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

TL;DR

This paper investigates the tail behavior of solutions to a fixed-point equation in branching processes and queues, revealing regular variation properties under heavy-tailed conditions with dependence.

Contribution

It extends tail asymptotics to dependent heavy-tailed scenarios in single- and multi-class branching processes and queues, providing new theoretical insights.

Findings

Tail asymptotics derived for dependent heavy-tailed variables

Regular variation properties established for fixed-point solutions

Applications demonstrated in multitype branching processes and multiclass queues

Abstract

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.