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Regular variation in a fixed-point problem for single- and multi-class branching processes and queues

Part of: Stochastic analysis Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  01 February 2019

Søren Asmussen  and
Sergey Foss
Søren Asmussen*
Affiliation:
Aarhus University
Sergey Foss*
Affiliation:
Heriot-Watt University and Novosibirsk State University
*
Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark. Email address: [email protected]
School of Mathematical and Computer Sciences, Heriot-Watt University, EH14 4AS, Edinburgh, UK. Research supported by RSF grant No. 17-11-01173.
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Abstract

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Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are 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 multi-type branching processes and busy periods in multi-class queues.

Keywords

Busy periodGalton‒Watson processintermediate regular variationmultivariate regular variationrandom recursionrandom sum

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K25: Queueing theory 60F10: Large deviations
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 47 - 61
Copyright
Copyright © Applied Probability Trust 2018 

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