1. For Whom The Booth Tolls Brian Camley Pascal Getreuer Brad Klingenberg

2. Problem Needless to say, we chose problem B. (We like a challenge)

3. What causes traffic jams? If there are not enough toll booths, queues will form If there are too many toll booths, a traffic jam will ensue when cars merge onto the narrower highway

4. Important Assumptions We minimize wait time Cars arrive uniformly in time (toll plazas are not near exits or on-ramps) Wait time is memoryless Cars and their behavior are identical

5. Queueing Theory We model approaching and waiting as an M|M|n queue

6. Queueing Theory Results The expected wait time for the n-server queue with arrival rate, service ,  = /  This shows how long a typical car will wait - but how often do they leave the tollbooths?

7. Queueing Theory Results The probability that d cars leave in time interval  t is: What about merging? This characterizes the first half of the toll plaza!

8. Merging

9. Simple Models We need to simply model individual cars to show how they merge… Cellular automata!

10. Nagel-Schreckenberg (NS) Standard rules for behavior in one lane: Each car has integer position x and velocity v

11. NS Behavior

12. NS Analytic Results Traffic flux J changes with density c in “inverse lambda” c J Hysteresis effect not in theory

13. Analytic and Computational

14. Empirical One-Lane Data Empirical data from Chowdhury, et al. Our computational and analytic results

15. Lane Changes Need a simple rule to describe merging This is consistent with Rickert et al.’s two-lane algorithm

16. Modeling Everything

17. Model Consistency

18. Total Wait Times

19. For Two Lanes Minimum at n = 4

20. For Three Lanes Minimum at n = 6

21. Higher n is left as an exercise for the reader It’s not always this simple - optimal n becomes dependent on arrival rate

22. Maximum at n = L + 1 The case n = L

23. Conclusions Our model matches empirical data and queueing theory results Changing the service rate doesn’t change results significantly We have a general technique for determining the optimum tollbooth number n = L is suboptimal, but a local minimum

24. Strengths and Weaknesses Strengths: Consistency Simplicity Flexibility Weaknesses: No closed form Computation time