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Model

The barrier waits for the maximum

The stage takes about twice what its average task time predicts, from variance alone.

Tasks
tasks behind the barrier
Variation
coefficient of variation: standard deviation ÷ mean

Recurring principle: Variance costs more than volume

How it behaves

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

c = 0.1c = 0.3c = 0.6
Data table
Stage time ÷ mean task time, by tasks behind a barrier
Tasks nseriesExpected max ÷ mean
2c = 0.11.12×
2.29c = 0.11.13×
2.62c = 0.11.14×
3c = 0.11.15×
3.44c = 0.11.16×
3.93c = 0.11.17×
4.5c = 0.11.17×
5.15c = 0.11.18×
5.9c = 0.11.19×
6.76c = 0.11.2×
7.73c = 0.11.2×
8.85c = 0.11.21×
10.1c = 0.11.22×
11.6c = 0.11.22×
13.3c = 0.11.23×
15.2c = 0.11.23×
17.4c = 0.11.24×
19.9c = 0.11.24×
22.8c = 0.11.25×
26.1c = 0.11.26×
29.9c = 0.11.26×
34.2c = 0.11.27×
39.2c = 0.11.27×
44.9c = 0.11.28×
51.4c = 0.11.28×
58.8c = 0.11.29×
67.3c = 0.11.29×
77.1c = 0.11.29×
88.2c = 0.11.3×
101c = 0.11.3×
116c = 0.11.31×
132c = 0.11.31×
152c = 0.11.32×
174c = 0.11.32×
199c = 0.11.33×
227c = 0.11.33×
260c = 0.11.33×
298c = 0.11.34×
341c = 0.11.34×
391c = 0.11.35×
447c = 0.11.35×
512c = 0.11.35×
586c = 0.11.36×
671c = 0.11.36×
768c = 0.11.36×
879c = 0.11.37×
1.01Kc = 0.11.37×
1.15Kc = 0.11.38×
1.32Kc = 0.11.38×
1.51Kc = 0.11.38×
1.73Kc = 0.11.39×
1.98Kc = 0.11.39×
2.27Kc = 0.11.39×
2.59Kc = 0.11.4×
2.97Kc = 0.11.4×
3.4Kc = 0.11.4×
3.89Kc = 0.11.41×
4.46Kc = 0.11.41×
5.1Kc = 0.11.41×
5.84Kc = 0.11.42×
6.69Kc = 0.11.42×
7.66Kc = 0.11.42×
8.76Kc = 0.11.43×
10Kc = 0.11.43×
11.5Kc = 0.11.43×
13.2Kc = 0.11.44×
15.1Kc = 0.11.44×
17.2Kc = 0.11.44×
19.7Kc = 0.11.44×
22.6Kc = 0.11.45×
25.9Kc = 0.11.45×
29.6Kc = 0.11.45×
33.9Kc = 0.11.46×
38.8Kc = 0.11.46×
44.4Kc = 0.11.46×
50.9Kc = 0.11.47×
58.2Kc = 0.11.47×
66.6Kc = 0.11.47×
76.3Kc = 0.11.47×
87.3Kc = 0.11.48×
100Kc = 0.11.48×
2c = 0.31.35×
2.29c = 0.31.39×
2.62c = 0.31.42×
3c = 0.31.44×
3.44c = 0.31.47×
3.93c = 0.31.5×
4.5c = 0.31.52×
5.15c = 0.31.54×
5.9c = 0.31.57×
6.76c = 0.31.59×
7.73c = 0.31.61×
8.85c = 0.31.63×
10.1c = 0.31.65×
11.6c = 0.31.66×
13.3c = 0.31.68×
15.2c = 0.31.7×
