Model

Universal Scalability Law

Coordination grows as N² and eventually reduces throughput.

C (N) = \frac{N}{1 + α ( N - 1 ) + βN ( N - 1 )}, N^{*} = \frac{1 - α}{β}

$C (N)$: Relative throughput; the throughput compared with one worker, when using N workers
$α$: Contention; waiting for shared resources (Amdahl's term)
$β$: Coherency; cost of keeping N participants consistent with each other
$N^{*}$: Peak; beyond it, each new worker reduces throughput

Trade-off: Coupling vs. coordination cost

How it behaves

One input on the horizontal axis, the others held fixed.

β = 0 (Amdahl: a ceiling)β = 0.0001 (a peak, then a fall)

Data table

Relative throughput by workers (α = 0.03)
Workers N	series	Throughput C(N) (× one worker)
1	β = 0 (Amdahl: a ceiling)	1×
5.99	β = 0 (Amdahl: a ceiling)	5.21×
11	β = 0 (Amdahl: a ceiling)	8.45×
16	β = 0 (Amdahl: a ceiling)	11×
20.9	β = 0 (Amdahl: a ceiling)	13.1×
25.9	β = 0 (Amdahl: a ceiling)	14.8×
30.9	β = 0 (Amdahl: a ceiling)	16.3×
35.9	β = 0 (Amdahl: a ceiling)	17.5×
40.9	β = 0 (Amdahl: a ceiling)	18.6×
45.9	β = 0 (Amdahl: a ceiling)	19.6×
50.9	β = 0 (Amdahl: a ceiling)	20.4×
55.9	β = 0 (Amdahl: a ceiling)	21.1×
60.9	β = 0 (Amdahl: a ceiling)	21.8×
65.8	β = 0 (Amdahl: a ceiling)	22.4×
70.8	β = 0 (Amdahl: a ceiling)	22.9×
75.8	β = 0 (Amdahl: a ceiling)	23.4×
80.8	β = 0 (Amdahl: a ceiling)	23.8×
85.8	β = 0 (Amdahl: a ceiling)	24.2×
90.8	β = 0 (Amdahl: a ceiling)	24.6×
95.8	β = 0 (Amdahl: a ceiling)	24.9×
101	β = 0 (Amdahl: a ceiling)	25.2×
106	β = 0 (Amdahl: a ceiling)	25.5×
111	β = 0 (Amdahl: a ceiling)	25.8×
116	β = 0 (Amdahl: a ceiling)	26.1×
121	β = 0 (Amdahl: a ceiling)	26.3×
126	β = 0 (Amdahl: a ceiling)	26.5×
131	β = 0 (Amdahl: a ceiling)	26.7×
136	β = 0 (Amdahl: a ceiling)	26.9×
141	β = 0 (Amdahl: a ceiling)	27.1×
146	β = 0 (Amdahl: a ceiling)	27.3×
151	β = 0 (Amdahl: a ceiling)	27.4×
156	β = 0 (Amdahl: a ceiling)	27.6×
161	β = 0 (Amdahl: a ceiling)	27.7×
166	β = 0 (Amdahl: a ceiling)	27.9×
171	β = 0 (Amdahl: a ceiling)	28×
176	β = 0 (Amdahl: a ceiling)	28.1×
181	β = 0 (Amdahl: a ceiling)	28.3×
186	β = 0 (Amdahl: a ceiling)	28.4×
191	β = 0 (Amdahl: a ceiling)	28.5×
196	β = 0 (Amdahl: a ceiling)	28.6×
201	β = 0 (Amdahl: a ceiling)	28.7×
205	β = 0 (Amdahl: a ceiling)	28.8×
210	β = 0 (Amdahl: a ceiling)	28.9×
215	β = 0 (Amdahl: a ceiling)	29×
220	β = 0 (Amdahl: a ceiling)	29.1×
225	β = 0 (Amdahl: a ceiling)	29.2×
230	β = 0 (Amdahl: a ceiling)	29.2×
235	β = 0 (Amdahl: a ceiling)	29.3×
240	β = 0 (Amdahl: a ceiling)	29.4×
