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Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment

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Abstract

We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle.

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References

  1. Atar, R., Dupuis, P.: Large deviations and queueing networks: methods for rate function identification. Stoch. Process. Appl. 84, 255–296 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Azencott, R., Ruget, G.: Mélanges d’équations différentielles et grands écarts à la loi des grands nombres. Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 1–54 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Azencott, R.: Grandes déviations et applications. In: Eighth Saint Flour Probability Summer School—1978. Lecture Notes in Math., vol. 774, pp. 1–176. Springer, Berlin (1980)

    Google Scholar 

  4. Baldi, P.: Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 151, 161–177 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser Boston, Boston (1997)

    Book  MATH  Google Scholar 

  6. Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Springer, Paris (1994)

    MATH  Google Scholar 

  7. Capuzzo-Dolcetta, I., Lions, P.-L.: Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318, 643–683 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Comets, F., Delarue, F., Schott, R.: Distributed algorithms in an ergodic Markovian environment. Random Struct. Algorithms 30, 131–167 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics, vol. 38. Springer, New York (1998)

    MATH  Google Scholar 

  10. Dupuis, P.: Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets. Stochastics 21, 63–96 (1987)

    MATH  MathSciNet  Google Scholar 

  11. Dupuis, P.: Large deviations analysis of some recursive algorithms with state dependent noise. Ann. Probab. 16, 1509–1536 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dupuis, P., Ellis, R.S.: The large deviations principle for a general class of queueing systems I. Trans. Am. Math. Soc. 347, 2689–2751 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dupuis, P., Ramanan, K.: A time-reversed representation for the tail probabilities of stationary reflected Brownian motion. Stoch. Process. Appl. 98, 253–287 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feng, J., Kurtz, T.: Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs, vol. 131. American Mathematical Society, Providence (2006)

    MATH  Google Scholar 

  15. Flajolet, P.: The evolution of two stacks in bounded space and random walks in a triangle. In: Proceedings of FCT’86. LNCS, vol. 233, pp. 325–340. Springer, Berlin (1986)

    Google Scholar 

  16. Freidlin, M., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften, vol. 260. Springer, New York (1984)

    MATH  Google Scholar 

  17. Guillotin-Plantard, N., Schott, R.: Distributed algorithms with dynamic random transitions. Random Struct. Algorithms 21, 371–396 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guillotin-Plantard, N., Schott, R.: Dynamic Random Walks. Theory and Applications. Elsevier, Amsterdam (2006)

    Google Scholar 

  19. Gulinsky, O., Veretennikov, A.: Large Deviations for Discrete-Time Processes with Averaging. VSP, Utrecht (1993)

    MATH  Google Scholar 

  20. Ignatiouk-Robert, I.: Large deviations for processes with discontinuous statistics. Ann. Probab. 33, 1479–1508 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ignatiouk-Robert, I.: Sample path large deviations and convergence parameters. Ann. Appl. Probab. 11, 1292–1329 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Knuth, D.E.: The Art of Computer Programming, vol. 1. Addison–Wesley, Reading (1973)

    Google Scholar 

  23. Lions, P.-L., Sznitman, A.-S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lions, P.-L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52, 793–820 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Louchard, G.: Some distributed algorithms revisited. Commun. Stat. Stoch. Models 4, 563–586 (1995)

    Article  MathSciNet  Google Scholar 

  26. Louchard, G., Schott, R.: Probabilistic analysis of some distributed algorithms. Random Struct. Algorithms 2, 151–186 (1991)

    MATH  MathSciNet  Google Scholar 

  27. Louchard, G., Schott, R., Tolley, M., Zimmermann, P.: Random walks, heat equations and distributed algorithms. Comput. Appl. Math. 53, 243–274 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Maier, R.: Colliding stacks: a large deviations analysis. Random Struct. Algorithms 2, 379–420 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  29. Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  30. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  31. Yao, A.: An analysis of a memory allocation scheme for implementing stacks. SIAM J. Comput. 10, 398–403 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 7, UFR de Mathématiques, Case 7012, 2, Place Jussieu, 75251, Paris Cedex 05, France

    Francis Comets & François Delarue

  2. IECN and LORIA, Université Henri Poincaré-Nancy 1, 54506, Vandoeuvre-lès-Nancy, France

    René Schott

Authors
  1. Francis Comets
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  2. François Delarue
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  3. René Schott
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Correspondence to Francis Comets.

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Comets, F., Delarue, F. & Schott, R. Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment. Appl Math Optim 60, 341–396 (2009). https://doi.org/10.1007/s00245-009-9079-8

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  • DOI: https://doi.org/10.1007/s00245-009-9079-8

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