Probabilistic Self-Stabilization with Two-State Machines

We present and prove the correctness of a probabilistic self-stabilizing algorithm that circulates a token around an asynchronous ring of identical two-state machines. The number of machines in the ring is odd, and communication is unidirectional. If the initial state of the ring has more than one token, execution of the algorithm results probabilistically in convergence to a state with one token. We analyze the expected convergence span, i.e. the average number of moves to stabilize from an abnormal state, for a few restricted cases. We also present simulation results for the worst case initial state of N tokens with N machines which has a move complexity of O(N 2) for small N.