2. Modeling Road Traffic Greg Pinkel Brad Ross Math 341 – Differential Equations December 1, 2008

3. Contents Introduction Model Revised Model Solving our model Conclusions

4. Introduction Modeling the behavior of cars in traffic Car Following Model Quick-Thinking-Driver (QTD) Car Following Model Major Results: –If lead car stops suddenly, system looks like a damped harmonic oscillator –Temporal Following Spacing should be ≥ 2 sec

5. Overview Purpose – traffic modeling Models –2-car model, x 0 position of leading car, x 1 position of following car (meters) –Acceleration of following car (m/s 2 ) –Sensitivity coefficients, λ (1/sec.) and β (1/sec. 2 ) –Reaction time T (seconds) –Preferred separation, d (meters) or temporal separation, τ (seconds) Original Model Quick Thinking Driver Model

6. Original Model λ : sensitivity coefficient, units s -1 Impact of and Addition of T is a Delay Differential Equation, very difficult to solve analytically

7. Quick Thinking Driver Model Ignores reaction time for driver β : sensitivity coefficient, units s -2 Assuming d is a constant “preferred separation” Impact of x 0 – x 1 d

8. Preferred Separation Tends to depend on speed (Think of real world situation) Relate preferred spatial separation to temporal separation by with a preferred temporal separation of τ seconds.

9. Review (and the 2 nd order D.E.) 1) 2)

10. Physically Reasonable Stopping With initial conditions, D = Separation of vehicles, x 0 –x 1, at time of x 0 sudden stop m/sm

11. Solving: Laplace Transforms and Mathematica!

12. The Solution: X 1 (t)

13. Major Results: Simple Harmonic Motion Have three distinct cases for different values of τ: Overdamped, (τ=1s) Underdamped, (τ=3s) Critically Damped, (τ=2s) U=10 m/s, D=10m, β=1s -2

14. Significance of Results Always have collision with the underdamped situation  To avoid collision: τ 2 ≥ 4, τ ≥ 2 seconds

15. Minimum Stopping Distance Collision is possible but will not necessarily happen: a βτ 2 > 4, βτ 2 = 4 Collision can be avoided by ensuring initial separation greater than D stop Derived by setting the solution of x(t) equal to zero with given conditions, D becomes D stop Having initial separation greater than this ensures that the collision will be avoided

16. Conclusion Based on conditions: βτ 2 > 4, βτ 2 = 4, and βτ 2 < 4 –collision is only avoidable for βτ 2 ≥ 4 –collision only avoided if x 0 – x 1 > D stop –Because βτ 2 < 4 always results in collision, maintain temporal time cushion at least 2 sec behind lead car