The algorithm guarantees termination within expected polynomial time. The Aspnes-Herlihy algorithm is a rather complex algorithm.
Processes move through a succession of asynchronous rounds, attempting to agree at each round.
At each round, the agreement attempt involves a distributed random walk. Properties of all three subprotocols are proved separately, and combined using general results about automaton composition. The probabilistic reasoning is isolated to a few small sections of the proof. These include ways to combine expectations for different complexity measures, to compose expected complexity properties, to convert probabilistic claims to deterministic claims, to use abstraction mappings to prove probabilistic properties, and to apply random walk theory in a distributed computational setting. We apply all of these techniques to analyze the expected complexity of the algorithm.