Modelling dualcarriageway traffic behaviour as a complex system: A proposal for discussion.
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5. Probability distributions
The assignment of each car its starting time headway and velocity characteristics as expected in realworld traffic is achieved using specific probability distributions, specifically exponential and Gaussian distributions. This exponential distribution is used for assigning time headway because the Poisson distribution gives the number of occurrences in a certain time period of an event, which occurs randomly but at a given rate, and the time between events that occur randomly but at a constant rate has an exponential distribution. An exponential distribution is often used to model the time between independent events that happen at a constant average rate. The distribution of car velocities is assumed to be Gaussian given the large number of independent variables (assumed to be random and uncorrelated) influencing the driver's desired velocity. The convergence of these processes to a Gaussian is due to the central limit theorem which states that the distribution of the mean of a sequence of random variables tends to a Gaussian distribution as the number of random variables increases indefinitely. The exponential distribution is used to assign the starting time headway as it is often used to model the time between independent events of a Poisson process, i.e. events that happen at a constant average rate, which can be achieved using several different methods for generating exponential variates [2] see [13] for a method of generating exponentially distributed random numbers. A Gaussian distribution is used to assign the starting velocity as many psychological measurements and physical phenomena can be approximated well by this probability distribution
6. Results
The development for assigning time headways and velocities is complete and velocity and position data sets relative to time for both the single lane and multiple lanes for small numbers of cars have also been achieved. Below are graphs for both velocity and position of the single lane model initially with just two cars and secondly for a larger car number with the interactions for cars as they move from one time regime to the other included. In the Figure 1 andFigure 2, there are just two cars and the car in front is travelling at the slower speed of the two. The second car, when it enters regime 2 (what is this regime?) of our model, it begins to decrease in speed nonlinearly according to the following equations(2) and (3).
In Figure 1, the leading car A, is travelling at 21 m/s, followed by car B, initially travelling at 32 m/s. This abrupt velocity change is a result of the car A, entering regime 2, and the car changes from its desired velocity to the velocity of the car in front as the time to the car in front decreases from 7.5 seconds to 2.5 second, at which time, car B, in this instance is travelling at the same velocity as the car A.
In Figure 4 position and time for 15 cars are displayed with the resultant velocity variation in Figure 3 highlighting the number of interactions that take place and how many steps it takes to solve a problem in term of time and space (how much memory it takes to solve a problem).
Another noticeable characteristic is, if the car that is joining the motorway and is immediately in the regime 2 window, then it can go through vivid changes in velocity. In this particular case the car joins the motorway with a gap of approximately 5.0s to the car in front and, it has to undergo a velocity change from 28.75 F1 to 27.20F1 , using a time step of 0.1m.s, causing the oscillatory instabilities that are observed due to deceleration of 15.5 F6. This oscillatory stability is clearly observed in Figure 5.
