Unreliable Failure Detectors for Reliable Distributed Systems

Unreliable Failure Detectors for Reliable Distributed Systems 265We now show that the above lower boundis tight:THEOREM A3.3. In asynchronous systems with m 5 n – f, Consensus can besolved using Y@(m).PROOF. Suppose m < n - f, and consider any failure detector QilE SPW(m).Since m, the number of mistakes made by ‘3, is less than the number of correctprocesses, there is at least one correct process that ~ never suspects. Thus, ~satisfies weak accuracy. By the definition of Y9(m ), Q also satisfies weakcompleteness. So ‘2 E W, and it can be used to solve Consensus (Corollary6.1.9).Suppose m = n - f. Even though $3 can now make a mistake about everycorrect process, it can still be used to solve Consensus (even if a majority of theprocesses are faulty). The corresponding algorithm uses rotating coordinators,and is similar to the one for OW given in Figure 6. Because of this similarity, weomit the details, ❑From the above two theorems:COROLLARY A3.4. in asynchronous systems with f z [n/21, Consensus can besolved using X%(m) if and only if m s n – f.We now turn our attention to solving Consensus using W ~(m ).THEOREM A3.5. In asynchronous ~stems with f ? rn/21, Consensus cannot besolved using W%(m) with m > 0.PROOF. In Theorem A3,2, we described a failure detector ~ that cannotbe used to solve Consensus in asynchronous systems with f = rn / 21. It is easyto verify that ~ makes at most one mistake about each correct process, and thus!2 E w%(l). ❑From Corollary A2.2, and the above theorem, we have:COROLLARY A3.6. In asynchronous systems with f ? [n/21, Consensus can besolved using W%(m) if and only if m = O.ACKNOWLEDGMENTS. We are deeply grateful to Vassos Hadzilacos for hiscrucial help in revising this paper. The comments and suggestions of theanonymous referees, Navin Budhiraja, and Bernadette Charron-Best, were alsoinstrumental in improving the paper. Finally, we would like to thank PrasadJayanti for greatly simplifying the algorithm in Figure 3.REFERENCESAMIR, Y., DOLEV, D,, KRAMER, S., ANDMALKI,D. 1991. Transis: A communication sub-system forhigh availability. Tech. Rep. CS91-13 (Nov.), Computer Science Department, The HebrewUniversity of .Ierusalem, Jerusalem, Israel.ATrIYA, H., BAR-N• Y, A., DOLEV, D., KOLLER, D., PELEG, D., AND REISCHUK, R. 1987. Achievablecases in an asynchronous environment. In Proceedings of the 28rh Symposium on Foundationsof Compufer Science (Oct.), IEEE Computer Society Press, Washington, D.C., pp. 337-346.ATTIYA, H,, DWORK, C., LYNCH, N., AND STOCKMEYER,L. 1991. Bounds on the time to reachagreement in the presence of timing uncertainty. Irr Proceedings of rhe 23rd ACM Symposium CMTheory of Computing (New Orleans, La,, May 6-8). ACM, New York, pp. 359-369.BEN-OR, M, 1983. Another advantage of free choice: Completely asynchronous agreement protocols.1n Proceedings of the 2nd ACM Symposium on Principles of Distributed Compuring (Montreal,Que.. Canada, Aug. 17-19). ACM, New York, pp. 27-30.