1. Ring Car Following Models by Sharon Gibson and Mark McCartney School of Computing & Mathematics, University of Ulster at Jordanstown

2.  Mathematical models which describe how individual drivers follow one another in a stream of traffic.  Many different approaches, including:  Fuzzy logic  Cellular Automata (CA)  Differential equations  Difference equations Car Following Models

3.  Classical stimulus response model (GHR model): where; x i (t) is the position of the i th vehicle at time t; T is the reaction or thinking time of the following driver; and the sensitivity coefficient is a measure of how strongly the following driver responds to the approach/recession of the vehicle in front.

4. Car Following Models  A simpler linear form of the GHR model (SGHR) can be expressed in terms of vehicle velocities as: where; u i (t) is the velocity of the i th vehicle at time t.

5. Ring Models  A model in which the last vehicle in the stream is itself being followed by the ‘lead’ (first) vehicle: Motivation:  ‘Real’ simulations re-use data  Idealised as a representation of outer rings  Mathematically interesting

6. A Simple Ring Model  If the driver of each vehicle has zero reaction time model simplifies to: Implication:  The steady state velocity of all vehicles can be found immediately once we have been given initial velocities.

7. A Simple Ring Model  Need to give the lead car a ‘preferred’ velocity profile, w 0 (t): where; the sensitivity coefficient  is a measure of how strongly the lead driver responds to his/her ‘preferred velocity’.

8. A Simple Ring Model  For n = 2, the transient velocity of the i th vehicle is of the form: where; and  The post transient velocity of the i th vehicle is dependent on the form of the preferred velocity.

9. A Simple Ring Model  Three forms of preferred velocity considered  Constant velocity,  Linearly increasing velocity,  Sinusoidal velocity, NB. The post transient results hold for a general n vehicles in the system.

10. where and where

11. Ring Model with Time Delay ♦This new ring model when the drivers reaction times are included can be expressed as: ♦We solve this system of Time Delay Differential Equations (TDDE) numerically using a RK4 routine

12. Approximating Time Delay  An approximate solution to the Time Delay Differential Equation (TDDE) form of the Ring Model can be found using a Taylor’s series expansion in time delay, T:

13. Approximating Time Delay  For n = 2, the transient velocity of the i th vehicle is of the form: where; and  If system is to reach steady state then:

14. Comparison of Zero Time Delay, Taylor’s Series Approximation & RK4 Numerical Methods

15. Stability of the Ring Model  System is locally stable if each car in the system eventually reaches a steady state velocity.  Non-oscillatory motion  Damped oscillatory motion  Stability criteria is dependent upon the number of vehicles in the system.  General criteria for n = 2:  Criteria for n > 2 currently under investigation – One of the boundaries obtained for n = 3:  Hypothesis: The stable region for each value of n > 2 is bounded by exactly 2 boundaries.

16. Stability of the Ring Model (n=2)

17. Stability of the Ring Model (n=3)

18. Stability of the Ring Model (n=5)

19. Future Work  Investigate discrete time models, as:  Easier to implement (Computationally faster)  Arguably more realistic  More likely to give rise to chaotic behaviour

20. Stability of the Ring Model (n=2) (Euler Method)

21. Questions?