Napkin Math #

Originally from https://github.com/sirupsen/napkin-math.

The goal of this project is to collect software, numbers, and techniques to quickly estimate the expected performance of systems from first-principles. For example, how quickly can you read 1 GB of memory? By composing these resources you should be able to answer interesting questions like: how much storage cost should you expect to pay for logging for an application with 100,000 RPS?

The best introduction to this skill is through my talk at SRECON.

The best way to practise napkin math in the grand domain of computers is to work on your own problems. The second-best is to subscribe to this newsletter where you’ll get a problem every few weeks to practise on. It should only take you a few minutes to solve each one as your facility with these techniques improve.

The archive of problems to practise with are here. The solution will be in the following newsletter.

Numbers #

Below are numbers that are rounded from runs on a GCP c2-standard-4 (Intel Cascade) and 2017 Macbook (2.8GHz, quad-core).

Note 1: Numbers have been rounded, which means they don’t line up perfectly. Note 2: Some throughput and latency numbers don’t line up (for ease of calculations see exact results e.g. here).

†: “Fast serialization/deserialization” is typically a simple wire-protocol that just dumps bytes, or a very efficient environment. Typically standard serialization such as e.g. JSON will be of the slower kind. We include both here as serialization/deserialization is a very, very broad topic with extremely different performance characteristics depending on data and implementation.

You can run this with RUSTFLAGS='-C target-cpu=native' cargo run --release -- --help. You won’t get the right numbers when you’re compiling in debug mode. You can help this project by adding new suites and filling out the blanks.

I am aware of some inefficiencies in this suite. I intend to improve my skills in this area, in order to ensure the numbers are the upper-bound of performance you may be able to squeeze out in production. I find it highly unlikely any of them will be more than 2-3x off, which shouldn’t be a problem for most users.

Cost Numbers #

Approximate numbers that should be consistent between Cloud providers.

Compression Ratios #

This is sourced from a few sources. [3] [4] [5] Note that compression speeds (but generally not ratios) vary by an order of magnitude depending on the algorithm and the level of compression (which trades speed for compression).

I typically ballpark that another x in compression ratio decreases performance by 10x. E.g. we can get a 2x ratio on English Wikipedia at ~200 MiB/s, and 3x at ~20MiB/s, and 4x at 1MB/s.

Techniques #

• Don’t overcomplicate. If you are basing your calculation on more than 6 assumptions, you’re likely making it harder than it should be.
• Keep the units. They’re good checksumming. Wolframalpha has terrific support if you need a hand in converting e.g. KiB to TiB.
• Calculate with exponents. A lot of back-of-the-envelope calculations are done with just coefficients and exponents, e.g. c * 10^e. Your goal is to get within an order of magnitude right–that’s just e. c matters a lot less. Only worrying about single-digit coefficients and exponents makes it much easier on a napkin (not to speak of all the zeros you avoid writing).
• Perform Fermi decomposition. Write down things you can guess at until you can start to hint at an answer. When you want to know the cost of storage for logging, you’re going to want to know how big a log line is, how many of those you have per second, what that costs, and so on.

Resources #

• [1]: https://eli.thegreenplace.net/2018/measuring-context-switching-and-memory-overheads-for-linux-threads/
• [2]: https://blog.tsunanet.net/2010/11/how-long-does-it-take-to-make-context.html
• [3]: https://cran.r-project.org/web/packages/brotli/vignettes/brotli-2015-09-22.pdf
• [4]: https://github.com/google/snappy
• [5]: https://quixdb.github.io/squash-benchmark/
• [6]: https://dl.acm.org/doi/10.1145/1879141.1879143
• [7]: https://en.wikipedia.org/wiki/Hard_disk_drive_performance_characteristics#Seek_times_&_characteristics
• [8]: https://github.com/simdjson/simdjson#performance-results
• [9]: https://github.com/protocolbuffers/protobuf/blob/master/docs/performance.md
• “How to get consistent results when benchmarking on Linux?”. Great compilation of various Kernel and CPU features to toggle for reliable bench-marking, e.g. CPU affinity, disabling turbo boost, etc. It also has resources on proper statistical methods for benchmarking.
• LLVM benchmarking tips. Similar to the above in terms of dedicating CPUs, disabling address space randomization, etc.
• Top-Down performance analysis methodology. Useful post about using toplev to find the bottlenecks. This is particularly useful for the benchmarking suite we have here, to ensure the programs are correctly written (I have not taken them through this yet, but plan to).
• Godbolt’s compiler explorer. Fantastic resource for comparing assembly between Rust and e.g. C with Clang/GCC.
• cargo-asm. Cargo extension to allow disassembling functions. Unfortunately the support for closure is a bit lacking, which requires some refactoring. It’s also very slow on even this simple program.
• Agner’s Assembly Guide. An excellent resource on writing optimum assembly, which will be useful to inspect the various functions for inefficiencies in our suite.
• Agner’s Instruction Tables. Thorough resource on the expected throughput for various instructions which is helpful to inspect the assembly.
• halobates.de. Useful resource for low-level performance by the author of toplev.
• Systems Performance (book). Fantastic book about analyzing system performance, finding bottlenecks, and understanding operating systems.
• io_uring. Best summary, it links to many resources.
• How Long Does It Takes To Make a Context Switch
• Integer Compression Comparisons
• Files are hard