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Heavy Traffic Methods in Wireless Systems: Towards Modeling Heavy Tails and Long Range Dependence

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Wireless Communications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 143))

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Abstract

Heavy traffic models for wireless queueing systems under short range dependence and light tail assumptions on the data traffic have been studied recently. We outline one such model considered by Buche and Kushner [7]. At the same time, similarly to what happened for wireline networks, the emergence of high capacity applications (multimedia, gaming) and inherent mechanisms (multi-access interference) of wireless networks have led to the growing evidence of long range dependence and heavy tail characteristics in data traffic. Extending heavy traffic methods under these assumptions presents significant challenges. We discuss an approach for extending the methods in [7] under a heavy tail assumption only. The corresponding heavy traffic model is based on (non-Gaussian) stable Lévy motion, not Brownian motion which is associated with a light tail assumption. When long range dependence is also present, a promising alternative approach and model based on a Poisson measure representation, motivated from Kurtz [17], are described. The corresponding heavy traffic model is now driven by fractional Brownian motion. As stochastic control analysis for stable Lévy motion or fractional Brownian motion is currently undeveloped, the queue limit models characterizing the wireless system can be studied only under given controls, such as stabilizing controls or else heuristic policies.

supported by Intelligent Automation Inc., Rockville, MD, through ARO grant W911NF-04-C-0138.

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References

  1. B. Ata, J.M. Harrison, and L.A. Shepp, Drift Rate Control of a Brownian Processing System, The Annals of Applied Probability, 15(2): 1145–1160, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  2. D.R. Basgeet, J. Irvine, A. Munro, P. Dugenie, D. Kaleshi, and O. Lazaro, Impact of Mobility on Aggregate Traffic in Mobile Multimedia System, The 5th International Symposium on Wireless Personal Multimedia Communications (IEEE), 2: 333–337, Oct. 2002.

    Google Scholar 

  3. R.F. Bass, Uniqueness in Law for Pure Jump Markov Processes, Probability Theory and Related Fields, 79: 271–287, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  4. R.F. Bass, Stochastic differential equations with jumps, Probability Surveys, 1: 1–19, 2004.

    Article  MathSciNet  Google Scholar 

  5. R.F. Bass, K. Burdzy, and Z.Q. Chen, Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Processes and their Applications, 111(1): 1–15, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Buche and H.J. Kushner, Rate of convergence for constrained stochastic approximation algorithms, SIAM Journal on Control and Optimization, 40: 1011–1041, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Buche and H.J. Kushner, Control of Mobile Communications With Time-Varying Channels in Heavy Traffic, IEEE Transactions on Automatic Control, 47(6): 992–1003, 2002.

    Article  MathSciNet  Google Scholar 

  8. R. Buche and H.J. Kushner, Control of Mobile Communication Systems With Time-Varying Channels via Stability Methods, IEEE Transactions on Automatic Control, 49(11): 1954–1962, 2004.

    Article  MathSciNet  Google Scholar 

  9. R.T. Buche and C. Lin, Heavy traffic control policies for wireless systems with time-varying channels, Proceedings, American Control Conference, 6: 3972–3974, 2005.

    Article  Google Scholar 

  10. S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986.

    MATH  Google Scholar 

  11. P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton University Press, 2002.

    Google Scholar 

  12. M. Izouierdo and D.S. Reeves, A survey of statistical source models for variable-bit-rate compressed video, Multimedia Systems, 7: 199–213, 1999.

    Article  Google Scholar 

  13. M. Jiang, M. Nikolic, S. Hardy, and L. Trajkovic, Impact of self-similarity on wireless data network performance, IEEE International Conference on Communications, 2: 477–481, 2001.

    Google Scholar 

  14. R. Kalden and S. Ibrahim, Searching for self-similarity in GPRS, Proceedings of the 5th annual Passive and Active Measurement Workshop (PAM 2004), Antibes Juan-les-Pins, France, April 19–20, 2004.

    Google Scholar 

  15. T. Komatsu, On the martingale problem for generators if stable processes with perturbations, Osaka J. of Mathematics, 21(1): 113–132, 2004.

    MathSciNet  Google Scholar 

  16. A. Krendzel, Y. Koucheryavy, J. Hariu, and S. Lopatin, Network Planning Problems in 3G/4G Wireless Systems, The 1st COST 290 Management Committee Meeting,Malta, Oct. 2004.

    Google Scholar 

  17. T.G. Kurtz, Limit Theorems for workload input models, Stochastic Neworks: Theory and Applications, Eds. F.P. Kelly, S. Zhachery & I. Ziedins, Oxford, 1996, pp. 119–139.

    Google Scholar 

  18. H.J. Kushner and L.F. Martins, Heavy Traffic Analysis of a Data Transmission System with many Independent Sources, SIAM J. on Appl. Math., 53(4): 1095–1122.

