Stage 08 · Automation: Amdahl's Law for Your Operations

Ariadne's eighth stage is "stop doing it by hand." This post is about how to know, before you build anything, how much that is worth.

The ceiling

Let $f$ be the fraction of a process that can be automated and $S$ the speedup applied to that fraction. Total speedup is

As $S \to \infty$, this converges to

This is Amdahl's law, in the form given by Hill and Marty Hill & Marty 2008. Some values worth memorising:

Gustafson's rebuttal, which is the useful half

Amdahl's law assumes a fixed problem. Gustafson objected that this is not how systems are actually used Gustafson 1988: "One does not take a fixed-size problem and run it on various numbers of processors except when doing academic research; in practice, the problem size scales."

Under scaled speedup, with $s'$ the serial fraction measured on the scaled system:

Linear rather than asymptotic. Gustafson's team measured speedups of 1021, 1020 and 1016 on a 1024-processor machine — figures Amdahl's law would forbid for a fixed problem.

So the right question at the start of an automation project is not "how much time will this save on today's volume." It is: what becomes possible at this cost per unit that was not possible before?

The 95% you did not budget for

There is a second ceiling, and it is about implementation rather than arithmetic.

Sculley and colleagues, writing from Google at NeurIPS, observed that "a mature system might end up being (at most) 5% machine learning code and (at least) 95% glue code" Sculley et al. 2015. They are describing ML systems, and the figure is an engineering judgement rather than a measurement — I flag that because it is quoted as a statistic constantly.

The pattern generalises exactly to business automation. The rule that decides which lead goes to which person is twenty lines. The glue — authenticating to four systems, reconciling contact records, handling the customer who used a different email, retrying the API that fails at 3am, alerting a human when it cannot proceed — is the actual project.

How I scope an automation

Four steps, in order, before writing code.

1. Time the process honestly. Not the happy path. Include the exceptions, because exceptions are where $1-f$ lives.

2. Split it into $f$ and $1-f$. Deterministic, rule-expressible steps are automatable. Steps requiring judgement about an ambiguous human situation are not — at least not without a human confirming, which keeps them in $1-f$ for planning purposes.

3. Compute the ceiling, then ask the Gustafson question. If $1/(1-f)$ is under 2×, the fixed-volume case is weak. Ask whether removing the constraint changes what you would attempt. Frequently the answer is yes, and that is the real justification.

4. Budget for the glue. If your estimate is the rule logic, multiply it. This is not padding; it is the observed shape of these systems.

The four principles on Ariadne's site include "keep humans in control" alongside "leave the keys." The next post is about why that first one is not a values statement but a hard-won engineering result — automating the easy parts reliably makes the remaining human job harder.

References

1. Amdahl, G. M. (1967). Validity of the single processor approach to achieving large scale computing capabilities. AFIPS '67 (Spring): Proceedings of the Spring Joint Computer Conference. https://doi.org/10.1145/1465482.1465560Contains no equations and one figure; the algebraic form came later.
2. Hill, M. D., & Marty, M. R. (2008). Amdahl's law in the multicore era. Computer, 41(7), 33–38. https://doi.org/10.1109/MC.2008.209The authoritative modern statement of the formula.
3. Gustafson, J. L. (1988). Reevaluating Amdahl's law. Communications of the ACM, 31(5), 532–533. https://doi.org/10.1145/42411.42415
4. Sculley, D., Holt, G., Golovin, D., Davydov, E., Phillips, T., Ebner, D., Chaudhary, V., Young, M., Crespo, J., & Dennison, D. (2015). Hidden technical debt in machine learning systems. Advances in Neural Information Processing Systems 28 (NeurIPS 2015), 2503–2511. https://papers.nips.cc/paper_files/paper/2015/hash/86df7dcfd896fcaf2674f757a2463eba-Abstract.htmlThe 5%/95% figure is a qualitative engineering judgement, not a measurement.