Unreliable Failure Detectors for Reliable Distributed Systems
Unreliable Failure Detectors for Reliable Distributed Systems
Unreliable Failure Detectors for Reliable Distributed Systems
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<strong>Unreliable</strong> <strong>Failure</strong> <strong>Detectors</strong> <strong>for</strong> <strong>Reliable</strong> <strong>Distributed</strong> <strong>Systems</strong> 263.YW(0)~ ~ E $??(atron.srnt) .....Con*nsuasolvablef#~(l) .. ...Consensus solvable ifT f < n<strong>for</strong> all J < n\Yq(2).....Conaensus solvable ifl ~ < n - I‘,WWltJ - 1)-c0n*n8us ~l=ble ifl f < ~~1+‘2Consensusnlvable<strong>for</strong>all~ n – f thenConsensus cannot be solved using Y%(m).PROOF (SKETCH). Consider an asynchronous system with f a [n/21 andassume m > n – f. We show that there is a failure detector 9 G ~~(m ) suchthat no algorithm solves Consensus using 9. We do so by describing the behaviorof a Strongly m-Mistaken failure detector Q such that <strong>for</strong> every algorithm A,there is a run R~ of A usin 9 that violates the specification of Consensus.Since 1 s n – ~ s m/2 7, we can partition the processes into three sets 110,II, and IIc,a,h,d, such that HO and 111 are non-empty sets containing n – fprocesses each, and I&h=d is a (possibly empty) set containing the remainingn – 2(n – f) processes. Hence<strong>for</strong>th, we only consider runs in which allprocesses in IIC,a,A,d crash at the beginning of the run. Let go E llo andq ~ c II ~. Consider the following two runs of A using $2:Run RO = (FO, Ho, l., SO, TO). All processes propose O. All processesin 110 are correct in FO, while all the f processes in HI U llcr.$h=~ crash in
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