On the expected time for Herman's probabilistic self-stabilizing algorithm

AbstractIn 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 unidirectional 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 local 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 ((π2-8)/8r(1-r))n2. Note that if r=12 then it is bounded by 0.936n2. Moreover, there exists a configuration whose expected time is Θ(n2). The method of the proof is based on the analysis of coalescing random walks