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Brief Announcement: Probabilistic Stabilization under Probabilistic Schedulers

  • Yukiko Yamauchi
  • Sébastien Tixeuil
  • Shuji Kijima
  • Masafumi Yamashita
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7611)

Abstract

Motivation. Roughly speaking, a weakly stabilizing system \(\cal S\) executed under a probabilistic scheduler ρ is probabilistically self-stabilizing, in the sense that any execution eventually reaches a legitimate execution with probability 1 [1-3]. Here ρ is a set of Markov chains, one of which is selected for \(\cal S\) by an adversary to generate as its evolution an infinite activation sequence to execute \(\cal S\). The performance measure is the worst case expected convergence time \(\tau_{{\cal S},M}\) when \(\cal S\) is executed under a Markov chain M ∈ ρ. Let \(\tau_{{\cal S},\rho} = \sup_{M \in \rho} \tau_{{\cal S},M}\). Then \(\cal S\) can be “comfortably” used as a probabilistically self-stabilizing system under ρ only if \(\tau_{{\cal S},\rho} < \infty\). There are \(\cal S\) and ρ such that \(\tau_{{\cal S},\rho} = \infty\), despite that \(\tau_{{\cal S},M} < \infty\) for any M ∈ ρ. Somewhat interesting is that, for some \(\cal S\), there is a randomised version \({\cal S}^*\) of \(\cal S\) such that \(\tau_{{\cal S}^*,\rho} < \infty\), despite that \(\tau_{{\cal S},\rho} = \infty\), i.e., randomization helps. This motivates a characterization of \(\cal S\) that satisfies \(\tau_{{\cal S}^*,\rho} < \infty\).

References

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    Gouda, M.G.: The Theory of Weak Stabilization. In: Datta, A.K., Herman, T. (eds.) WSS 2001. LNCS, vol. 2194, pp. 114–123. Springer, Heidelberg (2001)CrossRefGoogle Scholar
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    Herman, T.: Probabilistic self-stabilization. IPL 35(2), 63–67 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yukiko Yamauchi
    • 1
  • Sébastien Tixeuil
    • 2
  • Shuji Kijima
    • 1
  • Masafumi Yamashita
    • 1
  1. 1.Kyushu UniversityJapan
  2. 2.UPMC Sorbonne UniversitesFrance

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