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CHAPTER 3. KNOWN REVISIONS 69 maximise the number of failures handled by the systems. When using majorities, to handle f failures, the quorums must be of at least size f + 1 in a system of 2f + 1 acceptors. This approach quickly limits the scalability and fault tolerance of a system. The purpose of using strict majorities is to ensure that all quorums intersect, therefore it has been noted elsewhere [Lam78a, §1.4][Lam01a, §2.2][Lam05a][JRS11, §2] that majorities can be generalised to use any quorum system Q, provided that all quorums Q ∈ Q intersect. Therefore, we revise our definition of decided as follows: Definition 8. A proposal (e,v) is decided if the proposal (e,v) has been accepted by a quorum of acceptors. Formally, the quorum intersection requirement for Classic Paxos is specified as follows: ∀Q,Q′ ∈Q:Q∩Q′ ̸=∅ (3.1) A generalised version of the proposer algorithm is given by Algorithm 12. The acceptor algorithm remains unchanged. Figure 3.6 gives an example of this generalisation in practice. In this scenario, the sys- tem is comprised of 4 acceptors, A = {a1,a2,a3,a4} and the quorum system is Q = {{a1, a2}, {a1, a3}, {a1, a4}, {a2, a3, a4}}. Compared to strict majority quorums, which would require three acceptor to form a quorum, the proposer p1 in this example is able to complete both phases using a quorum of only two acceptors, a1 and a215. Strict majorities are just one example of a quorum set which satisfies Classic Paxos’s quorum intersection requirement. There are many quorum sets that could be utilised with Classic Paxos, offering different tradeoffs in sizes of quorums, number and diversity of quorums, number of participants as well as number and types of failures tolerated. The flexibility to choose a quorum set allows us to loosen the coupling between the number of acceptors, the number of acceptors participating in each phase and the number of failures tolerated. However, since all quorums are required to intersect, there remain fundamental limitations on what can be achieved. For example, Classic Paxos cannot reach a decision whilst any whole quorum has failed. As such, quorum systems other than strict majority are rarely utilised in practice. 3.12 Miscellaneous Other variants and optimisations include: 15Equally, acceptors a1 and a3 or acceptors a1 and a4 could have be used. Different quorums may be used by each phase.PDF Image | Distributed consensus
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