    Google Scholar 

  19. H.J. Kushner, Approximation and Weak Convergence Methods for Random Processes, MIT Press, 1984.

    Google Scholar 

  20. H.J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Second Edition, Springer, New York, 2001.

    Google Scholar 

  21. H.J. Kushner, Heavy Traffic Analysis of Controlled Queueing and Communication Networks, Springer, 2002.

    Google Scholar 

  22. H.J. Kushner, J. Yang, and D. Jarvis, “Controlled and optimally controlled multiplexing systems: A numerical exploration”, Queueing Systems, 20: 255–291, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  23. H.J. Kushner and G. Yin, Stochastic approximation and recursive algorithms and applications, second edition, Springer, New York, 2003.

    MATH  Google Scholar 

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

    Article  Google Scholar 

  25. T. Mikosch, S. Resnick, H. Rootzén, and A. Stegeman, Is Network Traffic Approximated by Stable Lévy Motion or Fractional Brownian Motion?, The Annals of Applied Probability, 12(1): 23–68, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Mikulevicius and H. Pragarauskas, On the martingale problem associated with nondegenerate Lévy operators, Lithuanian Mathematical Journal, 31(3): 297–311, 1992.

    Article  MathSciNet  Google Scholar 

  27. R. Narasimha and R. Rao, Modeling Variable Bit Rate Video On Wired and Wireless Networks Using Discrete-Time Self-Similar Systems, Proceedings, IEEE International Conference on Personal Wireless Communications, pp. 290–294, 2002.

    Google Scholar 

  28. K. Park and W. Willinger, Self-Similar Network Traffic and Performance Evaluation, J. Wiley & Sons, Inc., New York, 2000.

    Book  Google Scholar 

  29. V. Pipiras and M. Taqqu, Integration questions related to fractional Brownian motion, Probability Theory and Related Fields, 118: 251–291, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Pragarauskas and P.A. Zanzotto, On one-dimensional stochastic differential equations driven by stable processes, Lithuanian Mathematical Journal, 40(3): 277–295, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Shakkottai, R. Srikant, and A. Stoylar, Pathwise optimality of the exponential rule for wireless channels, Advances in Applied Probability, 36(4): 1021–1045, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  32. A.L. Stolyar, Max Weight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, The Annals of Applied Probability, 14(1): 1–53, February 2004.

    Google Scholar 

  33. D. Stroock, Diffusion processes associated with Lévy generators, Z. Warscheinlichkeitstheorie verw. Gebiete, 32: 209–244, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. Whitt, An overview of Brownian and non-Brownian FCLTs for the singleserver queue, Queueing Systems, 36: 39–70, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  35. W. Willinger, M.S. Taqqu, R. Sherman, and D.V. Wilson, Self-similarity through high-variability: Statistical analysis of ethernet LAN traffic at the source level, IEEE/ACM Trans. Networking, 5(1): 71–86, Feb. 1997.

    Google Scholar 

  36. P.A. Zanzotto, On stochastic differential equations driven by a Cauchy process and other stable Lévy motions, The Annals of Probability, 30(2): 802–825, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Zhang, M. Hu, and N.B. Shroff, Bursty Data Over CDMA: MAI Self Similarity, Rate Control, and Admission Control, Proceedings, IEEE INFOCOM, 1: 391–399, 2002.

    Google Scholar 

  38. J. Zhang and T. Konstantopoulos, Multiple-Access Interference Processes Are Self-Similar in Multimedia CDMA Cellular Networks, IEEE Transactions on Information Theory, 51(3): 1024–1038, 2005.

    Article  MathSciNet  Google Scholar 

  39. J.A. Zhao, B. Li, C.W. Kok, and I. Ahmad, MPEG-4 Video Transmission over Wireless Networks: A Link Level Performance Study, Wireless Networks, 10: 133–146, 2004.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, NC State University, Campus Box 8205, Raleigh, NC, 27695

    Robert T. Buche & Jim X. Zhang

  2. Statistics Department, Iowa State University, 303 Snedecor Hall, Ames, IA, 50011-1210

    Arka Ghosh

  3. Department of Statistics and OR, University of North Carolina, Smith Bldg, CB3260, Chapel Hill, NC, 27599

    Vladas Pipiras

Authors
  1. Robert T. Buche
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  2. Arka Ghosh
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  3. Vladas Pipiras
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  4. Jim X. Zhang
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, 36849-5201, USA

    Prathima Agrawal

  2. Network Advanced Technology, Motorola, Inc., Arlington Heights, IL, 60004, USA

    Philip J. Fleming

  3. Computing Sciences Research Center, Bell Laboratories Lucent Technologies, Murray Hill, NJ, 07974, USA

    Lisa Zhang

  4. Bell Laboratories, Lucent Technologies, Murray Hill, NJ, 07974-0636, USA

    Daniel Matthew Andrews

  5. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    George Yin

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Buche, R.T., Ghosh, A., Pipiras, V., Zhang, J.X. (2007). Heavy Traffic Methods in Wireless Systems: Towards Modeling Heavy Tails and Long Range Dependence. In: Agrawal, P., Fleming, P.J., Zhang, L., Andrews, D.M., Yin, G. (eds) Wireless Communications. The IMA Volumes in Mathematics and its Applications, vol 143. Springer, New York, NY. https://doi.org/10.1007/978-0-387-48945-2_3

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