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Power Law and Tsallis Entropy: Network Traffic and Applications

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Chaos, Nonlinearity, Complexity

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 206))

Abstract

A theoretical framework based on non-extensive Tsallis entropy is proposed to study the implication of long-range dependence in traffic process on network performance. Highlighting the salient features of Tsallis entropy, the axiomatic foundations of parametric entropies are also discussed. Possible application of nonextensive thermodynamics to study the macroscopic behavior of broadband network is outlined.

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References

  1. V. Lathora, A. Rapisarda and S. Ruffo, Lyapunov instablity and finite size efiects in a system with long range forces, Physical review letters 80 (1998), 692–695.

    Article  Google Scholar 

  2. C.K.Peng, S.V.Buldyrev, A.L. Goldberger, S. Havlin, F. Sciotino, M. Simons and H.E. Stanley, Long range correlations in nucleotide sequences, Nature 356 (1992), 168–170.

    Article  Google Scholar 

  3. G. Malescio, N.V. Dokholyan, S.V. Buldyrev and H.E. Stanley, e-print cond-mat/0005178.

    Google Scholar 

  4. C. Beck, G.S.Lewis and H.L. Swinney, Measuring nonextensitivity parameters in a turbulent Couette-Taylor flow, Phys. Rev. E 63, 035303 (2001).

    Google Scholar 

  5. Karmeshu and A. Krishnamchari, Sequence variability and long range dependence in DNA: An information theoretic perspective in neural information processing, Lecture Notes in Computer Science 3316 (2004), 1354–1361.

    Article  Google Scholar 

  6. A. H. Makse, S. Havlin and H.E. Stanley, Nature (London), 377 (608) (1995).

    Google Scholar 

  7. G.K. Zipf, Selective studies and the principle of relative frequency in language, Harvard University Press, Cambridge MA, 1937.

    Google Scholar 

  8. A. Robledo, Renormalization group, entropy optimization, and nonextensivity at criticality, Phys Rev. Lett. 83, 2289 (1999).

    Article  Google Scholar 

  9. D.B. Walton and J. Rafelski, Equilibrium distribution of heavy quarks in Fokker-Planck dynamics, Phys Rev. Lett. 84 (2000), 31–34.

    Article  Google Scholar 

  10. C. Tsallis, J.C. Anjos and E.P. Borges, e-print astro-ph/0203258.

    Google Scholar 

  11. H.S. Wio and S. Bouzat, Brazil journal of Physics, 29, 136 (1999).

    Google Scholar 

  12. E. Jaynes, Prior probabilities, IEEE Transactions on System Science Cybernetics SSC-4 (1968), 227–241.

    Article  Google Scholar 

  13. C.E. Shannon, A mathematical theory of communication, Bell system technical journal 27 (1948), 379–423.

    MathSciNet  MATH  Google Scholar 

  14. M. Gell-Mann and C. Tsallis, Nonextensive Entropy Interdisciplinary Applications- Preface, Oxford university press, 2004.

    Google Scholar 

  15. C. Tsallis, Possible generalization of Boltzmann-Gibbs Statistics, Journal of Statistical Physics, 52 (12) (1988), 479–487.

    Article  MathSciNet  MATH  Google Scholar 

  16. E.W. Montroll and M.F. Shlesinger, On the wonderful world of random walks, Nonequilibrium phenomena II from stochastics to hydrodynamics, ed. J.L. Lebowitz and E.W. Montroll, North-Holland physics publishing, 1984.

    Google Scholar 

  17. V. Latora, M. Baranger, A. Rapisarda and C. Tsallis, The rate of entropy increase at the edge of chaos. Phys. Letters A 273 (90), (2000).

    Google Scholar 

  18. S. Abe and A.K. Rajagopal, Rates of convergence of non-extensive statistical distributions to Levy distributions in full and half-spaces. Journal of physics A 33 (48) (2000), 8723–8732.

    Article  MathSciNet  Google Scholar 

  19. S. Abe and N. Suzuki, J. Geophys Res. 108(B2) (2003) 2113.

    Article  Google Scholar 

  20. S. Abe and N. Suzuki, Itineration of the internet over non equilibrium stationary states in Tsallis Statistics. Physical review E 67 (016106), (2003).

    Google Scholar 

  21. A. Erramilli, O. Narayan, W. Willinger, Experimental queueing analysis with LRD packet traffic, IEEE/ACM Transactions on Networking 4 (2) (1996), 209–223.

    Article  Google Scholar 

  22. W. Leland, M.S. Taqqu, W.Willinger and D.V. Wilson, On the selfsimilar nature of Ethernet traffic, IEEE/ACM Transactions on Networking 2 (1) (1994), 1–15.

    Article  Google Scholar 

  23. Mark E. Crovella and Azer Bestavros, Self-similarity in world wide web traffic : Evidence and possible causes, IEEE/ACM Transactions on Networking 5 (6) (1997), 835–846.

    Article  Google Scholar 

  24. D. Duffy, A. McIntosh, M. Rosenstein, W. Willinger, Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks, IEEE jounal on selected areas in communication 12 (3) (1994), 544–551.

    Article  Google Scholar 

  25. William Stallings, High speed networks and internets: Performance and quality of service, Second Edition, Pearson, 2002.

    Google Scholar 

  26. Paxon and Floyd, Wide area traffic : the failure of Poisson modelling, IEEE/ACM Transactions on Networking 3 (3) (1995), 226–244.

