DISC 2012: Distributed Computing pp 413-414

# 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)
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Herman, T.: Probabilistic self-stabilization. IPL 35(2), 63–67 (1990)

## Authors and Affiliations

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