Amdahl's Law
A formula bounding the speedup from parallelization by the fraction of work that stays sequential.
Definition
Amdahl's law states that the maximum speedup from parallelizing a program is limited by its serial fraction. If a fraction p of the work is parallelizable across N processors, the speedup is 1 / ((1 - p) + p/N).
Gustafson's law offers the optimistic counterpoint: in practice people use bigger machines to solve bigger problems, and the parallel portion often grows with problem size while the serial part stays fixed. Both laws are true; they answer different questions about scaling.
The law's sobering message, that a small serial fraction caps the achievable speedup no matter how many processors are added, redirects optimization toward shrinking that serial portion. Its optimistic counterpart, Gustafson's law, observes that larger machines are usually used for larger problems whose parallel work grows while the serial part stays fixed. Both are correct; which applies depends on whether the problem size is held constant or allowed to grow.
As N grows, the speedup approaches 1 / (1 - p): even a small serial portion caps the achievable gain.
Implications
- A 5 percent serial fraction limits speedup to at most 20x.
- The serial part, not the parallel part, becomes the bottleneck.
- Gustafson's law offers a more optimistic view for growing problem sizes.
Why it matters
Amdahl's law tempers expectations for parallel hardware and directs optimization effort toward reducing the serial fraction rather than simply adding processors.
Fusion connection
It guides how Kronos structures simulations, since minimizing inherently serial steps is what lets a plasma code keep scaling as more cluster nodes are added.