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# Traffic Jams

Best Answer superprismatic, 03 December 2013 - 01:47 AM

Spoiler for Here's the gist of my not-very-clever method of solving this:

Go to the full post

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Posted 26 November 2013 - 04:08 AM

You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams.
In terms of N, what is the expected number of clumps of cars?

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### #2 phil1882

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Posted 26 November 2013 - 12:37 PM

by simulation i get...

Spoiler for

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### #3 plasmid

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Posted 27 November 2013 - 05:27 AM

I find that answer surprising if true.

Let A be the slowest car in the group. Since every car behind A will eventually catch up to A and be stuck behind it, there will be one traffic jam with A and all subsequent cars, and howevermany traffic jams involving the cars in front of A. If there are M cars in front of A then there can be no more than M+1 traffic jams, and that's only in the special case where every car in front of A is traveling slower than the car in front of it (if you count a single unimpeded car to be a traffic jam). In general there will be less than M+1. So if car A is randomly distributed among the pack, M will on average be about N/2 and the total number of traffic jams should be less than that.
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### #4 phil1882

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Posted 27 November 2013 - 05:05 PM

yes, i tihnk i ran my simulation wrong. trying to think of a way to fix it.

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### #5 bonanova

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Posted 28 November 2013 - 10:22 AM

Spoiler for approximately

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #6 superprismatic

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Posted 29 November 2013 - 09:44 PM

Spoiler for apparently

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### #7 bonanova

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Posted 02 December 2013 - 03:34 PM

Spoiler for apparently

I agree. Nice!

Is there a clever derivation?

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #8 superprismatic

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Posted 03 December 2013 - 01:47 AM   Best Answer

Spoiler for Here's the gist of my not-very-clever method of solving this:

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### #9 bonanova

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Posted 03 December 2013 - 01:49 PM

Spoiler for Here's the gist of my not-very-clever method of solving this:

Spoiler for Nice!

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

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Posted 03 December 2013 - 07:22 PM

Spoiler for stirling of the second kind

Edited by BMAD, 03 December 2013 - 07:23 PM.

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