Phase synchronization on asynchronous uniform rings with odd size

This paper proposes a self-stabilizing phase synchronization protocol for uniform rings with an odd size. Nodes in the ring work asynchronously and proceed in a cyclic sequence of K phases, where K is even. The phase values of all the nodes are required to be no more than one apart. A system state which satisfies the requirement is therefore called a legitimate state. The proposed protocol guarantees that no matter with which initial state the system may start, the ring stabilizes eventually at a state after which the closure property on the legitimate state holds. Phase values should never go backward. The closure property on the legitimate states commonly used in previous works on self-stabilization cannot capture this requirement. This paper defines two terms, legitimate step and illegitimate step, to address this issue. An execution step that brings the ring from a legitimate state to another legitimate state in a way that the phase values of the nodes only advance is called a legitimate step. An execution step that observes the closure property on the legitimate states but makes some phase values go backward is modeled as an illegitimate step. It is shown that, for the proposed protocol, only a finite number of illegitimate steps are possible. After all possible illegitimate steps have occurred, the closure property on the legitimate steps holds