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
Full-text PDF
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Additional Information
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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References
- S. Asmussen, Applied probability and queues, second edition, Applications of Mathematics (New York), vol. 51, Springer, New York, 2003. MR 1978607
- S. Asmussen and S. Foss, Regular variation in a fixed-point problem for single- and multiclass branching processes and queues, Adv. in Appl. Probab. 50 (2018), no. A, 47–61. MR 3905090
- A. Benassi, S. Jaffard, and D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90. MR 1462329
- A. Benassi, S. Cohen and J. Istas, Identification and properties of real harmonizable fractional Lévy motions, Bernoulli 8 (2002), no. 1, 97–115. MR 1884160
- H. Biermé, M. M. Meerschaert and H. P. Scheffler, Operator scaling stable random fields, Stochastic Process. Appl. 117 (2007), no. 3, 312–332. MR 2290879
- L. Breiman, On some limit theorems similar to the arc-sin law, Theory Probab. Appl. 10 (1965), 351–360. MR 0184274
- D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl. 49 (1994), no. 1, 75–98. MR 1258283
- S. Cohen, and J. Istas, Fractional fields and applications, Mathématiques & Applications (Berlin), vol. 73, Springer, Heidelberg, 2013. MR 3088856
- C. Dombry and I. Kaj, The on-off network traffic model under intermediate scaling, Queueing Syst. 69 (2011), no. 1, 29–44. MR 2835229
- R. Gaigalas, A Poisson bridge between fractional Brownian motion and stable Lévy motion, Stochastic Process. Appl. 116 (2006), no. 3, 447–462. MR 2199558
- R. Gaigalas and I. Kaj, Convergence of scaled renewal processes and a packet arrival model, Bernoulli 9 (2003), no. 4, 671–703. MR 1996275
- M. G. Genton, O. Perrin and M. S. Taqqu, Self-similarity and Lamperti transformation for random fields, Stoch. Models 23 (2007), no. 3, 397–411. MR 2341075
- D. Heath, S. Resnick and G. Samorodnitsky, Heavy tails and long range dependence in ON/OFF processes and associated fluid models, Math. Oper. Res. 23 (1998), no. 1, 145–165. MR 1606462
- I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. MR 0322926
- A. Iksanov, Renewal theory for perturbed random walks and similar processes, Probability and its Applications, Birkhäuser/Springer, Cham, 2016. MR 3585464
- A. Iksanov, A. Marynych and M. Meiners, Asymptotics of random processes with immigration I: Scaling limits, Bernoulli 23 (2017), no. 2, 1233–1278. MR 3606765
- I. Kaj, Limiting fractal random processes in heavy-tailed systems, In: Levy-Vehel, J., Lutton, E. (eds.), Fractals in Engineering, New Trends in Theory and Applications, Springer, London, 2005, pp. 199–218.
- I. Kaj and M. S. Taqqu, Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach, In: In and out of equilibrium 2, Progr. Probab., 60, Birkhäuser, Basel, 2008, pp. 383–427. MR 2477392
- R. Leipus, V. Paulauskas and D. Surgailis, On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise, J. Appl. Probab. 43 (2006), no. 2, 421–440. MR 2248574
- R. Leipus and D. Surgailis, On long-range dependence in regenerative processes based on a general ON/OFF scheme, J. Appl. Probab. 44 (2007), no. 2, 379–392. MR 2340205
- R. Leipus, A. Philippe, V. Pilipauskaitė, and D, Surgailis, Sample covariances of random-coefficient $\mathrm {AR}(1)$ panel model, Electron. J. Stat. 13 (2019), no. 2, 4527–4572. MR 4029802
- A. Marynych and G. Verovkin, A functional limit theorem for random processes with immigration in the case of heavy tails, Mod. Stoch. Theory Appl. 4 (2017), no. 2, 93–108. MR 3668776
- A. De Meyer and J. L. Teugels, On the asymptotic behaviour of the distributions of the busy period and service time in $M/G/1$, J. Appl. Probab. 17 (1980), no. 3, 802–813. MR 580039
- T. Mikosch, S. Resnick, H. Rootzén and A. Stegeman, Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab. 12 (2002), no. 1, 23–68. MR 1890056
- T. Mikosch and G. Samorodnitsky, Scaling limits for cumulative input processes, Math. Oper. Res. 32 (2007), no. 4, 890–918. MR 2363203
- T. Mikosch and O. Wintenberger, Precise large deviations for dependent regularly varying sequences, Probab. Theory Related Fields 156 (2013), no. 3-4, 851–887. MR 3078288
- K. Park and W. Willinger, Self-similar network traffic and performance evaluations, Wiley, New York, 2000.
