About Amdahl's Law
Amdahl's Law, formulated by computer architect Gene Amdahl in 1967, describes the theoretical limit on how much a task can speed up when only part of it is parallelized across multiple processors. It is one of the foundational results in parallel computing: it explains why throwing more processors at a problem produces diminishing returns once the non-parallelizable (serial) portion becomes the bottleneck.
The formula
For a task where a fraction P can be run in parallel (0 ≤ P ≤ 1) and the remaining (1 − P) must run serially, the speedup achieved with N processors is:
- Speedup: S(N) = 1 / ((1 − P) + P/N)
- Efficiency: E(N) = S(N) / N — how much of each processor's theoretical contribution is actually realized
- Maximum speedup: as N → ∞, S(N) → 1 / (1 − P) — the ceiling set purely by the serial fraction
This calculator also converts speedup into a concrete parallel run time: if the task takes T seconds on a single processor, it takes T / S(N) seconds with N processors.
How to read the results
- A small increase in P (the parallelizable fraction) matters far more than adding processors once N is already large — this is the core insight of Amdahl's Law.
- Efficiency near 100% means you are getting close to linear scaling; efficiency well under 50% means additional processors are mostly wasted on this workload.
- The maximum theoretical speedup is a hard ceiling: no number of processors can beat 1 / (1 − P) for a fixed task, because the serial portion always has to run start-to-finish on one processor.
Frequently Asked Questions
What is Amdahl's Law?
Amdahl's Law describes the maximum theoretical speedup of a task when only part of it can be parallelized. The formula is S(N) = 1 / ((1 - P) + P/N), where P is the fraction of the task that can be run in parallel and N is the number of processors. The remaining fraction, (1 - P), must run serially and limits how much speedup is possible.
What is the maximum possible speedup?
As the number of processors N approaches infinity, speedup approaches 1 / (1 - P). This is the theoretical ceiling set entirely by the serial (non-parallelizable) portion of the task. For example, if 90% of a task is parallelizable (P = 0.9), the maximum possible speedup is 1 / 0.1 = 10x, no matter how many processors you add.
How is Amdahl's Law different from Gustafson's Law?
Amdahl's Law assumes the total workload size is fixed and asks how much faster a fixed task finishes with more processors, which is why speedup saturates. Gustafson's Law instead assumes the workload grows with the number of processors, modeling scaled-up problems rather than a fixed one, and generally shows more optimistic scaling for large systems.