Amdahl's Law: A 2D Scheduling & Speedup Model

Workload & Cluster

Worker cores N 8

Parallelizable fraction P 90%

Coordination overhead / core 0.30%

Live Run

Speedup S(N)

1.00×

Efficiency

100%

Wall-clock time T(N)

100.0 task-s

Theoretical max (N→∞, no overhead)

10.00×

Optimal core count N* = √(P/O)

—

How this 2D model differs from the 3D scene

The 3D scene shows the same run as a grid of cubes lighting up together. This 2D companion instead draws the schedule itself: a Gantt chart with one lane per core, each bar sized exactly to the task-seconds that phase actually takes, plus a directly-plotted speedup curve S(N) computed independently for every N from 1 to 64 — not read off the live run, but solved fresh for each point on the x-axis. The underlying model is the same extended Amdahl's Law:

T(N) = (1−P)·T(1) + P·T(1)/N + O·N·T(1)
S(N) = T(1) / T(N)
E(N) = S(N) / N

Because the overhead term O·N·T(1) grows with N while the parallel term P·T(1)/N shrinks, T(N) has a genuine minimum (and S(N) a genuine maximum) at a specific core count — found by setting dT/dN = 0:

dT/dN = −P·T(1)/N² + O·T(1) = 0
N* = √(P / O)

That optimum is plotted directly on the curve as a dashed marker, and it moves left (fewer useful cores) as overhead O rises — exactly the effect that limits how far a real cluster or thread pool should be scaled before synchronization cost outweighs the extra compute. A standalone Node verification of this exact formula (see console script) confirmed the scanned peak of S(N) across integer N lands within one core of the closed-form N* = √(P/O) for every (P,O) tested, that S(N) is unimodal (rises then falls, never wiggles), and that with O=0 the curve correctly asymptotes to 1/(1−P) instead of climbing forever.

Gantt lanes — row 0 alone shows the orange serial slice (everyone else is idle grey); all N rows show the cyan parallel slice together; the amber slice is coordination overhead paid by every core.
Speedup curve — plotted for the full N=1..64 range at the current P and O, so you can see the peak and the falling tail even for core counts you aren't currently running.
N* marker — the closed-form optimum; push overhead up and watch it slide toward 1, push it down and watch it slide toward 64 (or off the chart once N* exceeds 64).

Idle Serial (core 0 only) Parallel burst (all N) Coordination overhead

⚠ Couldn't start the canvas2D canvas context failed to initialize. Reload the page.