The time of completion of a linear birth-growth model

Consider the following birth-growth model in R. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.