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Product Form Steady-State Distribution for Stochastic Automata Networks with Domino Synchronizations

  • Jean-Michel Fourneau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5261)

Abstract

We present a new kind of synchronization which allows Continuous Time Stochastic Automata Networks (SAN) to have a product form steady-state distribution. Unlike previous models on SAN with product form solutions, our model allows synchronization between three automata but functional rates are not allowed. The synchronization is not the usual ”Rendez-Vous” but an ordered list of transitions. Each transition may fail. When a transition fails, the synchronization ends but all the transitions already executed are kept. This synchronization is related to the triggered customer movement between queues in a network and this class of SAN is a generalization of Gelenbe’s networks with triggered customer movement.

Keywords

Tensor Product Product Form Geometric Distribution Transition Probability Matrice Negative Customer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Boucherie, R.: A Characterization of independence for competing Markov chains with applications to stochastic Petri nets. IEEE Trans. Software Eng. 20(7), 536–544 (1994)CrossRefGoogle Scholar
  2. 2.
    Boujdaine, F., Fourneau, J.M., Mikou, N.: Product Form Solution for Stochastic Automata Networks with synchronization. In: 5th Process Algebra and Performance Modeling Workshop, Twente, Netherlands (1997)Google Scholar
  3. 3.
    Buchholz, P., Dayar, T.: Comparison of Multilevel Methods for Kronecker-based Markovian Representations. Computing Journal 73(4), 349–371 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dayar, T., Gusak, O., Fourneau, J.M.: Stochastic Automata Networks and Near Complete Decomposability. SIAM Journal and Applications 23, 581–599 (2002)MathSciNetGoogle Scholar
  5. 5.
    Dayar, T., Gusak, O., Fourneau, J.M.: Iterative disaggregation for a class of lumpable discrete-time SAN. Performance Evaluation, 2003 53(1), 43–69 (2003)CrossRefGoogle Scholar
  6. 6.
    Donnatelli, S.: Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Performance Evaluation 18, 21–36 (1993)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fernandes, P., Plateau, B., Stewart, W.J.: Efficient Descriptor-Vector Multiplications in Stochastic Automata Networks. JACM, 381–414 (1998)Google Scholar
  8. 8.
    Fourneau, J.M.: Domino Synchronization: product form solution for SANs. Studia Informatica  23, 4(51), 173–190Google Scholar
  9. 9.
    Fourneau, J.M., Plateau, B., Stewart, W.: Product form for Stochastic Automata Networks. In: Proc. of ValueTools 2007, Nantes, France (2007)Google Scholar
  10. 10.
    Fourneau, J.M., Plateau, B., Stewart, W.: An Algebraic Condition for Product Form in Stochastic Automata Networks without Synchronizations. Performance Evaluation (to appear, 2008)Google Scholar
  11. 11.
    Fourneau, J.M.: Discrete Time Markov chains competing over resources: product form steady-state distribution. In: QEST 2008 (to appear, 2008)Google Scholar
  12. 12.
    Fourneau, J.M., Quessette, F.: Graphs and Stochastic Automata Networks. In: Proc. of the 2nd International Workshop on the Numerical Solution of Markov Chains, Raleigh, USA (1995)Google Scholar
  13. 13.
    Gelenbe, E.: Product form queueing networks with negative and positive customers. Journal of Applied Probability 28, 656–663 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gelenbe, E.: G-networks with triggered customer movement. Journal of Applied Probability 30, 742–748 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Harrison, P., Hillston, J.: Exploiting Quasi-reversible Structures in Markovian Process Algebra Models. Computer Journal 38(7), 510–520 (1995)CrossRefGoogle Scholar
  16. 16.
    Hillston, J.: A compositional approach to Performance Modeling, Ph.D Thesis, University of Edinburgh (1994)Google Scholar
  17. 17.
    Hillston, J., Thomas, N.: Product Form Solution for a Class of PEPA Models. Performance Evaluation 35(3-4), 171–192 (1999)zbMATHCrossRefGoogle Scholar
  18. 18.
    Kloul, L., Hillston, J.: An efficient Kronecker representation for PEPA models. In: de Alfaro, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165. Springer, Heidelberg (2001)Google Scholar
  19. 19.
    Plateau, B.: On the Stochastic Structure of Parallelism and Synchronization Models for Distributed Algorithms. In: Proc. ACM Sigmetrics Conference on Measurement and Modeling of Computer Systems, Austin, Texas (August 1985)Google Scholar
  20. 20.
    Plateau, B., Fourneau, J.M., Lee, K.H.: PEPS: A Package for Solving Complex Markov Models of Parallel Systems. In: Proceedings of the 4th Int. Conf. on Modeling Techniques and Tools for Computer Performance Evaluation, Majorca, Spain (September 1988)Google Scholar
  21. 21.
    Plateau, B., Stewart, W.J.: Stochastic Automata Networks: Product Forms and Iterative Solutions, Inria Report 2939, FranceGoogle Scholar
  22. 22.
    Lazar, A., Robertazzi, T.: Markovian Petri Net Protocols with Product Form Solution. Performance Evaluation 12, 66–77 (1991)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Stewart, W.J., Atif, K., Plateau, B.: The numerical solution of Stochastic Automata Networks. European Journal of Operation Research 86(3), 503–525 (1995)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jean-Michel Fourneau
    • 1
    • 2
  1. 1.PRiSM, Université de Versailles-Saint-Quentin, CNRS, UniverSudVersaillesFrance
  2. 2.INRIA Projet Mescal, LIG, CNRSMontbonnotFrance

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