Asynchronous approximate agreement in partially connected networks

An important problem in fault-tolerant distributed systems is maintaining agreement between nonfaulty processes in the presence of undiagnosed faults. To achieve agreement, processes must exchange their local opinions of a particular value, and then vote on the values received to arrive at a consensus. Approximate agreement defines a condition in which it is not necessary for consensus values to be identical, but only to agree within a predefined tolerance. Approximate agreement can be achieved through a sequence of convergent voting rounds, in which the range of values held by nonfaulty processes is reduced in each round. Simple expressions have been obtained for the convergence rate and fault tolerance of a broad family of locally convergent voting algorithms. in the presence of mixed fault-modes. However, the results apply only to synchronous systems, in which there is a known finite bound on computation and communications times. This article addresses convergent voting algorithms in asynchronous partially connected systems, in which no bound exists on process operations and communication delays. It is shown that these systems can be treated as a variant of synchronous partially connected systems. The result is tighter bounds on convergence rates and fault tolerance than is possible with previous analysis of asynchronous systems.