245	β = 0 (Amdahl: a ceiling)	29.5×
250	β = 0 (Amdahl: a ceiling)	29.5×
255	β = 0 (Amdahl: a ceiling)	29.6×
260	β = 0 (Amdahl: a ceiling)	29.7×
265	β = 0 (Amdahl: a ceiling)	29.7×
270	β = 0 (Amdahl: a ceiling)	29.8×
275	β = 0 (Amdahl: a ceiling)	29.8×
280	β = 0 (Amdahl: a ceiling)	29.9×
285	β = 0 (Amdahl: a ceiling)	29.9×
290	β = 0 (Amdahl: a ceiling)	30×
295	β = 0 (Amdahl: a ceiling)	30×
300	β = 0 (Amdahl: a ceiling)	30.1×
305	β = 0 (Amdahl: a ceiling)	30.1×
310	β = 0 (Amdahl: a ceiling)	30.2×
315	β = 0 (Amdahl: a ceiling)	30.2×
320	β = 0 (Amdahl: a ceiling)	30.3×
325	β = 0 (Amdahl: a ceiling)	30.3×
330	β = 0 (Amdahl: a ceiling)	30.4×
335	β = 0 (Amdahl: a ceiling)	30.4×
340	β = 0 (Amdahl: a ceiling)	30.4×
345	β = 0 (Amdahl: a ceiling)	30.5×
350	β = 0 (Amdahl: a ceiling)	30.5×
355	β = 0 (Amdahl: a ceiling)	30.6×
360	β = 0 (Amdahl: a ceiling)	30.6×
365	β = 0 (Amdahl: a ceiling)	30.6×
370	β = 0 (Amdahl: a ceiling)	30.7×
375	β = 0 (Amdahl: a ceiling)	30.7×
380	β = 0 (Amdahl: a ceiling)	30.7×
385	β = 0 (Amdahl: a ceiling)	30.8×
390	β = 0 (Amdahl: a ceiling)	30.8×
395	β = 0 (Amdahl: a ceiling)	30.8×
400	β = 0 (Amdahl: a ceiling)	30.8×
1	β = 0.0001 (a peak, then a fall)	1×
5.99	β = 0.0001 (a peak, then a fall)	5.19×
11	β = 0.0001 (a peak, then a fall)	8.38×
16	β = 0.0001 (a peak, then a fall)	10.8×
20.9	β = 0.0001 (a peak, then a fall)	12.8×
25.9	β = 0.0001 (a peak, then a fall)	14.3×
30.9	β = 0.0001 (a peak, then a fall)	15.5×
35.9	β = 0.0001 (a peak, then a fall)	16.5×
40.9	β = 0.0001 (a peak, then a fall)	17.3×
45.9	β = 0.0001 (a peak, then a fall)	18×
50.9	β = 0.0001 (a peak, then a fall)	18.5×
55.9	β = 0.0001 (a peak, then a fall)	18.9×
60.9	β = 0.0001 (a peak, then a fall)	19.3×
65.8	β = 0.0001 (a peak, then a fall)	19.5×
70.8	β = 0.0001 (a peak, then a fall)	19.7×
75.8	β = 0.0001 (a peak, then a fall)	19.9×
80.8	β = 0.0001 (a peak, then a fall)	20×
85.8	β = 0.0001 (a peak, then a fall)	20.1×
90.8	β = 0.0001 (a peak, then a fall)	20.1×
95.8	β = 0.0001 (a peak, then a fall)	20.2×
101	β = 0.0001 (a peak, then a fall)	20.2×
106	β = 0.0001 (a peak, then a fall)	20.1×
111	β = 0.0001 (a peak, then a fall)	20.1×
116	β = 0.0001 (a peak, then a fall)	20.1×
121	β = 0.0001 (a peak, then a fall)	20×
126	β = 0.0001 (a peak, then a fall)	19.9×
131	β = 0.0001 (a peak, then a fall)	19.8×
136	β = 0.0001 (a peak, then a fall)	19.8×
141	β = 0.0001 (a peak, then a fall)	19.7×
146	β = 0.0001 (a peak, then a fall)	19.6×
151	β = 0.0001 (a peak, then a fall)	19.5×
