  We present a mean-field model that describes droplet growth due to
condensation and collisions and droplet loss due to fallout. The model allows
for an effective numerical simulation. We study how the rain initiation time
depends on different parameters. We also present a simple model that allows one
to estimate the rain initiation time for turbulent clouds with an inhomogeneous
concentration of cloud condensation nuclei. In particular, we show that
over-seeding even a part of a cloud by small hygroscopic nuclei one can
substantially delay the onset of precipitation.
