Cellular automata for traffic simulation Nagel-Schreckenberg model

In this project, traffic simulation according to the cellular automaton of the Nagel-Schreckenberg model (1992) with different boundary conditions. The sudden occurrence of traffic jams is successfully realised as well as boundary induced phases and phase transitions are observed in the Asymmetric Simple Exclusion Process. The extension to the Velocity Dependent Randomization model leads to metastabile high flow states and hysteresis of the flow. The impact of speed limits on the probability of the formation of traffic jams is investigated.
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Cellular automata for traffic simulation Nagel-Schreckenberg model Project report in Computational Physics Cellular automata for trac simulation Nagel-Schreckenberg model Torsten Held Stefan Bittihn Bonn, 17th March 2011 Abstract In this project, trac is simulated according to the cellular automaton of the Nagel-Scheckenberg model (1992) with dierent boundary conditions. The sudden occurrence of trac jams is successfully realised as well as boundary induced phases and phase transitions are observed in the Asymmetric Simple Exclusion Process. The extension to the Velocity Dependent Randomization model leads to metastabile high ow states and hysteresis of the ow. The impact of speed limits on the probability of the formation of trac jams is investigated. Furthermore, the eects of on- and o-ramps and trac lights are analysed.Contents1 Introduction 22 The Nagel-Schreckenberg model 2 2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Parameters and transfer to reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The Asymmetric Simple Exclusion Process 34 Metastability and hysteresis in the Velocity-Dependent-Randomization-model 4 4.1 Control by initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 Control by on- and o-ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3 Lifetime of the metastable phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Further applications 7 5.1 The eects of on- and o-ramps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 The eects of trac lights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Summary 91 IntroductionThe aim of trac-simulation-algorithms is to gain an understanding of (road-)trac including its variousphenomena, e.g. the dependence of the dierent trac parameters as ow and density or the formationof trac jams. With the help of a suitable simulation, one can make predections about the development of real tracsituations and furthermore use the results to optimise trac plannings. The rst attempts to simulate trac date back into the 1950s. A very important step foreward was theNagel-Schreckenberg model (NaSch model) which was invented by Kai Nagel and Michael Schreckenbergin 1992. It was the rst model to take into account the imperfect bahaviour of human drivers and wasthus the rst model to explain the spontanious formation of trac jams. The NaSch model is the basisof this project. An interesting application of the (extended) NaSch model is for example the OSLIM project [1] whichsimulates and predicts the trac of North-Rhine-Westphalia online and in real time.2 The Nagel-Schreckenberg modelThe basic NaSch model [2] is a probabilistic cellular automaton: It contains a one-lane-road with discretepositions (cells). Also time (rounds) and integral velocities 0, ..., vmax are discrete. Every round, rsteach car updates its velocity dependent on the position of the next car ahead and then every car movesaccording to its velocity. The updating consists of 4 steps: 1. Acceleration: vn → min(vn + 1, vmax ) 2. Deceleration: vn → min(vn , dn − 1) 3. Randomization: vn → max(vn − 1, 0) with probability p 4. Movement: xn → xn + vnThe acceleration step is given by the attempt to drive as fast as possible within the speed limit vmax .Every car has the same target velocity vmax . The acceleration is 1. The deceleration step is to avoidcrashes: A car will not drive on or pass the position of the car driving ahead with distance dn . Therandomization step leads to an additional deceleration of 1 with probability p and is due to severalbehaviours of human drivers: The rst one is an overreaction at braking and keeping a too large distanceto the car in front. Secondly, when dn increases, one might have a delay in the acceleration process. As alast point, at maximum velocity and free lane, one has a probability of sudden deceleration by distraction.The randomization is the basis for the formation of jams, because o ...