Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 831909, 35 pages
http://dx.doi.org/10.1155/2012/831909
Research Article

Stability and Probability 1 Convergence for Queueing Networks via Lyapunov Optimization

Electrical Engineering Department, University of Southern California, 3740 McClintock Avenue, Room 500, Los Angeles, CA 90089-2565, USA

Received 15 August 2011; Revised 17 March 2012; Accepted 11 April 2012

Academic Editor: P. G. L. Leach

Copyright © 2012 Michael J. Neely. 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. L. Tassiulas and A. Ephremides, “Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Transactions on Automatic Control, vol. 37, no. 12, pp. 1936–1948, 1992. View at Publisher · View at Google Scholar
  2. L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallel queues with randomly varying connectivity,” IEEE Transactions on Information Theory, vol. 39, no. 2, pp. 466–478, 1993. View at Publisher · View at Google Scholar
  3. M. J. Neely, Dynamic power allocation and routing for satellite and wireless networks with time varying channels [Ph.D. thesis], Massachusetts Institute of Technology, LIDS, Cambridge, Mass, USA, 2003.
  4. L. Georgiadis, M. J. Neely, and L. Tassiulas, “Resource allocation and cross-layer control in wireless networks,” Foundations and Trends in Networking, vol. 1, no. 1, pp. 1–144, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, Mass, USA, 1995.
  6. D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, UK, 1991.
  7. S. Ross, Introduction to Probability Models, Academic Press, New York, NY, USA, 8th edition, 2002.
  8. R. Gallager, Discrete Stochastic Processes, Kluwer Academic, Boston, Mass, USA, 1996.
  9. S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer, London, UK, 1993.
  10. J. G. Dai, “On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models,” The Annals of Applied Probability, vol. 5, no. 1, pp. 49–77, 1995. View at Google Scholar
  11. M. J. Neely, “Energy optimal control for time-varying wireless networks,” IEEE Transactions on Information Theory, vol. 52, no. 7, pp. 2915–2934, 2006. View at Publisher · View at Google Scholar
  12. M. J. Neely, “Universal scheduling for networks with arbitrary traffic, channels, and mobility,” in Proceedings of the 49th IEEE Conference on Decision and Control (CDC '10), pp. 1822–1829, Atlanta, Ga, USA, December 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. R. Agrawal and V. Subramanian, “Optimality of certain channel aware scheduling policies,” in Proceedings of the 40th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Ill, USA, October 2002.
  14. H. Kushner and P. Whiting, “Asymptotic properties of proportional-fair sharing algorithms,” in Proceedings of the 40th Annual Allerton Conference on Communication, Control, and Computing, May 2002.
  15. A. L. Stolyar, “Maximizing queueing network utility subject to stability: greedy primal-dual algorithm,” Queueing Systems, vol. 50, no. 4, pp. 401–457, 2005. View at Publisher · View at Google Scholar
  16. Q. Li and R. Negi, “Scheduling in wireless networks under uncertainties: a greedy primal-dual approach,” Tech. Rep., 2010, http://arxiv.org/abs/1001.20502010. View at Google Scholar
  17. A. Eryilmaz and R. Srikant, “Fair resource allocation in wireless networks using queue-length-based scheduling and congestion control,” IEEE/ACM Transactions on Networking, vol. 15, no. 6, pp. 1333–1344, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. X. Lin and N. B. Shroff, “Joint rate control and scheduling in multihop wireless networks,” in Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 1484–1489, Paradise Island, The Bahamas, December 2004. View at Scopus
  19. N. McKeown, A. Mekkittikul, V. Anantharam, and J. Walrand, “Achieving 100% throughput in an input-queued switch,” IEEE Transactions on Communications, vol. 47, no. 8, pp. 1260–1267, 1999. View at Publisher · View at Google Scholar · View at Scopus
