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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

   
 
 

 

Aggregation of network traffic and anisotropic scaling of random fields


Authors: Remigijus Leipus, Vytautė Pilipauskaitė and Donatas Surgailis
Journal: Theor. Probability and Math. Statist. 108 (2023), 77-126
MSC (2020): Primary 62M10, 60G22; Secondary 60G15, 60G18, 60G52, 60H05
DOI: https://doi.org/10.1090/tpms/1188
Published electronically: May 2, 2023
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Abstract: We discuss joint spatial-temporal scaling limits of sums $A_{\lambda ,\gamma }$ (indexed by $(x,y) \in \mathbb {R}^2_+$) of large number $O(\lambda ^{\gamma })$ of independent copies of integrated input process $X = \{X(t), t \in \mathbb {R}\}$ at time scale $\lambda$, for any given $\gamma >0$. We consider two classes of inputs $X$: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that the normalized random fields $A_{\lambda ,\gamma }$ tend to an $\alpha$-stable Lévy sheet $(1< \alpha <2)$ if $\gamma < \gamma _0$, and to a fractional Brownian sheet if $\gamma > \gamma _0$, for some $\gamma _0>0$. We also prove an ‘intermediate’ limit for $\gamma = \gamma _0$. Our results extend the previous works of R. Gaigalas and I. Kaj [Bernoulli 9 (2003), no. 4, 671–703] and T. Mikosch, S. Resnick, H. Rootzén and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23–68] and other papers to more general and new input processes.


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Additional Information

Remigijus Leipus
Affiliation: Vilnius University, Faculty of Mathematics and Informatics, Institute of Applied Mathematics, Naugarduko 24, 03225 Vilnius, Lithuania
Email: remigijus.leipus@mif.vu.lt

Vytautė Pilipauskaitė
Affiliation: Aalborg University, Department of Mathematical Sciences, Skjernvej 4A, 9220 Aalborg, Denmark
Email: vypi@math.aau.dk

Donatas Surgailis
Affiliation: Vilnius University, Faculty of Mathematics and Informatics, Institute of Applied Mathematics, Naugarduko 24, 03225 Vilnius, Lithuania
Email: donatas.surgailis@mif.vu.lt

Keywords: Heavy tails, long-range dependence, self-similarity, shot-noise process, regenerative process, superimposed network traffic, joint spatial-temporal limits, anisotropic scaling of random fields, scaling transition, intermediate limit, Telecom process, stable Lévy sheet, fractional Brownian sheet, renewal process, large deviations, ON/OFF process, M/G/$\infty$ queue, M/G/1/0 queue, M/G/1/$\infty$ queue
Received by editor(s): December 23, 2021
Accepted for publication: November 4, 2022
Published electronically: May 2, 2023
Dedicated: Dedicated to the memory of M. Yadrenko
Article copyright: © Copyright 2023 Taras Shevchenko National University of Kyiv