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

Abstract |
References |
Similar Articles |
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.

References
- Søren Asmussen,
*Applied probability and queues*, 2nd ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003. Stochastic Modelling and Applied Probability. MR **1978607**
- Søren Asmussen and Sergey Foss,
*Regular variation in a fixed-point problem for single- and multi-class branching processes and queues*, Adv. in Appl. Probab. **50** (2018), no. A, 47–61. MR **3905090**, DOI 10.1017/apr.2018.69
- Albert Benassi, Stéphane Jaffard, and Daniel Roux,
*Elliptic Gaussian random processes*, Rev. Mat. Iberoamericana **13** (1997), no. 1, 19–90 (English, with English and French summaries). MR **1462329**, DOI 10.4171/RMI/217
- Albert Benassi, Serge Cohen, and Jacques Istas,
*Identification and properties of real harmonizable fractional Lévy motions*, Bernoulli **8** (2002), no. 1, 97–115. MR **1884160**
- Hermine Biermé, Mark M. Meerschaert, and Hans-Peter Scheffler,
*Operator scaling stable random fields*, Stochastic Process. Appl. **117** (2007), no. 3, 312–332. MR **2290879**, DOI 10.1016/j.spa.2006.07.004
- L. Breiman,
*On some limit theorems similar to the arc-sin law*, Teor. Verojatnost. i Primenen. **10** (1965), 351–360 (English, with Russian summary). 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**, DOI 10.1016/0304-4149(94)90113-9
- Serge Cohen and Jacques Istas,
*Fractional fields and applications*, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 73, Springer, Heidelberg, 2013. With a foreword by Stéphane Jaffard. MR **3088856**, DOI 10.1007/978-3-642-36739-7
- Clément Dombry and Ingemar Kaj,
*The on-off network traffic model under intermediate scaling*, Queueing Syst. **69** (2011), no. 1, 29–44. MR **2835229**, DOI 10.1007/s11134-011-9231-4
- Raimundas Gaigalas,
*A Poisson bridge between fractional Brownian motion and stable Lévy motion*, Stochastic Process. Appl. **116** (2006), no. 3, 447–462. MR **2199558**, DOI 10.1016/j.spa.2005.10.003
- Raimundas Gaigalas and Ingemar Kaj,
*Convergence of scaled renewal processes and a packet arrival model*, Bernoulli **9** (2003), no. 4, 671–703. MR **1996275**, DOI 10.3150/bj/1066223274
- Marc G. Genton, Olivier Perrin, and Murad S. Taqqu,
*Self-similarity and Lamperti transformation for random fields*, Stoch. Models **23** (2007), no. 3, 397–411. MR **2341075**, DOI 10.1080/15326340701471018
- David Heath, Sidney Resnick, and Gennady 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**, DOI 10.1287/moor.23.1.145
- I. A. Ibragimov and Yu. V. Linnik,
*Independent and stationary sequences of random variables*, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR **0322926**
- Alexander Iksanov,
*Renewal theory for perturbed random walks and similar processes*, Probability and its Applications, Birkhäuser/Springer, Cham, 2016. MR **3585464**, DOI 10.1007/978-3-319-49113-4
- Alexander Iksanov, Alexander Marynych, and Matthias Meiners,
*Asymptotics of random processes with immigration I: Scaling limits*, Bernoulli **23** (2017), no. 2, 1233–1278. MR **3606765**, DOI 10.3150/15-BEJ776
- 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.
- Ingemar Kaj and Murad S. Taqqu,
*Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach*, In and out of equilibrium. 2, Progr. Probab., vol. 60, Birkhäuser, Basel, 2008, pp. 383–427. MR **2477392**, DOI 10.1007/978-3-7643-8786-0_{1}9
- Remigijus Leipus, Vygantas Paulauskas, and Donatas 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**, DOI 10.1239/jap/1152413732
- Remigijus Leipus and Donatas 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**, DOI 10.1239/jap/1183667408
- Remigijus Leipus, Anne Philippe, Vytautė Pilipauskaitė, and Donatas Surgailis,
*Sample covariances of random-coefficient $\textrm {AR}(1)$ panel model*, Electron. J. Stat. **13** (2019), no. 2, 4527–4572. MR **4029802**, DOI 10.1214/19-EJS1632
- Alexander Marynych and Glib 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**, DOI 10.15559/17-VMSTA76
- 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**, DOI 10.2307/3212973
- Thomas Mikosch, Sidney Resnick, Holger Rootzén, and Alwin Stegeman,
*Is network traffic approximated by stable Lévy motion or fractional Brownian motion?*, Ann. Appl. Probab. **12** (2002), no. 1, 23–68. MR **1890056**, DOI 10.1214/aoap/1015961155
