Recurrence in Self-Stabilization

Self-stabilization ensures that a system converges to a legitimate execution in finite time, where a legitimate execution comprises a sequence of configurations satisfying some safety condition. In this work, we investigate the notion of recurrence, which denotes how frequently a condition is satisfied in an execution of a system. We use this notion in self-stabilization to address the convergence of a system to a behavior that guarantees a minimum recurrence of some condition. We apply this notion to show how the design of distributed mutual exclusion algorithms can be altered to achieve a high service time under various convergence time and space complexities. As a particular contribution, we present a self-stabilizing mutual exclusion algorithm that has optimal service time together with optimal stabilization time complexity of (D/2 - 1) for synchronous executions and under any topology, where D is the diameter of the topology. In addition, we rectify an earlier proof stating that (D/2) is a lower bound, to conclude that (D/2 - 1) is optimal for synchronous executions.