On the one dimensional poisson random geometric graph

Given a poisson process on a bounded interval,its random geometric graph is the graph whose vertices are the points of the poisson process,and edges exist between two points if and only if their distance is less than a fixed given threshold. we compute explicitly the distribution of the number of connected components of this graph. the proof relies on inverting some laplace transforms. © 2011 l. decreusefond and e. ferraz.