Non-triviality in a totally asymmetric one-dimensional Boolean percolation model on a half-line

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space covered a.s., or volume fraction strictly less than one. The aim of this work to demonstrate third possibility model with totally asymmetric grains on half-line: a.s. no unbounded component, but An explicit condition given characterizing existence an occupied component.