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Multidimensional Systems and Signal Processing

, Volume 30, Issue 1, pp 413–449 | Cite as

Ergodic optimization of stochastic differential systems in wireless networks

  • Hana BailiEmail author
Article
  • 24 Downloads

Abstract

The present work is the second article in a couple of intertwined papers. They form complementary items on the same subject. They both address the problem of joint power allocation and time slot scheduling in a wireless communication system with time varying traffic. The system is handled by a single base station transmitting over time varying channels. The operating time horizon is divided into time slots, and a fixed amount of power is available at each time slot. The mobile users share each time slot and the power available at this time slot. Since many wireless network applications have stringent delay requirements, designing high-performance resource allocation algorithms to achieve minimum possible delay is of great importance, and this is the main objective of the work presented in this paper. The delay performance of a resource allocation algorithm can be characterized by the average delay experienced by the data transmitted in the network. We propose a heavy traffic analysis for the physical system on hand, i.e., appropriate re-scaling that leads to a diffusion approximation of the system in the sense of weak convergence. The approximate diffusion is constrained or bounded in the K-dimensional positive orthant. We establish the convergence result of the heavy traffic analysis, and then a closed form solution to the resource optimization problem is provided. Here the solution relies on the ergodicity of the approximate diffusion.

Keywords

Heavy traffic analysis Weak convergence to a diffusion process Optimal control Ergodic theorems 

References

  1. Baili, H. (2015). Optimization of hybrid stochastic differential systems in communications networks. Nonlinear Analysis: Hybrid Systems, 17(3), 25–43.MathSciNetzbMATHGoogle Scholar
  2. Baili, H. (2016). Optimal scheduling and power allocation in wireless networks with heavy traffic: The infinite time horizon case. International Journal of Systems Science: Operations & Logistics.  https://doi.org/10.1080/23302674.2016.1250969
  3. Buche, R., & Kushner, H. J. (2002). Control of mobile communications with time varying channels in heavy traffic. IEEE Transactions on Automatic Control, 47(6), 992–1003.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Budhiraja, A., & Lee, C. (2009). Stationary distribution convergence for generalized Jackson networks in heavy traffic. Mathematics of Operations Research, 34, 45–56.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Davis, M. H. A. (1993). Markov models and optimization. London: Chapman & Hall.CrossRefzbMATHGoogle Scholar
  6. Gamarnik, D., & Stolyar, A. L. (2012). Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: Asymptotics of the stationary distribution. Queueing Systems, 71, 25–51.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gamarnik, D., & Zeevi, A. (2006). Validity of heavy traffic steady-state approximation in generalized Jackson networks. The Annals of Applied Probability, 16, 56–90.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gardiner, C. W. (1985). Handbook of stochastic methods for physics, chemistry and the natural sciences (2nd ed.). Berlin: Springer.CrossRefGoogle Scholar
  9. Gurvich, I. (2014a). Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Mathematics of Operations Research, 39, 121–162.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gurvich, I. (2014b). Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. The Annals of Applied Probability, 24, 2527–2559.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Harrison, J. M., & Nguyen, V. (1993). Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems, 13, 5–40.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Harrison, J. M., & Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Annals of Probability, 9, 302–308.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Harrison, J. M., & Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics, 22, 77–115.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Krylov, N. V. (1995). Introduction to the theory of diffusion processes. Providence, RI: American Mathematical Society.zbMATHGoogle Scholar
  15. Leon-Garcia, A. (2008). Probability, statistics, and random processes for electrical engineering (3rd ed.). Englewood Cliffs, NJ: Prentice Hall.Google Scholar
  16. Miyazawa, M. (2015). Diffusion approximation for stationary analysis of queues and their networks: A review. Journal of the Operations Research Society of Japan, 58, 104–148.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Risken, H. (1989). The Fokker–Planck equation: Methods of solution and applications (2nd ed.). Berlin: Springer.zbMATHGoogle Scholar
  18. Serfozo, R. (1999). Introduction to stochastic networks (1st ed.). Berlin: Springer.CrossRefzbMATHGoogle Scholar
  19. Skorokhod, A. V. (1989). Asymptotic methods in the theory of stochastic differential equations. New York: American Mathematical Society.zbMATHGoogle Scholar
  20. Skorokhod, A. V. (2005). Basic principles and applications of probability theory. Berlin: Springer.zbMATHGoogle Scholar
  21. Skorokhod, A. V., Hoppensteadt, F. C., & Salehi, H. (2002). Random perturbation methods with applications in science and engineering. New York: Springer.CrossRefzbMATHGoogle Scholar
  22. Wu, W., Arapostathis, A., & Shakkottai, S. (2006). Optimal power allocation for a time-varying wireless channel under heavy-traffic approximation. IEEE Transactions on Automatic Control, 51(4), 580–594.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Signals and SystemsCentraleSupélec, Université Paris-SaclayGif-sur-YvetteFrance

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