Last updated: 2026-06-29

Gustafson's Law

Gustafson's law estimates the speedup of a parallel program when the size of the problem grows together with the number of processors. It reframes the scaling question asked by Amdahl's law: instead of fixing the workload and shortening the run, it fixes the time budget and asks how much more work a larger machine can complete. The calculator takes the parallel fraction of the run and a processor count and returns the scaled speedup and the parallel efficiency.

Fixed time instead of fixed work

When practitioners buy a bigger cluster, they often do not run the same job faster — they run a bigger job in the same time. A weather model moves to a finer grid, a renderer produces more frames, a training run ingests more data. In this setting the parallel portion of the work expands to fill the added processors, while the serial overhead (startup, coordination, final aggregation) stays roughly constant.

Gustafson's law captures this. Let $p$ be the fraction of the run time that is parallel work as measured on the parallel system, and $1 - p$ the serial fraction shared by every processor. The scaled speedup on $N$ processors is:

$S = (1 - p) + p \cdot N$

Because the parallel term grows linearly with $N$, the predicted speedup keeps rising as processors are added, rather than flattening against a ceiling.

Worked example

Suppose 95% of the run on a 16-processor system is parallel work. The scaled speedup is:

So the 16-processor system completes about 15.25× the work of a single processor in the same wall-clock time, for a parallel efficiency of $15.25 / 16 \approx 95\%$. Contrast this with the fixed-size view, where the same 95% parallel fraction limits speedup to roughly 9.1× on 16 processors and 20× in the limit.

Strong scaling versus weak scaling

The difference between the two laws is the difference between strong scaling and weak scaling. Strong scaling — the regime of Amdahl's Law Calculator — keeps the problem fixed and is bounded by the serial fraction. Weak scaling — the regime of Gustafson's law — grows the problem with the hardware and scales nearly linearly. Neither law is universally correct; the right choice depends on whether your workload is fixed or expands to use the resources you give it.

Because the parallel fraction is measured on the scaled system and depends on problem size, treat the result as an estimate of expected behavior rather than a precise guarantee. Estimating speedup under both Gustafson's and Amdahl's assumptions brackets the likely range.