Amdahl’s law & parallel scaling
The serial fraction limits the speedup of a fixed workload.
More cores cannot speed up the part you cannot split.
Cannot be divided here.
Add more cores
The serial fraction limits total speedup. Extra cores cannot accelerate the part that must run sequentially.
What this model includes
Fixed total work, perfect parallel scaling, no scheduling or communication overhead. This is an optimistic bound.
What happens inside
Split fixed work
If fraction p parallelizes perfectly across N workers, normalized time is (1−p)+p/N. Speedup is its reciprocal. The serial fraction sets the limit as workers increase. This assumes fixed total work and ignores scheduling, synchronization, and communication overhead.
Check the assumptions
Real scaling can be worse because of contention and bandwidth, or show apparent superlinear effects when a larger combined cache changes the working set. Scaling the problem size asks a different question from fixed-work speedup. Measure complete work, not only the parallel kernel.
speedup = 1 / ((1 − p) + p / N)What this means for your code
Low-level engineer
Find lock contention, serial setup, and shared-memory bottlenecks. Account for core topology and work balance.
Software developer
Parallelize only after identifying useful independent work. Reduce the serial section and task overhead as well as increasing workers.
Read the actual specifications
These references supply the underlying contracts and implementation details. The diagrams here are simplified teaching models.