- V. Pilipauskaitė, V. Skorniakov and D. Surgailis, Joint temporal and contemporaneous aggregation of random-coefficient $\mathrm {AR}(1)$ processes with infinite variance, Adv. in Appl. Probab. 52 (2020), no. 1, 237–265. MR 4092813
- V. Pilipauskaitė and D. Surgailis, Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes, Stochastic Process. Appl. 124 (2014), no. 2, 1011–1035. MR 3138604
- V. Pilipauskaitė and D. Surgailis, Joint aggregation of random-coefficient $\mathrm {AR}(1)$ processes with common innovations, Statist. Probab. Lett. 101 (2015), 73–82. MR 3332835
- V. Pilipauskaitė and D. Surgailis, Anisotropic scaling of the random grain model with application to network traffic, J. Appl. Probab. 53 (2016), no. 3, 857–879. MR 3570099
- V. Pilipauskaitė and D. Surgailis, Scaling transition for nonlinear random fields with long-range dependence, Stochastic Process. Appl. 127 (2017), no. 8, 2751–2779. MR 3660890
- V. Pilipauskaitė and D. Surgailis, Scaling limits of linear random fields on $\mathbb Z^2$ with general dependence axis, In and out of equilibrium 3, Progr. Probab., 77, Birkhäuser/Springer, Cham, 2021, pp. 683–710. MR 4237288
- V. Pipiras, M. S. Taqqu and J. B. Levy, Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed, Bernoulli 10 (2004), no. 1, 121–163. MR 2044596
- V. Pipiras and M. S. Taqqu, Small and large scale asymptotics of some Lévy stochastic integrals, Methodol. Comput. Appl. Probab. 10 (2008), no. 2, 299–314. MR 2399685
- J. W. Pratt, On interchanging limits and integrals, Ann. Math. Statist. 31 (1960), 74–77. MR 123673
- D. Puplinskaitė and D. Surgailis, Aggregation of autoregressive random fields and anisotropic long-range dependence, Bernoulli 22 (2016), no. 4, 2401–2441. MR 3498033
- D. Puplinskaitė and D. Surgailis, Scaling transition for long-range dependent Gaussian random fields, Stochastic Process. Appl. 125 (2015), no. 6, 2256–2271. MR 3322863
- B. S. Rajput and J. Rosiński, Spectral representations of infinitely divisible processes, Probab. Theory Related Fields 82 (1989), no. 3, 451–487. MR 1001524
- G. Samorodnitsky, Stochastic processes and long range dependence, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2016. MR 3561100
- K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge Univ. Press, Cambridge, 1999. MR 1739520
- D. Surgailis, Scaling transition and edge effects for negatively dependent linear random fields on $\mathbb {Z}^2$, Stochastic Process. Appl. 130 (2020), no. 12, 7518–7546. MR 4167214
- M.S. Taqqu, W. Willinger and R. Sherman, Proof of a fundamental result in self-similar traffic modeling, Comput. Commun. Rev. 27 (1997), no. 2, 5–23.
- H. Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer, New York, 2000. MR 1741181
- W. Willinger, M.S. Taqqu, M. Leland and D. Wilson, Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level, IEEE/ACM Trans. Networking* 5 (1997), 71–86.
- B. Zwart, Tail asymptotics for the busy period in the $GI/G/1$ queue, Math. Oper. Res. 26 (2001), no. 3, 485–493. MR 1849881
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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