  The TASEP is a paradigmatic model of out-of-equilibrium statistical physics,
for which many quantities have been computed, either exactly or by approximate
methods. In this work we study two new kinds of observables that have some
relevance in biological or traffic models. They represent the probability for a
given clearance zone of the lattice to be empty (for the first time) at a given
time, starting from a step density profile. Exact expressions are obtained for
single-time quantities, while more involved history-dependent observables are
studied by Monte Carlo simulation, and partially predicted by a
phenomenological approach.
