I don't have an answer either, except to note that this setup is very similar to the core of my dissertation. This stochastic distribution of random-length intervals models the problem of genomic mapping; if you have blasted many copies of some genetic material into fragments, how do you characterize the fragmentations that can uniquely be reconstructed (up to complete reversal) -- vs. fragmentations that are ambiguous (can be consistently reconstructed in multiple ways)?
Then there's the question of, OK, you've sampled fragments of DNA with a distribution that you believe gives you sufficient probability of being unambiguous, so how do you actually figure out how to order the fragments into a map? If you further cut the fragments at all occurrences of a few short DNA sequences (restriction sites for digestion enzymes), overlaping areas will yield common distributions of sub-fragment lengths, and you can begin to see who overlaps who, and start to piece the whole thing together.
If you are cool with postscript, you can find two papers at this link (search for Settergren), or I have pdf'd them here and there. Or,you could just download the whole dissertation.