IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v550y2020ics0378437119322058.html
   My bibliography  Save this article

Multi-fractional generalized Cauchy process and its application to teletraffic

Author

Listed:
  • Li, Ming

Abstract

The contributions given in this paper are in two aspects. The first is to introduce a novel random function, which we call the multi-fractional generalized Cauchy (mGC) process. The second is to dissertate its application to network traffic for studying the multi-fractal behavior of traffic on a point-by-point basis. The introduced mGC process is with the time varying fractal dimension D(t) and the time varying Hurst parameter H(t). The representations of the autocorrelation function (ACF) and the power spectrum density (PSD) of the mGC process are proposed. Besides, the asymptotic expressions of the ACF and PSD of the mGC process are presented. The computation formula of D(t) is given. The mGC model may be a new tool to describe the multi-fractal behavior of traffic. Precisely, it may be used to reveal the local irregularity or local self-similarity (LSS), which is a small-time scale behavior of traffic, and global long-term persistence or long-range dependence (LRD), which is a large-time scale behavior of traffic, on a point-by-point basis. The cast study with real traffic traces exhibits that the variance of D(t) is much greater than that of H(t). Thus, the present mGC model may provide a novel way to explain the fact that traffic has highly local irregularity while its LRD is robust.

Suggested Citation

  • Li, Ming, 2020. "Multi-fractional generalized Cauchy process and its application to teletraffic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).
    • Handle: RePEc:eee:phsmap:v:550:y:2020:i:c:s0378437119322058
    DOI: 10.1016/j.physa.2019.123982
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437119322058
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2019.123982?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ayache, Antoine & Roueff, François & Xiao, Yimin, 2009. "Linear fractional stable sheets: Wavelet expansion and sample path properties," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1168-1197, April.
    2. de Coninck, Joël & Dunlop, François & Huillet, Thierry, 2008. "On the correlation structure of some random point processes on the line," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(4), pages 725-744.
    3. B. Podobnik & D. F. Fu & H. E. Stanley & P. Ch. Ivanov, 2007. "Power-law autocorrelated stochastic processes with long-range cross-correlations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 56(1), pages 47-52, March.
    4. Izabella Lokshina, 2016. "Study on estimating probabilities of buffer overflow in high-speed communication networks," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 62(2), pages 289-302, June.
    5. Li, Ming & Zhao, Wei, 2012. "Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6268-6278.
    6. Li, Ming, 2017. "Record length requirement of long-range dependent teletraffic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 472(C), pages 164-187.
    7. Muniandy, S.V. & Lim, S.C. & Murugan, R., 2001. "Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 407-428.
    8. Budhiraja, Amarjit & Liu, Xin, 2011. "Multiscale diffusion approximations for stochastic networks in heavy traffic," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 630-656, March.
    9. Cattani, Carlo & Ciancio, Armando, 2016. "On the fractal distribution of primes and prime-indexed primes by the binary image analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 222-229.
    10. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2003. "A theory of power-law distributions in financial market fluctuations," Nature, Nature, vol. 423(6937), pages 267-270, May.
    11. Li, Ming & Li, Jia-Yue, 2017. "Generalized Cauchy model of sea level fluctuations with long-range dependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 309-335.
    12. Li, Ming & Lim, S.C., 2008. "Modeling network traffic using generalized Cauchy process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2584-2594.
    13. I. Lubashevsky, 2011. "Truncated Lévy flights and generalized Cauchy processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 82(2), pages 189-195, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Heydari, M.H. & Razzaghi, M. & Rouzegar, J., 2022. "Chebyshev cardinal polynomials for delay distributed-order fractional fourth-order sub-diffusion equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    2. Li, Ming, 2021. "Generalized fractional Gaussian noise and its application to traffic modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 579(C).