    Article  Google Scholar 

  27. S. Abe, Tsallis entropy: How unique?, preprint (2001).

    Google Scholar 

  28. C. Beck and F. Schlogl, Thermodynamics of chotic systems - an introduction, Cambridge University Press, 1993.

    Google Scholar 

  29. S. Abe, A note on the q-deformation theoretic aspect of the genealized entropies in non-extensive physics, Phys Letters A 224 (1997) 326.

    Google Scholar 

  30. F. Jackson, Mess. Math, 38 (1909) 57.

    Google Scholar 

  31. F. Jackson, Quart. J. Pure Appl. Math, 41 (1910) 1993.

    Google Scholar 

  32. J.N.Kapur and H.K.Kesavan, Entropy optimization principles with application, Academic Press Inc, London, 1992.

    Google Scholar 

  33. Shachi Sharma, Krishanmachari and Karmeshu, Validity of Jaynes’ entropy concentration theorem: Tsallis and other generalized entropy measures, preprint (2005 a).

    Google Scholar 

  34. S. Martin, G. Morison, W. Nailon and T. Durrani, Fast and accurate image registration using Tsallis entropy and simultaneous perturbation stochastic approximation, Electonics Letters, 40 (10) (2004).

    Google Scholar 

  35. J. Havrda and F. Charvat, Quantification method of classification processes: concept of structural α-entropy, Kybernetika 3 (1967), 30–35.

    MathSciNet  MATH  Google Scholar 

  36. C. Tsallis, Non-extensive statistical mechanics: construction and physical interpretation, in Nonextensive Entropy Interdisciplinary Applications ed. M. Gellmann and C. Tsallis, Oxford university press, 2004, 1–52.

    Google Scholar 

  37. A. Renyi, On measures of entropy and information, Proceedings Fourth Berkeley Symp. Math. Statist and Prob., University of California Press 1 (1961), 547–561.

    Google Scholar 

  38. A.M. Mathai and P.N. Rathie, Basic concepts in information theory and statistics, Johan Wiley and sons, 1975.

    Google Scholar 

  39. S. Abe, Stability of Tsallis entropy and instabilities of Renyi and normalized Tsallis entropies: A basis for q-exponential distributions, physical review E 66 (046134) (2002).

    Google Scholar 

  40. A.I. Khinchin, Mathematical foundations of information theory, Dover publications, 1957.

    Google Scholar 

  41. D.K. Fadeev, On the concept of entropy of a finite probabilistic scheme (Russian), Uspeki Mat. Nauk 11 (1956), 227–231.

    Google Scholar 

  42. Hiroki Suyari, Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy, IEEE transactions on information theory, 50 (8) (2004), 1783–1787.

    Article  MathSciNet  Google Scholar 

  43. Karmeshu and N.R. Pal, Uncertainty, entropy and maximum entropy principle-an overview, in Entropy measures, maximum entropy principle and emerging applications, ed. Karmeshu, Springer-Verlag, 2003.

    Google Scholar 

  44. B. Forte and C.T. Ng, On a characterization of the entropies of degree β, Utilitas Mathematica 4 (1973), 193–205.

    MathSciNet  MATH  Google Scholar 

  45. Karmeshu and Shachi Sharma, Queue length distribution of network packet traffic: Tsallis entropy maximization with fractional moments, accepted in IEEE communiation letters, 2005.

    Google Scholar 

  46. Karmeshu and Shachi Sharma, Long tail behavior of queue lengths in broadband networks: Tsallis entropy framework, preprint (2005 b).

    Google Scholar 

  47. T.M. Apostol, Introduction to analytic number theory, Springer-verlag, Newyork, 1976.

    MATH  Google Scholar 

  48. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, 1983.

    Google Scholar 

  49. S. Abe and A.K. Rajgopal, Micro canonical foundation for systems with power-law distributions. Journal of physics A 33 (48), 8733–8738 (2000).

    Article  Google Scholar 

  50. S. Abe and G.B. Bagci, Necessity of q-expectation value in nonextensive statistical mechanics. Physical review E 71 (016139), (2005).

    Google Scholar 

  51. Shachi Sharma and Karmeshu, Asymptotic power law characteristics in queueing systems-using maximum Tsallis entropy principle, preprint (2005 c).

    Google Scholar 

  52. J.Y. Hui and E. Karasan, A thermodynamic theory of broadband networks with application to dynamic routing, IEEE journal of selected areas in communications 13 (6) (1995), 991–1003.

    Article  Google Scholar 

  53. V.E.Benes, Mathematical theory of connecting networks and telephone traffic, NewYork, Academic, 1965.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi, 110067, India

    Karmeshu & Shachi Sharma

  2. Centre for Development of Telematics, New Delhi, 110030, India

    Shachi Sharma

Authors
  1. Karmeshu
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  2. Shachi Sharma
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Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Nuclear Engineering and Technology Programme Indian Institute of Technology Kanpur, Kanpur, 208016, India

    A. Sengupta Professor

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Karmeshu, Sharma, S. (2006). Power Law and Tsallis Entropy: Network Traffic and Applications. In: Sengupta, A. (eds) Chaos, Nonlinearity, Complexity. Studies in Fuzziness and Soft Computing, vol 206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31757-0_5

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  • DOI: https://doi.org/10.1007/3-540-31757-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31756-2

  • Online ISBN: 978-3-540-31757-9

  • eBook Packages: EngineeringEngineering (R0)

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