Brief Announcement: Probabilistic Stabilization under Probabilistic Schedulers

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 $\tau_{{\cal S}^*,\rho} < \infty$.

This work is supported in part by MEXT/IPSJ KAKENHI (21650002, 22300004, 23700019, 24104003, and 24650008), ANR project SHAMAN and JSPS fellowship.