Construct a Poisson point process of density one on a line of length $L$.
Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the total region they can see is the diameter of a circle of radius $r$). Let each point of the process lie at the midpoint of this diameter.
The two images below depict the situation when the point process contains 5 points (top) and 8 points (bottom).
Now, this partitions the line. I have labelled each partition with an integer equal to the number of points that have it in their "sights". For example, in the top picture one of the partitions is visible to four nodes on the line.
Say $P(k=n)$ is the probability that there are $n$ of partitions of label $k$. What is this function?
It would also be nice to know what the sum of the lengths of all regions of a particular label is. For example, in the top diagram there are three partitions of label 2, but what is the distribution of the length of the concatenation of all three partitions?
I imagine there are already some results on this particular sort of partitioning of an interval?
I know that:
The partitions are all exponentially distributed random variables.