    3. Angelini, Daniele & Bianchi, Sergio, 2023. "Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    4. Heydari, M.H. & Razzaghi, M., 2021. "A numerical approach for a class of nonlinear optimal control problems with piecewise fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Li, Ming & Wang, Anqi, 2020. "Fractal teletraffic delay bounds in computer networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    6. Hosseininia, M. & Heydari, M.H. & Avazzadeh, Z., 2022. "Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Ming & Li, Jia-Yue, 2017. "Generalized Cauchy model of sea level fluctuations with long-range dependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 309-335.
    2. Li, Ming, 2017. "Record length requirement of long-range dependent teletraffic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 472(C), pages 164-187.
    3. Li, Ming & Zhang, Peidong & Leng, Jianxing, 2016. "Improving autocorrelation regression for the Hurst parameter estimation of long-range dependent time series based on golden section search," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 189-199.
    4. Li, Ming & Wang, Anqi, 2020. "Fractal teletraffic delay bounds in computer networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    5. Li, Ming, 2021. "Generalized fractional Gaussian noise and its application to traffic modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 579(C).
    6. Dutta, Srimonti & Ghosh, Dipak & Samanta, Shukla, 2014. "Multifractal detrended cross-correlation analysis of gold price and SENSEX," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 195-204.
    7. Grobys, Klaus, 2023. "A Fractal and Comparative View of the Memory of Bitcoin and S&P 500 Returns," Research in International Business and Finance, Elsevier, vol. 66(C).
    8. Wang, Yudong & Wu, Chongfeng & Pan, Zhiyuan, 2011. "Multifractal detrending moving average analysis on the US Dollar exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3512-3523.
    9. Arshad, Shaista & Rizvi, Syed Aun R. & Ghani, Gairuzazmi Mat & Duasa, Jarita, 2016. "Investigating stock market efficiency: A look at OIC member countries," Research in International Business and Finance, Elsevier, vol. 36(C), pages 402-413.
    10. Bai, Man-Ying & Zhu, Hai-Bo, 2010. "Power law and multiscaling properties of the Chinese stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1883-1890.
    11. Gao, Yan & Gao, Yao, 2015. "Statistical properties of short-selling and margin-trading activities and their impacts on returns in the Chinese stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 293-307.
    12. Dutta, Srimonti & Ghosh, Dipak & Chatterjee, Sucharita, 2016. "Multifractal detrended Cross Correlation Analysis of Foreign Exchange and SENSEX fluctuation in Indian perspective," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 188-201.
    13. Song, Wanqing & Li, Ming & Li, Yuanyuan & Cattani, Carlo & Chi, Chi-Hung, 2019. "Fractional Brownian motion: Difference iterative forecasting models," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 347-355.
    14. Li, Ming & Sun, Xichao & Xiao, Xi, 2019. "Revisiting fractional Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 56-62.
    15. Li, Ming & Zhao, Wei, 2012. "Quantitatively investigating the locally weak stationarity of modified multifractional Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6268-6278.
    16. Abduraimova, Kumushoy, 2022. "Contagion and tail risk in complex financial networks," Journal of Banking & Finance, Elsevier, vol. 143(C).
    17. Juan C. Henao-Londono & Sebastian M. Krause & Thomas Guhr, 2021. "Price response functions and spread impact in correlated financial markets," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(4), pages 1-20, April.
    18. Igor Fedotenkov, 2020. "A Review of More than One Hundred Pareto-Tail Index Estimators," Statistica, Department of Statistics, University of Bologna, vol. 80(3), pages 245-299.
    19. Giorgio Fagiolo & Mauro Napoletano & Andrea Roventini, 2008. "Are output growth-rate distributions fat-tailed? some evidence from OECD countries," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 23(5), pages 639-669.
    20. Peng Yue & Qing Cai & Wanfeng Yan & Wei-Xing Zhou, 2020. "Information flow networks of Chinese stock market sectors," Papers 2004.08759, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:550:y:2020:i:c:s0378437119322058. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.