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Rényi's Parking Constants comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same.

Are there any published results when the distribution is non-uniform?

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    If you don't get any good answers here in the next day or so, you might try this at Math Overflow.2011-01-02
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    This was asked recently @ MathOverflow and didn't get much response - http://mathoverflow.net/questions/50924/random-parking-problem-on-a-probability-distribution2011-02-15
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    @Dinesh @Mike Spivey: The question was asked on both sites the same day. @Chao Xu: You might care to specify whether the reference provided on the MO page mentioned by @Dinesh satisfied you or not and, if not, why it did not.2011-02-15

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Here is one: Jean-François Marckert, Parking with density, Random Structures and Algorithms 18 (4), 364-380 (2001). A preprint version is available on this page.