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<p><b>Abstract</b>—An important problem in fault-tolerant distributed systems is maintaining agreement between nonfaulty processes in the presence of undiagnosed faults. To achieve agreement, processes 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. Rather, it is only necessary that they agree to within a predefined tolerance.</p><p>Approximate Agreement can be achieved through a sequence of convergent voting rounds, in which the range of values held by nonfaulty nodes is reduced in each round. Recent research has revealed simple expressions for the convergence rate and fault tolerance of a broad family of convergent voting algorithms called <it>Mean-Subsequence-Reduced</it> (MSR) algorithms. These results were derived under the Thambidurai and Park hybrid fault model comprised of <it>asymmetric</it>, <it>symmetric</it>, and <it>benign</it> faults. However, these results apply only to <it>synchronous</it> systems, in which there is a known finite bound on computation and communications times. This paper extends the previous results to <it>asynchronous</it> systems, in which no such bound exists. In addition, we introduce two new hybrid fault models which further differentiate between <it>omissive</it> faults and <it>transmissive</it> faults. The new fault models permit tighter bounds on the fault-tolerance of asynchronous systems to be derived.</p>
Approximate agreement, Byzantine agreement, clock synchronization, convergent voting algorithms, fault-tolerant multiprocessors, hybrid faults.

R. Kieckhafer and M. Azadmanesh, "New Hybrid Fault Models for Asynchronous Approximate Agreement," in IEEE Transactions on Computers, vol. 45, no. , pp. 439-449, 1996.
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