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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 bothdependence between $Q$ and $N$ and the tails to be of the same order ofmagnitude. Similar results are derived for a $K$-class version withapplications to multitype branching processes and busy periods in multiclassqueues.

Regular Variation in a Fixed-Point Problem for Single- and Multiclass Branching Processes and Queues