Abstract
In this article we investigate the expected time for Herman’s probabilistic self-stabilizing algorithm in distributed systems: Suppose that the number of identical processes in a ring, say n, is odd and n≥ 3. If the initial configuration of the ring is not “legitimate”, that is, the number of tokens differs from one, then execution of the algorithm made up of synchronous probabilistic procedures with a parameter 0<r<1 results in convergence to a legitimate configuration with a unique token (Herman’s algorithm). We then show that the expected time of the convergence is less than \(\frac{\pi^2-8}{8r(1-r)}n^2\). Moreover there exists a configuration whose expected time is Θ(n 2). The method of the proof is based on the analysis of coalescing random walks.
This work was supported by Grant-in-Aid for Young Scientists (B) No. 14740077 from MEXT Japan.
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Nakata, T. (2004). On the Expected Time for Herman’s Probabilistic Self-stabilizing Algorithm. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_45
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DOI: https://doi.org/10.1007/978-3-540-27798-9_45
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