156	β = 0.0001 (a peak, then a fall)	19.3×
161	β = 0.0001 (a peak, then a fall)	19.2×
166	β = 0.0001 (a peak, then a fall)	19.1×
171	β = 0.0001 (a peak, then a fall)	19×
176	β = 0.0001 (a peak, then a fall)	18.9×
181	β = 0.0001 (a peak, then a fall)	18.8×
186	β = 0.0001 (a peak, then a fall)	18.6×
191	β = 0.0001 (a peak, then a fall)	18.5×
196	β = 0.0001 (a peak, then a fall)	18.4×
201	β = 0.0001 (a peak, then a fall)	18.3×
205	β = 0.0001 (a peak, then a fall)	18.1×
210	β = 0.0001 (a peak, then a fall)	18×
215	β = 0.0001 (a peak, then a fall)	17.9×
220	β = 0.0001 (a peak, then a fall)	17.7×
225	β = 0.0001 (a peak, then a fall)	17.6×
230	β = 0.0001 (a peak, then a fall)	17.5×
235	β = 0.0001 (a peak, then a fall)	17.4×
240	β = 0.0001 (a peak, then a fall)	17.2×
245	β = 0.0001 (a peak, then a fall)	17.1×
250	β = 0.0001 (a peak, then a fall)	17×
255	β = 0.0001 (a peak, then a fall)	16.9×
260	β = 0.0001 (a peak, then a fall)	16.8×
265	β = 0.0001 (a peak, then a fall)	16.6×
270	β = 0.0001 (a peak, then a fall)	16.5×
275	β = 0.0001 (a peak, then a fall)	16.4×
280	β = 0.0001 (a peak, then a fall)	16.3×
285	β = 0.0001 (a peak, then a fall)	16.2×
290	β = 0.0001 (a peak, then a fall)	16.1×
295	β = 0.0001 (a peak, then a fall)	15.9×
300	β = 0.0001 (a peak, then a fall)	15.8×
305	β = 0.0001 (a peak, then a fall)	15.7×
310	β = 0.0001 (a peak, then a fall)	15.6×
315	β = 0.0001 (a peak, then a fall)	15.5×
320	β = 0.0001 (a peak, then a fall)	15.4×
325	β = 0.0001 (a peak, then a fall)	15.3×
330	β = 0.0001 (a peak, then a fall)	15.2×
335	β = 0.0001 (a peak, then a fall)	15.1×
340	β = 0.0001 (a peak, then a fall)	15×
345	β = 0.0001 (a peak, then a fall)	14.9×
350	β = 0.0001 (a peak, then a fall)	14.8×
355	β = 0.0001 (a peak, then a fall)	14.7×
360	β = 0.0001 (a peak, then a fall)	14.6×
365	β = 0.0001 (a peak, then a fall)	14.5×
370	β = 0.0001 (a peak, then a fall)	14.4×
375	β = 0.0001 (a peak, then a fall)	14.3×
380	β = 0.0001 (a peak, then a fall)	14.2×
385	β = 0.0001 (a peak, then a fall)	14.1×
390	β = 0.0001 (a peak, then a fall)	14×
395	β = 0.0001 (a peak, then a fall)	13.9×
400	β = 0.0001 (a peak, then a fall)	13.8×

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Physical Substrate & Performance Laws
Why do trade-offs exist at all?
$C (N) = \frac{N}{1 + α ( N - 1 ) + β \cdot N \cdot ( N - 1 )}$
Read it in place →
Core Data Structures
How do we choose a data structure when the real cost is set by the hardware, not by Big-O?
$C (N) = N / (1 + α (N - 1) + βN (N - 1))$
Read it in place →