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

Part of: Stochastic analysis Markov processes Special processes Limit theorems

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 

References

[1]Aldous, D. J. and Bandyopadhyay, A. (2005).A survey of max-type recursive distributional equations.Ann. Appl. Prob. 15,1047–1110.Google Scholar
[2]Alsmeyer, G. and Meiners, M. (2013).Fixed points of the smoothing transform: two-sided solutions.Prob. Theory Relat. Fields 155,165–199.Google Scholar
[3]Alsmeyer, G.,Biggins, J. D. and Meiners, M. (2012).The functional equation of the smoothing transform.Ann. Prob. 40,2069–2105.Google Scholar
[4]Basrak, B.,Davis, R. A. and Mikosch, T. (2002).A characterization of multivariate regular variation.Ann. Appl. Prob. 12,908–920.Google Scholar
[5]Bingham, N. H. and Doney, R. A. (1974).Asymptotic properties of supercritical branching processes. I. The Galton-Watson process.Adv. Appl. Prob. 6,711–731.Google Scholar
[6]Bingham, N. H. and Doney, R. A. (1975).Asymptotic properties of supercritical branching processes. II. Crump-Mode and Jirina processes.Adv. Appl. Prob. 7,66–82.Google Scholar
[7]Buraczewski, D.,Damek, E. and Mikosch, T. (2016).Stochastic Models with Power-Law Tails.Springer.Google Scholar
[8]Buraczewski, D.,Damek, E.,Mentemeier, S. and Mirek, M. (2013).Heavy tailed solutions of multivariate smoothing transforms.Stoch. Process. Appl. 123,1947–1986.Google Scholar
[9]De Meyer, A. and Teugels, J. L. (1980).On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1.J. Appl. Prob. 17,802–813.Google Scholar
[10]Denisov, D.,Foss, S. and Korshunov, D. (2010).Asymptotics of randomly stopped sums in the presence of heavy tails.Bernoulli 16,971–994.Google Scholar
[11]Durrett, R. and Liggett, T. M. (1983).Fixed points of the smoothing transformation.Z. Wahrscheinlichkeitsth. 64,275–301.Google Scholar
[12]Dwass, M. (1969).The total progeny in a branching process and a related random walk.J. Appl. Prob. 6,682–686.Google Scholar
[13]Embrechts, P.,Klüppelberg, C. and Mikosch, T. (1997).Modelling Extremal Events.Springer,Berlin.Google Scholar
[14]Ernst, P. A.,Asmussen, S. and Hasenbein, J. J. (2018).Stability and tail asymptotics in a multiclass queue with state-dependent arrival rates.Queueing Systems 90,207–224.Google Scholar
[15]Faÿ, G.,González-Arévalo, B.,Mikosch, T. and Samorodnitsky, G. (2006).Modeling teletraffic arrivals by a Poisson cluster process.Queueing Systems 54,121–140.Google Scholar
[16]Foss, S. (1984).Queues with customers of several types. In Advances in Probability Theory: Limit Theorems & Related Problems, ed. A. A. Borovkov,Optimization Software, Publications Division,New York, pp. 348–377.Google Scholar
[17]Foss, S. and Zachary, S. (2003).The maximum on a random time interval of a random walk with long-tailed increments and negative drift.Ann. Appl. Prob. 13,37–53.Google Scholar
[18]Foss, S.,Korshunov, D. and Zachary, S. (2013).An Introduction to Heavy-Tailed and Subexponential Distributions,2nd edn.Springer,New York.Google Scholar
[19]Jagers, P. (1975).Branching Processes with Biological Applications.John Wiley,New York.Google Scholar
[20]Jelenković, P. R. and Olvera-Cravioto, M. (2010).Information ranking and power laws on trees.Adv. Appl. Prob. 42,1057–1093.Google Scholar
[21]Meerschaert, M. M. and Scheffler, H.-P. (2001).Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice.John Wiley,New York.Google Scholar
[22]Resnick, S. I. (1987).Extreme Values, Regular Variation, and Point Processes (Ser. Appl. Prob. Trust 4).Springer,New York.Google Scholar
[23]Resnick, S. I. (2007).Heavy-Tail Phenomena: Probabilistic and Statistical Modeling.Springer,New York.Google Scholar
[24]Seneta, E. (1969).Functional equations and the Galton‒Watson process.Adv. Appl. Prob. 1,1–42.Google Scholar
[25]Seneta, E. (1974).Regularly varying functions in the theory of simple branching processes.Adv. Appl. Prob. 6,408–420.Google Scholar
[26]Vatutin, V. A. (1977).Limit theorems for critical Markov branching processes with several types of particles and infinite second moments.Mat. Sbornik 32,215–225.Google Scholar
[27]Volkovich, Y. and Litvak, N. (2010).Asymptotic analysis for personalized web search.Adv. Appl. Prob. 42,577–604.Google Scholar
[28]Zwart, A. P. (2001).Tail asymptotics for the busy period in the GI/G/1 queue.Math. Operat. Res. 26,485–493.Google Scholar