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For Whom The Booth Tolls

For Whom The Booth Tolls. Brian Camley Pascal Getreuer Brad Klingenberg. Problem. Needless to say, we chose problem B. (We like a challenge). What causes traffic jams?. If there are not enough toll booths, queues will form

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For Whom The Booth Tolls

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  1. For Whom The Booth Tolls Brian CamleyPascal GetreuerBrad 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: This characterizes the first half of the toll plaza! What about merging?

  8. Merging

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

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

  11. NS Behavior

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

  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. Minimum at n = 4 For Two Lanes

  20. Minimum at n = 6 For Three Lanes

  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

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