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bonanova

Traffic jams

Question

At 5-second intervals, 1024 automobiles enter a straight (and otherwise empty) single-lane highway traveling at initial speeds chosen at random from the interval [50, 70] miles per hour. Cars may not pass nor collide with other cars. When a slower car is encountered, a car must simply reduce its speed, and for the purposes of this puzzle we may consider the cars in such a case become permanently attached, traveling at the slower car's speed.

Eventually there will be N clusters of cars. What is the expected value of N? (Equivalently, what is the expected cluster size?)

Edited by bonanova
change to 1024 cars

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2 answers to this question

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okay, not too proud to start this off with some probably way simplistic maths:

Spoiler

six or seven clusters.  if you can have a fraction of a cluster would guess 6 5/9.

Think the time and speed intervals are red herrings.  Once the slowest car enters the highway all other vehicles will line up behind/cluster with it, eventually.  The slowest cars expected position would be 50.  Then the same scenario repeats so the second slowest expected at 25; then 12 1/2, 6 1/4, 3 1/8, 1 5/9.  But can you have a fraction of a car thus six or seven or on average 6 5/9?

 

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