameter Calibration and Freeway Dynamics Component Development. Report No. FHWA/RD-80/106. U.S. Department of Transportation, Oct Halati, A., L. Henry, S. Walker. CORSIM-Corridor Traffic Simulation Model, Traffic Congestion and Traffic Safety Conference, Chicago, ASCE, pp , Benekohal, R. F., and J. Treiterer. CARSIM: Car Following Model for Simulation of Traffic in Normal and Stop-and-Go Conditions. Transportation Research Record 1194, TRB, National Research Council, Washington, D.C., pp , Yang, Q., and H. N. Koutsopoulos. A microscopic traffic simulator for evaluation of dynamic traffic management systems. Transportation Research C, 4 (3): , Nagel, K. High-speed microsimulations of traffic flow, Ph.D. Dissertation, University of Cologne, Germany, Krauss, S. Microscopic Modeling of Traffic Flow: Investigation of Collision Free Vehicle Dynamics, Ph.D. Dissertation, University of Cologne, Germany, 1998.

13 Bham Nagel, K. From Particle Hopping Models to Traffic Flow Theory, Transportation Research Record 1644, Washington D.C., pp. 1-9, Ponzlet, Martin. Validation of a CA-model for traffic simulation of the Northrhine- Westphalia Motorway Network, Planning and Transport Research and Computation (International) Co. Meeting (4th: 1996: Brunel University). Seminars D & E. Transportation planning methods. Part 1. London: PTRC Education and Research Services, Krauss, S. Wagner, P. and Gawron, C. Continuous limit of the Nagel-Schreckenberg model, Physical Review E, vol. 54, no. 4, pp , Krauss, S. Wagner, P. and Gawron, C. Metastable states in a microscopic model of traffic flow, Physical Review E, vol. 55, no. 5, pp , Knospe W., Santen L., Schadschneider A., Schreckenberg M. Towards a realistic microscopic description of highway traffic. Journal of Physics A-Mathematical & General, vol.33, no.48, pp. L477-85, Nagel, K., and M. Schreckenberg. A Cellular automaton model for freeway traffic, Journal de Physique I, France,, 1, Emmerich, H., and E. Rank. Investigating Traffic Flow in the Presence of Hindrances by Cellular Automata, Physica A 16(4), pp , Nagatani, T. Jamming transition in the traffic-flow model with two-level crossings, Phys. Rev. E 48(5), Sahimi, M. Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulated annealing, Rev. Mod. Phys. 65 (4), pp. 1393, Herman, R., and R. B. Potts. Single Lane Traffic Theory and Experiment. Proc. Symposium on the Theory of Traffic Flow, Warren, Mich., Treiterer, J. Investigation of Traffic Dynamics by Aerial Photogrammetry Techniques. Final Report EES78, Transportation Research Center, Department of Civil Engineering, Ohio State University, Feb Bham, G. H. and R. F. Benekohal. Development, Evaluation and Comparison of Acceleration Models. Paper No , 81 st Annual Conference of the Transportation Research Board, Washington, D.C., Bham, G. H. and R. F. Benekohal. Development, Evaluation and Comparison of Acceleration Models. In review Journal of Transportation Engineering, American Society of Civil Engineers, Bham, G. H. and R. F. Benekohal. Measuring Traffic Congestion using Space Occupancy in Real Time for ITS Applications, 7 th International Conference on Applications of Advanced Technologies in Transportation (AATT7), Boston, Aug Standard Performance Evaluation Corporation. Spec Benchmark Summaries: and Accessed July 31, Winsum, W. V., and A. Heino. Choice of Time-Headway in Car-Following and the Role of Time to Collision Information in Braking. Ergonomics, Vol. 39, pp , Wagner, D. B. Power Programming with Mathematica: The Kernel. McGraw-Hill, N.Y., Special Report 09: Highway Capacity Manual, Transportation Research Board, National Research Council, Washington, D.C., 1994.

14 Bham 14 Table 1. Comparison of Execution times for CELLSIM and STCA CELLSIM STCA Ratio Number Time (seconds) Number Time (seconds) Exec. of Execution Sim. of Execution Sim. Time Vehicles Simulation 'A' /Exec. Vehicles Simulation 'B' /Exec. 'A/B' * # * # * extrapolated, # estimated 60 Simulation Time / Execution Time CELLSIM STCA No. of Vehicles in Simulation Figure 1. Ratio of Simulation Time/Execution Time versus Number of Vehicles in Simulation