Engineering
Amdahl's Law parallel speedup, asymptotic ceiling & Gustafson's Law formulas
Calculate multi-core / GPU parallel speedup 1 / ((1 - p) + p / N), infinite-core serial bottleneck limit 1 / (1 - p), and Gustafson's scaled speedup.
Separate fixed-workload and scaled-workload speedup assumptions
Amdahl uses the parallel fraction of the single-worker execution time for a fixed workload. Its ideal speedup is 1/((1-p)+p/N). With p=0.8 and N=8 the result is 3.33; for p<1 the infinite-worker limit is 1/(1-p). At p=1 there is no finite asymptotic ceiling in this ideal model.
Gustafson uses the parallel fraction of elapsed time measured on the parallel machine for a scaled workload. It is a different fraction: (1-q)+N*q. Supply q independently in C2; using the same number as p is a hypothetical comparison, not two predictions for the same measured workload. Both formulas omit communication, imbalance, scheduling and memory overhead.
| Parallel Scaling Metric (p = 80% Parallel, N = 8 Cores) | Formula | Speedup |
|---|---|---|
| Amdahl's Law (8 Cores) | 1 / ((1 - 0.80) + 0.80 / 8) | 3.33x (41.7% efficiency) |
| Amdahl Asymptotic Ceiling (N -> inf) | 1 / (1 - 0.80) | 5.00x absolute max |
| Gustafson's Scaled Speedup (8 Cores) | (1 - 0.80) + 8 * 0.80 | 6.60x |
| Amdahl at p = 95% Parallel on 16 Cores | 1 / (0.05 + 0.95 / 16) | 9.14x (Max = 20.0x) |
Amdahl's Law speedup across N cores or workers
A2=ParallelFraction p (0.80) and B2=ProcessorCores N (8).
=IF(OR(COUNT(A2:B2)<>2,A2<0,A2>1,B2<1,B2<>INT(B2)),"Invalid inputs",ROUND(1 / ((1 - A2) + (A2 / B2)), 2))Asymptotic max speedup ceiling and Gustafson's Law
The ceiling applies only for p<1. The separate Gustafson check uses C2, the parallel elapsed-time fraction of the scaled workload, rather than reusing the Amdahl fraction in A2.
=IF(OR(NOT(ISNUMBER(A2)),A2<0,A2>1),"Invalid inputs",IF(A2=1,"No finite ceiling",ROUND(1/(1-A2),2)))Try the example
Use the original values or change them to test your understanding.