Model

What a majority quorum buys

A quorum is the only fan-out that does not amplify the tail. Replication multiplies availability only if failures are independent.

A_{majority} = i \geq ⌈(N + 1) /2 ⌉ \sum (i N) p^{i} (1 - p)^{N - i}

$p$: Node availability; probability one replica is up
$N$: Replicas; members of the group

Recurring principle: Independence is assumed

How it behaves

One input on the horizontal axis, the others held fixed.

needs a majorityneeds all (unanimity)

Data table

Unavailability by group size, each node up 99% (independent)
Members N	series	Unavailability
1	needs a majority	0.01
1.1	needs a majority	0.01
1.2	needs a majority	0.01
1.3	needs a majority	0.01
1.4	needs a majority	0.01
1.5	needs a majority	0.02
1.6	needs a majority	0.02
1.7	needs a majority	0.02
1.8	needs a majority	0.02
1.9	needs a majority	0.02
2	needs a majority	0.02
2.1	needs a majority	0.02
2.2	needs a majority	0.02
2.3	needs a majority	0.02
2.4	needs a majority	0.02
2.5	needs a majority	0
2.6	needs a majority	0
2.7	needs a majority	0
2.8	needs a majority	0
2.9	needs a majority	0
3	needs a majority	0
3.1	needs a majority	0
3.2	needs a majority	0
3.3	needs a majority	0
3.4	needs a majority	0
3.5	needs a majority	0.001
3.6	needs a majority	0.001
3.7	needs a majority	0.001
3.8	needs a majority	0.001
3.9	needs a majority	0.001
4	needs a majority	0.001
4.1	needs a majority	0.001
4.2	needs a majority	0.001
4.3	needs a majority	0.001
4.4	needs a majority	0.001
4.5	needs a majority	0
4.6	needs a majority	0
4.7	needs a majority	0
4.8	needs a majority	0
4.9	needs a majority	0
5	needs a majority	0
5.1	needs a majority	0
5.2	needs a majority	0
5.3	needs a majority	0
5.4	needs a majority	0
5.5	needs a majority	0
5.6	needs a majority	0
5.7	needs a majority	0
5.8	needs a majority	0
5.9	needs a majority	0
6	needs a majority	0
6.1	needs a majority	0
6.2	needs a majority	0
6.3	needs a majority	0
6.4	needs a majority	0
6.5	needs a majority	0
6.6	needs a majority	0
6.7	needs a majority	0
6.8	needs a majority	0
6.9	needs a majority	0
7	needs a majority	0
7.1	needs a majority	0
7.2	needs a majority	0
7.3	needs a majority	0
7.4	needs a majority	0
7.5	needs a majority	0
7.6	needs a majority	0
7.7	needs a majority	0
7.8	needs a majority	0
7.9	needs a majority	0
8	needs a majority	0
8.1	needs a majority	0
8.2	needs a majority	0
8.3	needs a majority	0
8.4	needs a majority	0
8.5	needs a majority	0
8.6	needs a majority	0
8.7	needs a majority	0
8.8	needs a majority	0
8.9	needs a majority	0
9	needs a majority	0
1	needs all (unanimity)	0.01
1.1	needs all (unanimity)	0.01
1.2	needs all (unanimity)	0.01
1.3	needs all (unanimity)	0.01
1.4	needs all (unanimity)	0.01
1.5	needs all (unanimity)	0.02
1.6	needs all (unanimity)	0.02
1.7	needs all (unanimity)	0.02
1.8	needs all (unanimity)	0.02
1.9	needs all (unanimity)	0.02
2	needs all (unanimity)	0.02
2.1	needs all (unanimity)	0.02
2.2	needs all (unanimity)	0.02
2.3	needs all (unanimity)	0.02
2.4	needs all (unanimity)	0.02
2.5	needs all (unanimity)	0.03
2.6	needs all (unanimity)	0.03
2.7	needs all (unanimity)	0.03
2.8	needs all (unanimity)	0.03
2.9	needs all (unanimity)	0.03
3	needs all (unanimity)	0.03
3.1	needs all (unanimity)	0.03
3.2	needs all (unanimity)	0.03
3.3	needs all (unanimity)	0.03
3.4	needs all (unanimity)	0.03
3.5	needs all (unanimity)	0.039
3.6	needs all (unanimity)	0.039
3.7	needs all (unanimity)	0.039
3.8	needs all (unanimity)	0.039
3.9	needs all (unanimity)	0.039
4	needs all (unanimity)	0.039
4.1	needs all (unanimity)	0.039
4.2	needs all (unanimity)	0.039
4.3	needs all (unanimity)	0.039
4.4	needs all (unanimity)	0.039
4.5	needs all (unanimity)	0.049
4.6	needs all (unanimity)	0.049
4.7	needs all (unanimity)	0.049
4.8	needs all (unanimity)	0.049
4.9	needs all (unanimity)	0.049
5	needs all (unanimity)	0.049
5.1	needs all (unanimity)	0.049
5.2	needs all (unanimity)	0.049
5.3	needs all (unanimity)	0.049
5.4	needs all (unanimity)	0.049
5.5	needs all (unanimity)	0.059
5.6	needs all (unanimity)	0.059
5.7	needs all (unanimity)	0.059
5.8	needs all (unanimity)	0.059
5.9	needs all (unanimity)	0.059
6	needs all (unanimity)	0.059
6.1	needs all (unanimity)	0.059
6.2	needs all (unanimity)	0.059
6.3	needs all (unanimity)	0.059
6.4	needs all (unanimity)	0.059
6.5	needs all (unanimity)	0.068
6.6	needs all (unanimity)	0.068
6.7	needs all (unanimity)	0.068
6.8	needs all (unanimity)	0.068
6.9	needs all (unanimity)	0.068
7	needs all (unanimity)	0.068
7.1	needs all (unanimity)	0.068
7.2	needs all (unanimity)	0.068
7.3	needs all (unanimity)	0.068
7.4	needs all (unanimity)	0.068
7.5	needs all (unanimity)	0.077
7.6	needs all (unanimity)	0.077
7.7	needs all (unanimity)	0.077
7.8	needs all (unanimity)	0.077
7.9	needs all (unanimity)	0.077
8	needs all (unanimity)	0.077
8.1	needs all (unanimity)	0.077
8.2	needs all (unanimity)	0.077
8.3	needs all (unanimity)	0.077
8.4	needs all (unanimity)	0.077
8.5	needs all (unanimity)	0.087
8.6	needs all (unanimity)	0.087
8.7	needs all (unanimity)	0.087
8.8	needs all (unanimity)	0.087
8.9	needs all (unanimity)	0.087
9	needs all (unanimity)	0.087

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Distributed Systems Coordination
Who is allowed to decide, and how does everyone else learn the decision without asking?
$3 \times 0.9 9^{2} \times 0.01 + 0.9 9^{3} = 0.9997$
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