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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 262581, 16 pages
http://dx.doi.org/10.1155/2013/262581
Research Article

Continuum Modeling and Control of Large Nonuniform Wireless Networks via Nonlinear Partial Differential Equations

1Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373, USA
2Department of Statistics and Operation Research, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3260, USA
3Department of Statistics, Colorado State University, Fort Collins, CO 80523-1373, USA

Received 4 January 2013; Revised 27 February 2013; Accepted 8 March 2013

Academic Editor: Lan Xu

Copyright © 2013 Yang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Y. Zhang, E. K. P. Chong, J. Hannig, and D. J. Estep, “Continuum limits of Markov chains with application to network modeling,” http://arxiv.org/abs/1106.4288.
  2. Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, “On continuum limits of markov chains and network modeling,” in Proceedings of the 49th IEEE Conference on Decision and Control (CDC '10), pp. 6779–6784, Atlanta, Ga, USA, December 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. E. K. P. Chong, D. Estep, and J. Hannig, “Continuum modeling of large networks,” International Journal of Numerical Modelling, vol. 21, no. 3, pp. 169–186, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. N. J. Burch, Continuum modeling of stochastic wireless sensor networks [M.S. thesis], Colorado State University, 2008.
  5. N. Burch, E. Chong, D. Estep, and J. Hannig, “Analysis of routing protocols and interference-limited communication in large wireless networks,” Journal of Engineering Mathematics, pp. 1–17, 2012.
  6. H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press Series in Signal Processing, Optimization, and Control, 6, MIT Press, Cambridge, Mass, USA, 1984. View at Zentralblatt MATH · View at MathSciNet
  7. H. Robbins and S. Monro, “A stochastic approximation method,” Annals of Mathematical Statistics, vol. 22, pp. 400–407, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Kiefer and J. Wolfowitz, “Stochastic estimation of the maximum of a regression function,” Annals of Mathematical Statistics, vol. 23, pp. 462–466, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Benaïm, “Dynamics of stochastic approximation algorithms,” in Séminaire de Probabilités, XXXIII, vol. 1709 of Lecture Notes in Mathematics, pp. 1–68, Springer, Berlin, Germany, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. L. Lai, “Stochastic approximation,” The Annals of Statistics, vol. 31, no. 2, pp. 391–406, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. G. Kurtz, “Solutions of ordinary differential equations as limits of pure jump Markov processes,” Journal of Applied Probability, vol. 7, pp. 49–58, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. W. R. Darling, “Fluid limits of pure jump Markov processes: a practical guide,” http://arxiv.org/abs/math/0210109.
  13. R. McVinish and P. Pollett, “The deterministic limit of heterogeneous density dependent Markov chains,” submitted to Annals of Applied Probability.
  14. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, NY, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  15. P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  16. P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388–404, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. Grossglauser and D. N. C. Tse, “Mobility increases the capacity of ad hoc wireless networks,” IEEE/ACM Transactions on Networking, vol. 10, no. 4, pp. 477–486, 2002. View at Publisher · View at Google Scholar · View at Scopus
  18. J. D. Herdtner and E. K. P. Chong, “Throughput-storage tradeoff in ad hoc networks,” in Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM '05), vol. 4, pp. 2536–2542, March 2005. View at Scopus
  19. M. Benaïm and J.-Y. L. Boudec, “A class of mean field interaction models for computer and communication systems,” Performance Evaluation, pp. 11–12, 2008.
  20. P. E. Caines, “Bode lecture: mean field stochastic control,” in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, 2009.
  21. D. A. Dawson, J. Tang, and Y. Q. Zhao, “Balancing queues by mean field interaction,” Queueing Systems, vol. 49, no. 3-4, pp. 335–361, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. Graham and P. Robert, “Self-adaptive congestion control for multiclass intermittent connections in a communication network,” Queueing Systems, vol. 69, no. 3-4, pp. 237–257, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. C. S. Chang, “Sample path large deviations and intree networks,” Queueing Systems, vol. 20, no. 1-2, pp. 7–36, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. N. G. Duffield, “A large deviation analysis of errors in measurement based admission control to buffered and bufferless resources,” Queueing Systems, vol. 34, no. 1-4, pp. 131–168, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Ahn and J. Jeon, “Analysis of G/D/1 queueing systems with inputs satisfying large deviation principle under weak* topology,” Queueing Systems, vol. 40, no. 3, pp. 295–311, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. D. Blount, “Law of large numbers in the supremum norm for a chemical reaction with diffusion,” The Annals of Applied Probability, vol. 2, no. 1, pp. 131–141, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C.-f. Huang and H. Pagès, “Optimal consumption and portfolio policies with an infinite horizon: existence and convergence,” The Annals of Applied Probability, vol. 2, no. 1, pp. 36–64, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. G.-L. Xu and S. E. Shreve, “A duality method for optimal consumption and investment under short-selling prohibition. II. Constant market coefficients,” The Annals of Applied Probability, vol. 2, no. 2, pp. 314–328, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. M. Burger, P. A. Markowich, and J.-F. Pietschmann, “Continuous limit of a crowd motion and herding model: analysis and numerical simulations,” Kinetic and Related Models, vol. 4, no. 4, pp. 1025–1047, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. H. Chen and W. Whitt, “Diffusion approximations for open queueing networks with service interruptions,” Queueing Systems, vol. 13, no. 4, pp. 335–359, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. J. M. Harrison and V. Nguyen, “Brownian models of multiclass queueing networks: current status and open problems,” Queueing Systems, vol. 13, no. 1–3, pp. 5–40, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. H. Chen, O. Kella, and G. Weiss, “Fluid approximations for a processor-sharing queue,” Queueing Systems, vol. 27, no. 1-2, pp. 99–125, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. R.-R. Chen and S. Meyn, “Value iteration and optimization of multiclass queueing networks,” Queueing Systems, vol. 32, no. 1–3, pp. 65–97, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. R. C. Hampshire, M. Harchol-Balter, and W. A. Massey, “Fluid and diffusion limits for transient sojourn times of processor sharing queues with time varying rates,” Queueing Systems, vol. 53, no. 1-2, pp. 19–30, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. F. J. Piera, R. R. Mazumdar, and F. M. Guillemin, “Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit,” Queueing Systems, vol. 58, no. 1, pp. 3–27, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover, Mineola, NY, USA, 1996. View at MathSciNet
  37. J.-H. He, “Asymptotic Methods for Solitary Solutions and Compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar
  38. G. R. Liu and S. S. Quek, The Finite Element Method: A Practical Course, Butterworth-Heinemann, 2003.
  39. A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, John Wiley & Sons, Chichester, UK, 1980. View at MathSciNet
  40. J. C. Robinson, An Introduction to Ordinary Differential Equations, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet