Unreliable Failure Detectors for Reliable Distributed Systems

240 T. D. CHANDRA AND S. TOUEGEvery procese p executes the foliowin~procedurevp4-(L,l,...VP(P]4- IJpAP t VPpropose,l){p’s estimateof the pruposed values}Phase 1: {astmchnmous rounds r~, 1< rP < n – 1}forrptltorb-lsend (rP, AP, p) to allwait until ~q : received (rp, Aq, q) or ~ ~ 9P] {que~ the ~ailum detector}nwgsP(rP] t {(rp,Aq,II)I r-+ed (rp, AQ~ q)}Apt(L, L,...,l)fork+ ltonif Vp[k] = 1 and 3(rP, Aq,q) E msgsp[rP]Vp[k] t Aq[k]with A~[k] # 1 thenAP[k] + AQ[k]Pheee 2: send VP to allwait until ~q : received VQ or q 6 ‘3P] {que~ the faike detector}lastm9gsP t {Vq I received VQ}forktl tonif WQ G hmtnasgsp with Vq[k] = 1 then Vp[k] t 1Phase 3: decide( first non-l component of Vp)FIG.5. Solving Consensus using any Q c 9.The algorithm runs through three phases. In Phase 1, processes execute n – 1asynchronous rounds (rP denotes the current round number of process p ) duringwhich they broadcast and relay their proposed values. Each process p waits untilit receives a round r message from every process that is not in 9P, beforeproceeding to round r + 1. Note that while p is waiting for a message from q inround r, it is possible that q is added to 9P. If this occurs, p stops waiting for q‘smessage and proceeds to round r + 1.By the end of Phase 2, correct processes agree on a vector based on theproposed values of all processes. The ith element of this vector either containsthe proposed value of process pi or 1. We will show that this vector contains theproposed value of at least one process. In Phase 3, correct processes decide thefirst nontrivial component of this vector.Let R = (F, H9, 1, S, T) be any run of the algorithm in Figure 5 using9 G Y in which all correct processes propose a value. We have to show that thetermination, uniform validity, agreement and uniform integrity properties ofConsensus hold.Note that VP[q] is p‘s current estimate of q‘s proposed value. Furthermore,AP[q] = Uqat the end of round r if and only if p receives Vq, the value proposedby q, for the first time in round r.LEMMA 6.1.1. For all p and q, and in all phases, VP[q] is either v~ or 1.PROOF. Obvious from the algorithm. •lLEMMA 6.1.2. Eveiy correct process eventual~ reaches Phase 3.