  20. M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, P. Whiting, and R. Vijayakumar, “Providing quality of service over a shared wireless link,” IEEE Communications Magazine, vol. 39, no. 2, pp. 150–154, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. E. Leonardi, M. Mellia, F. Neri, and M. Ajmone Marsan, “Bounds on average delays and queue size averages and variances in input-queued cell-based switches,” in Proceedings of the 20th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM '01), pp. 1095–1103, Anchorage, Alaska, USA, April 2001. View at Scopus
  22. M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing for time-varying wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 1, pp. 89–103, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. N. Kahale and P. E. Wright, “Dynamic global packet routing in wireless networks,” in Proceedings of the 16th IEEE Annual Conference on Computer Communications (INFOCOM '97), pp. 1414–1421, April 1997. View at Scopus
  24. S. Shakkottai, R. Srikant, and A. L. Stolyar, “Pathwise optimality of the exponential scheduling rule for wireless channels,” Advances in Applied Probability, vol. 36, no. 4, pp. 1021–1045, 2004. View at Publisher · View at Google Scholar
  25. S. P. Meyn, “Stability and asymptotic optimality of generalized maxweight policies,” SIAM Journal on Control and Optimization, vol. 47, no. 6, pp. 3259–3294, 2008/09. View at Publisher · View at Google Scholar
  26. A. L. Stolyar, “Maxweight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic,” The Annals of Applied Probability, vol. 14, no. 1, pp. 1–53, 2004. View at Publisher · View at Google Scholar
  27. D. Shah and D. Wischik, “Optimal scheduling algorithms for input-queued switches,” in Proceedings of the 25th IEEE International Conference on Computer Communications (INFOCOM '06), pp. 1–11, Barcelona, Spain, April 2006. View at Publisher · View at Google Scholar · View at Scopus
  28. T. Ji, E. Athanasopoulou, and R. Srikant, “Optimal scheduling policies in small generalized switches,” in Proceedings of the 28th IEEE Conference on Computer Communications (INFOCOM '09), pp. 2921–2925, Rio De Janiero, Brazil, April 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. V. J. Venkataramanan and X. Lin, “Structural properties of LDP for queue-length based wireless scheduling algorithms,” in Proceedings of the 45th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Ill, USA, September 2007.
  30. V. Lau and C. H. Koh, “Tradeoff analysis of delay-power-CSIT quality of generalized dynamic backpressure algorithm for energy efficient OFDM systems,” in Proceedings of the IEEE International Symposium on Information Theory, pp. 1287–1291, St Petersburg, Russia, August 2011.
  31. P. W. Glynn and S. P. Meyn, “A Liapounov bound for solutions of the Poisson's equation,” The Annals of Probability, vol. 24, no. 2, pp. 916–931, 1996. View at Publisher · View at Google Scholar
  32. A. Shwartz and A. M. Makowski, “Comparing policies in Markov decision processes: Mandl's lemma revisited,” Mathematics of Operations Research, vol. 15, no. 1, pp. 155–174, 1990. View at Publisher · View at Google Scholar
  33. A. M. Makowski and A. Shwartz, “The Poisson equation for countable Markov chains: probabilistic methods and interpretations,” in Handbook of Markov Decision Processes: Methods and Applications, E. A. Feinberg and A. Shwartz, Eds., vol. 40, pp. 269–303, Kluwer Academic, Boston, Mass, USA, 2002. View at Publisher · View at Google Scholar
  34. S. M. Ross, Introduction to Stochastic Dynamic Programming, Academic Press, New York, NY, USA, 1995.
  35. Y. S. Chow, “On a strong law of large numbers for martingales,” Annals of Mathematical Statistics, vol. 38, no. 2, article 610, 1967. View at Google Scholar
  36. M. J. Neely, Stochastic Network Optimization with Application to Communication and Queueing Systems, Morgan & Claypool, 2010.
  37. L. Huang and M. J. Neely, “Delay reduction via Lagrange multipliers in stochastic network optimization,” IEEE Transactions on Automatic Control, vol. 56, no. 4, pp. 842–857, 2011. View at Publisher · View at Google Scholar