  We consider a system of independent one-dimensional random walkers where new
particles are added at the origin at fixed rate whenever there is no older
particle present at the origin. A Poisson ansatz leads to a semi-linear lattice
heat equation and predicts that starting from the empty configuration the total
number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction
and also describe the asymptotic macroscopic profile of the particle
configuration.

Probability