We consider the layer-by-layer growth on vicinal surfaces when the sep
aration between the geometrical steps is of the order of an adatom dif
fusion length with respect to irreversible nucleation. The surface dif
fusion is strongly anisotropic, therefore the steps become rough. We d
erive a type of diffusion equation for kinks on the roughened steps an
d obtain a distribution function of the lengths of the terraces betwee
n two adjacent steps. This enables us to calculate the distribution of
quasi 1D nuclei on the terraces. We predict a sharp increase in the n
umber of non-equilibrium surface vacancies in the range of parameters
where the nucleation takes over from 'step flow' growth. The theory is
applied to the (110) vicinal surface of silicon. Calculated results a
re compared to some previously published and new experimental data.