Probabilistic self-stabilization
Information Processing Letters
Token management schemes and random walks yield self-stabilizing mutual exclusion
PODC '90 Proceedings of the ninth annual ACM symposium on Principles of distributed computing
ACM Computing Surveys (CSUR)
Stabilizing time-adaptive protocols
Theoretical Computer Science
Self-stabilization
Self-stabilizing systems in spite of distributed control
Communications of the ACM
Two-State Self-Stabilizing Algorithms for Token Rings
IEEE Transactions on Software Engineering
Coupling and self-stabilization
Distributed Computing - Special issue: DISC 04
On the expected time for Herman's probabilistic self-stabilizing algorithm
Theoretical Computer Science
An elementary proof that Herman's Ring is Θ(N2)
Information Processing Letters
On stabilization in Herman's algorithm
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
On stabilization in Herman's algorithm
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
A tighter bound for the self-stabilization time in Herman's algorithm
Information Processing Letters
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Herman's algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration. It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worst-case expected time to stabilization (with respect to the initial configuration) is Θ(N2). Our first contribution is to give an upper bound of 0.64N2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N2) convergence bound in this case. Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman's algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case.