Gustafson's Law (Scaled Speedup)

Amdahl assumes the job stays the same size while the machine grows. John Gustafson pointed out in 1988 that nobody actually works that way. Give a scientist a machine ten times bigger and they run a finer mesh, not the same mesh ten times faster. Under that assumption the speedup is $S = (1-p) + pN$, which is linear in the processor count and has no ceiling at all: 95% parallel on 21 processors gives exactly 20x, and on 1000 processors it gives 950x.

The two laws are not in conflict, and this is where people get confused. They measure $p$ at different moments. Amdahl's $p$ is the parallel fraction of the ORIGINAL serial run; Gustafson's is the parallel fraction of the run time on the LARGE machine, after the problem has been scaled up. Feed one law the other's number and you will get a confidently wrong answer. The engineering names for the two regimes are strong scaling and weak scaling, and a benchmark that does not say which one it measured is not telling you anything.

What follows from this is less obvious than the algebra. Gustafson's law says the value of a big machine depends on whether your problem has a knob you can turn. Rendering, climate models and training runs all do. A build system, a database migration or a nightly report usually does not, and for those Amdahl is the law that applies and the extra cores sit idle.