Different Monotonicity Definitions in Stochastic Modelling

  • Imène Kadi
  • Nihal Pekergin
  • Jean-Marc Vincent
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5513)

Abstract

In this paper we discuss different monotonicity definitions applied in stochastic modelling. Obviously, the relationships between the monotonicity concepts depends on the relation order that we consider on the state space. In the case of total ordering, the stochastic monotonicity used to build bounding models and the realizable monotonicity used in perfect simulation are equivalent to each other while in the case of partial order there is only implication between them. Indeed, there are cases of partial order, where we can’t move from the stochastic monotonicity to the realizable monotonicity, this is why we will try to find the conditions for which there are equivalences between these two notions. In this study, we will present some examples to give better intuition and explanation of these concepts.

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References

  1. 1.
    Bolch, G., Greiner, S., de Meer, H., Trivedi, K.: Queueing Networks and Markov Chains. John Wiley & Sons, Chichester (1998)CrossRefMATHGoogle Scholar
  2. 2.
    Borovkov, A.A., Foss, S.: Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math. 34 (1994)Google Scholar
  3. 3.
    Diaconis, P., Freedman, D.: Iterated random functions. SIAM Review 41(1), 45–76 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fill, J.A., Machida, M.: An interruptible algorithm for perfect sampling via markov chains. In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, New York, USA, pp. 688–695 (1997)Google Scholar
  5. 5.
    Fill, J.A., Machida, M.: Realizable monotonicity and inverse probability transform. Technical report, Department of Mathematical sciences. The Johns Hopkins university (2000)Google Scholar
  6. 6.
    Fourneau, J.M., Kadi, I., Pekergin, N., Vienne, J., Vincent, J.M.: Perfect simulation and monotone stochastic bounds. In: ValueTools 2007: Proceedings of the 2nd international conference on Performance 249-263evaluation methodologies and tools, pp. 1–10, ICST (2007)Google Scholar
  7. 7.
    Fourneau, J.M., Pekergin, N.: An algorithmic approach to stochastic bounds. In: Calzarossa, M.C., Tucci, S. (eds.) Performance 2002. LNCS, vol. 2459, pp. 64–88. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Glasserman, P., Yao, D.: Monotone Structure in Discrete-Event Systems. John Wiley & Sons, Chichester (1994)MATHGoogle Scholar
  9. 9.
    Olle Haggstrom. Finite Markov Chains and algorithmic applications, Matematisk Statistik, Chalmers teknisk hogshola och Goteborgs universitet (2001)Google Scholar
  10. 10.
    Massey, W.A.: Stochastic ordering for markov processes on partially ordered spaces. Math. Oper. Res. 12(2), 350–367 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Vincent, J.-M., Fernandes, P., Webber, T.: Perfect simulation of stochastic automata networks. In: Al-Begain, K., Heindl, A., Telek, M. (eds.) ASMTA 2008. LNCS, vol. 5055, pp. 249–263. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms 9(1&2), 223–252 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Stenflo, O.: Ergodic theorems fory Iterated Function Systems controlled by stochastic sequences. Doctoral thesis n. 14, Umea university (1998)Google Scholar
  14. 14.
    Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. John Wiley & Sons, ChichesterGoogle Scholar
  15. 15.
    Vincent, J.-M.: Perfect simulation of queueing networks with blocking and rejection. In: SAINT-W 2005: Proceedings of the 2005 Symposium on Applications and the Internet Workshops, Trento, Italy, pp. 268–271 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Imène Kadi
    • 1
  • Nihal Pekergin
    • 2
  • Jean-Marc Vincent
    • 3
  1. 1.PRiSM, University Versailles-Saint-QuentinFrance
  2. 2.LACL, University Paris-EstCréteilFrance
  3. 3.LIG, project-INRIA MESCALMontbonnotFrance

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