  We consider a fixed-point equation for a non-negative integer-valued random
variable, that appears in branching processes with state-independent
immigration. A similar equation appears in the analysis of a single-server
queue with a homogeneous Poisson input, feedback and permanent customer(s).
  It is known that the solution to this equation uniquely exists under mild
first and logarithmic moments conditions. We find further the tail asymptotics
of the distribution of the solution when the immigration size and branch size
distributions are heavy-tailed. We assume that the distributions of interest
are dominantly varying and have a long tail. This class includes, in
particular, (intermediate, extended) regularly varying distributions.
  We consider also a number of generalisations of the model.
