  Tail asymptotics of the solution $R$ to a fixpoint problem of type $R =_{st}
Q + \sum_1^N R_m$ is derived under heavy-tailed conditions allowing both
dependence between $Q$ and $N$ and the tails to be of the same order of
magnitude. Similar results are derived for a $K$-class version with
applications to multitype branching processes and busy periods in multiclass
queues.
