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