- Thomas Mikosch and Gennady Samorodnitsky,
*Scaling limits for cumulative input processes*, Math. Oper. Res. **32** (2007), no. 4, 890–918. MR **2363203**, DOI 10.1287/moor.1070.0267
- Thomas Mikosch and Olivier Wintenberger,
*Precise large deviations for dependent regularly varying sequences*, Probab. Theory Related Fields **156** (2013), no. 3-4, 851–887. MR **3078288**, DOI 10.1007/s00440-012-0445-0
- K. Park and W. Willinger,
*Self-similar network traffic and performance evaluations,* Wiley, New York, 2000.
- Vytautė Pilipauskaitė, Viktor Skorniakov, and Donatas Surgailis,
*Joint temporal and contemporaneous aggregation of random-coefficient $\textrm {AR}(1)$ processes with infinite variance*, Adv. in Appl. Probab. **52** (2020), no. 1, 237–265. MR **4092813**, DOI 10.1017/apr.2019.59
- Vytautė Pilipauskaitė and Donatas Surgailis,
*Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes*, Stochastic Process. Appl. **124** (2014), no. 2, 1011–1035. MR **3138604**, DOI 10.1016/j.spa.2013.10.004
- Vytautė Pilipauskaitė and Donatas Surgailis,
*Joint aggregation of random-coefficient $\textrm {AR}(1)$ processes with common innovations*, Statist. Probab. Lett. **101** (2015), 73–82. MR **3332835**, DOI 10.1016/j.spl.2015.03.002
- Vytautė Pilipauskaitė and Donatas Surgailis,
*Anisotropic scaling of the random grain model with application to network traffic*, J. Appl. Probab. **53** (2016), no. 3, 857–879. MR **3570099**, DOI 10.1017/jpr.2016.45
- Vytautė Pilipauskaitė and Donatas Surgailis,
*Scaling transition for nonlinear random fields with long-range dependence*, Stochastic Process. Appl. **127** (2017), no. 8, 2751–2779. MR **3660890**, DOI 10.1016/j.spa.2016.12.011
- Vytautė Pilipauskaitė and Donatas Surgailis,
*Scaling limits of linear random fields on $\Bbb Z^2$ with general dependence axis*, In and out of equilibrium 3. Celebrating Vladas Sidoravicius, Progr. Probab., vol. 77, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 683–710. MR **4237288**
- Vladas Pipiras, Murad S. Taqqu, and Joshua 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**, DOI 10.3150/bj/1077544606
- Vladas Pipiras and Murad S. Taqqu,
*Small and large scale asymptotics of some Lévy stochastic integrals*, Methodol. Comput. Appl. Probab. **10** (2008), no. 2, 299–314. MR **2399685**, DOI 10.1007/s11009-007-9052-4
- John W. Pratt,
*On interchanging limits and integrals*, Ann. Math. Statist. **31** (1960), 74–77. MR **123673**, DOI 10.1214/aoms/1177705988
- Donata Puplinskaitė and Donatas Surgailis,
*Aggregation of autoregressive random fields and anisotropic long-range dependence*, Bernoulli **22** (2016), no. 4, 2401–2441. MR **3498033**, DOI 10.3150/15-BEJ733
- Donata Puplinskaitė and Donatas Surgailis,
*Scaling transition for long-range dependent Gaussian random fields*, Stochastic Process. Appl. **125** (2015), no. 6, 2256–2271. MR **3322863**, DOI 10.1016/j.spa.2014.12.011
- Balram S. Rajput and Jan Rosiński,
*Spectral representations of infinitely divisible processes*, Probab. Theory Related Fields **82** (1989), no. 3, 451–487. MR **1001524**, DOI 10.1007/BF00339998
- Gennady Samorodnitsky,
*Stochastic processes and long range dependence*, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2016. MR **3561100**, DOI 10.1007/978-3-319-45575-4
- Ken-iti Sato,
*Lévy processes and infinitely divisible distributions*, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR **1739520**
- Donatas Surgailis,
*Scaling transition and edge effects for negatively dependent linear random fields on $\Bbb {Z}^2$*, Stochastic Process. Appl. **130** (2020), no. 12, 7518–7546. MR **4167214**, DOI 10.1016/j.spa.2020.08.005
- 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.
- Hermann Thorisson,
*Coupling, stationarity, and regeneration*, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR **1741181**, DOI 10.1007/978-1-4612-1236-2
- 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.
- A. P. Zwart,
*Tail asymptotics for the busy period in the $GI/G/1$ queue*, Math. Oper. Res. **26** (2001), no. 3, 485–493. MR **1849881**, DOI 10.1287/moor.26.3.485.10584

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**

Similar Articles

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2020):
62M10,
60G22,
60G15,
60G18,
60G52,
60H05

Retrieve articles in all journals
with MSC (2020):
62M10,
60G22,
60G15,
60G18,
60G52,
60H05

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