Consensus in the Presence of Timing Uncertainty :

We consider the time complexity of reaching agreement in a semi-synchronous model of distributed systems, in the presence of omission and Byzantine failures. In our semi-synchronous model, processes have inexact knowledge about the time to perform certain primitive actions: messages arrive within time d of when they are sent and the time between two consecutive steps of any process is in the known interval c 1 ; c 2 ]. We use C = c 2 =c 1 as a measure of the timing uncertainty. A simple adaptation of the synchronous lower bound shows that at least time (f + 1)d is required to tolerate f failures; time (f + 1)Cd is suucient for stopping or omission failures by directly simulating synchronous rounds. By strengthening the algorithm for stopping failures of Attiya, Dwork, Lynch, and Stockmeyer ((1]), we derive an algorithm for omission failures that has minimal dependency on the uncertainty factor C. If fewer than half the processes are faulty then the running time is 4(f + 1)d + Cd, which is within a factor of 4 of optimal and may be much faster than direct rounds simulation if C is large. If more than half the processes are faulty, then the running time is shown to be greater by approximately a factor of min(f n?f ; p C). 0 Finally, we present a general simulation for n 3f + 1 that tolerates Byzantine failures and simulates any round-based synchronous algorithm at a cost of time 2Cd + d per round.