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My name is Marc Brooker. I've been writing code, reading code, and living vicariously through computers for as long as I can remember. I like to build things that work. I also dabble in brewing, cooking and skiing.

I'm currently an engineer at Amazon Web Services (AWS) in Seattle, where I lead engineering on AWS Lambda and our other serverless products. Before that, I worked on EC2 and EBS.
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Quorum Availability

It's counterintuitive, but is it right?

In our paper Millions of Tiny Databases, we say this about the availability of quorum systems of various sizes:

As illustrated in Figure 4, smaller cells offer lower availability in the face of small numbers of uncorrelated node failures, but better availability when the proportion of node failure exceeds 50%. While such high failure rates are rare, they do happen in practice, and a key design concern for Physalia.

And this is what Figure 4 looks like:

The context here is that a cell is a Paxos cluster, and the system needs a majority quorum for the cluster to be able to process requests1. A cluster of one box needs one box available, five boxes need three available and so on. The surprising thing here is the claim that having smaller clusters is actually better if the probability of any given machine failing is very high. The paper doesn't explain it well, and I've gotten a few questions about it. This post attempts to do better.

Let's start by thinking about what happens for a cluster of one machine (n=1), in a datacenter of N machines (for very large N). We then fail each machine independently with probability p. What is the probability that our one machine failed? That's trivial: it's p. Now, let's take all N machines and put them into a cluster of n=N. What's the probability that a majority of the cluster is available? For large N, it's 1 for p < 0.5, and 0 for p > 0.5. If less than half the machines fail, less than half have failed. If more than half the machines fail, more than half have failed. Ok?

Notice how a cluster size of 1 is worse than N up until p = 0.5 then better after. Peleg and Wool say:

... for 0 < p < ½ the most available NDC2 is shown to be the "democracy" (namely, the minimal majority system), while the "monarchy" (singleton system) is least available. Due to symmetry, the picture reverses for ½ < p < 1.

Here, the minimal majority system is the one I'd call a majority quorum, and is used by Physalia (and, indeed, most Paxos implementations). The monarchy is where you have one leader node.

What about real practical cluster sizes like n=3, 5, and 7? There are three ways we can do this math. In The Availability of Quorum Systems, Peleg and Wool derive closed-form solutions to this problem3. Our second approach is to observe that the failures of the nodes are Bernoulli trials with probability p, and therefore we can read the answer to "what is the probability that 0 or 1 of 3 fail for probability p" from the distribution function of the binomial distribution. Finally, we can be lazy and do it with Monte Carlo. That's normally my favorite method, because it's easier to include correlation and various "what if?" questions as we go.

Whichever way you calculate it, what do you expect it to look like? For small n you may expect it to be closer in shape to n=1, and for large n you may expect it to approach the shape of n=N. If that's what you expect, you'd be right.

I'll admit that I find this result deeply deeply counter-intuitive. I think it's right, because I've approached it multiple ways, but it still kind of bends my mind a little. That may just be me. I've discussed it with friends and colleagues over the years, and they seem to think it matches their intuition. It's counter-intuitive to me because it suggests that smaller n (smaller clusters, or smaller cells in Physalia's parlance) is better for high p! If you think a lot of your boxes are going to fail, you may get better availability (not durability, though) from smaller clusters.

Weird.

Correlation to the rescue!

It's not often that my statistical intuition is saved by introducing correlation, but in this case it helps. I'd argue that, in practice, that you only lose machines in an uncorrelated Bernoulli trial way for small p. Above a certain p, it's likely that the failures have some shared cause (power, network, clumsy people, etc) and so the failures are likely to be correlated in some way. In which case, we're back into the game we're playing with Physalia of avoiding those correlated failures by optimizing placement.

In many other kinds of systems, like ones you deploy across multiple datacenters (we'd call that regional in AWS, deployed across multiple availability zones), you end up treating the datacenters as units of failure. In that case, for 3 datacenters you'd pick something like n=9 because you can keep quorum after the failure of an entire datacenter (3 machines) and any one other machine. As soon as there's correlation, the math above is mostly useless and the correlation's cause is all that really matters.

Availability also isn't the only thing to think about with cluster size for quorum systems. Durability, latency, cost, operations, and contention on leader election also come into play. Those are topics for another post (or section 2.2 of Millions of Tiny Databases).

Updates

JP Longmore sent me this intuitive explanation, which makes a lot of sense:

Probability of achieving a quorum will increase when removing 2 nodes from a cluster, each with failure rate p>.5, since on average you're removing 2 bad nodes instead of 2 good nodes. Other cases with 1 good node & 1 bad node don't change the outcome (quorum/not). Repeat reasoning till N=1 or all remaining nodes have p<=.5 (if failure rate isn’t uniform).

Footnotes

  1. Physalia uses a very naive Paxos implementation, intentionally optimized for testability and simplicity. The quorum intersection requirements of Paxos (or Paxos-like protocols) are more subtle than this, and work like Heidi Howard et al's Flexible Paxos has been pushing the envelope here recently. Flexible Paxos: Quorum intersection revisited is a good place to start.
  2. Here, an NDC is a non-dominated coterie, and a coterie is a set of groups of nodes (like {{a, b}, {b, c}, {a, c}}). See Definition 2.2 in How to Assign Votes in a Distributed System for the technical definition of domination. What's important, though, is that for each dominated coterie there's a non-dominated coterie that provides the same mutual exclusion properties, but superior availability under partitions. The details are not particularly important here, but are very interesting if you want to do tricky things with quorum intersection.
  3. Along with a whole lot of other interesting facts about quorums, majority quorums and other things. It's a very interesting paper. Another good read in this space is Garcia-Molina and Barbara's How to Assign Votes in a Distributed System, which both does a better job than Peleg and Wool of defining the terms it uses, but also explores the general idea of assigning votes to machines, rather than simply forming quorums of machines. As you read it, it's worth remembering that it predates Paxos, and many of the terms might not mean what you expect.