  We consider subcritical branching processes with immigration which evolve
under the influence of a random environment and study the tail distribution of
life periods of such processes defined as the length of the time interval
between the moment when first invader (or invaders) came to an empty site until
the moment when the site becomes empty again. We prove that the tail
distribution decays with exponential rate. The main tools are the change of
measure and some conditional limit theorems for random walks.