17.4c = 0.31.72×
19.9c = 0.31.73×
22.8c = 0.31.75×
26.1c = 0.31.77×
29.9c = 0.31.78×
34.2c = 0.31.8×
39.2c = 0.31.81×
44.9c = 0.31.83×
51.4c = 0.31.84×
58.8c = 0.31.86×
67.3c = 0.31.87×
77.1c = 0.31.88×
88.2c = 0.31.9×
101c = 0.31.91×
116c = 0.31.92×
132c = 0.31.94×
152c = 0.31.95×
174c = 0.31.96×
199c = 0.31.98×
227c = 0.31.99×
260c = 0.32×
298c = 0.32.01×
341c = 0.32.02×
391c = 0.32.04×
447c = 0.32.05×
512c = 0.32.06×
586c = 0.32.07×
671c = 0.32.08×
768c = 0.32.09×
879c = 0.32.1×
1.01Kc = 0.32.12×
1.15Kc = 0.32.13×
1.32Kc = 0.32.14×
1.51Kc = 0.32.15×
1.73Kc = 0.32.16×
1.98Kc = 0.32.17×
2.27Kc = 0.32.18×
2.59Kc = 0.32.19×
2.97Kc = 0.32.2×
3.4Kc = 0.32.21×
3.89Kc = 0.32.22×
4.46Kc = 0.32.23×
5.1Kc = 0.32.24×
5.84Kc = 0.32.25×
6.69Kc = 0.32.26×
7.66Kc = 0.32.27×
8.76Kc = 0.32.28×
10Kc = 0.32.29×
11.5Kc = 0.32.3×
13.2Kc = 0.32.31×
15.1Kc = 0.32.32×
17.2Kc = 0.32.33×
19.7Kc = 0.32.33×
22.6Kc = 0.32.34×
25.9Kc = 0.32.35×
29.6Kc = 0.32.36×
33.9Kc = 0.32.37×
38.8Kc = 0.32.38×
44.4Kc = 0.32.39×
50.9Kc = 0.32.4×
58.2Kc = 0.32.41×
66.6Kc = 0.32.41×
76.3Kc = 0.32.42×
87.3Kc = 0.32.43×
100Kc = 0.32.44×
2c = 0.61.71×
2.29c = 0.61.77×
2.62c = 0.61.83×
3c = 0.61.89×
3.44c = 0.61.94×
3.93c = 0.61.99×
4.5c = 0.62.04×
5.15c = 0.62.09×
5.9c = 0.62.13×
6.76c = 0.62.17×
7.73c = 0.62.21×
8.85c = 0.62.25×
10.1c = 0.62.29×
11.6c = 0.62.33×
13.3c = 0.62.36×
15.2c = 0.62.4×
17.4c = 0.62.43×
19.9c = 0.62.47×
22.8c = 0.62.5×
26.1c = 0.62.53×
29.9c = 0.62.56×
34.2c = 0.62.59×
39.2c = 0.62.63×
44.9c = 0.62.65×
51.4c = 0.62.68×
58.8c = 0.62.71×
67.3c = 0.62.74×
77.1c = 0.62.77×
88.2c = 0.62.8×
101c = 0.62.82×
116c = 0.62.85×
132c = 0.62.88×
152c = 0.62.9×
174c = 0.62.93×
199c = 0.62.95×
227c = 0.62.98×
260c = 0.63×
298c = 0.63.03×
341c = 0.63.05×
391c = 0.63.07×
447c = 0.63.1×
512c = 0.63.12×
586c = 0.63.14×
671c = 0.63.16×
768c = 0.63.19×
879c = 0.63.21×
1.01Kc = 0.63.23×
1.15Kc = 0.63.25×
1.32Kc = 0.63.27×
1.51Kc = 0.63.3×
1.73Kc = 0.63.32×
1.98Kc = 0.63.34×
2.27Kc = 0.63.36×
2.59Kc = 0.63.38×
2.97Kc = 0.63.4×
3.4Kc = 0.63.42×
3.89Kc = 0.63.44×
4.46Kc = 0.63.46×
5.1Kc = 0.63.48×
5.84Kc = 0.63.5×
6.69Kc = 0.63.52×
7.66Kc = 0.63.54×
8.76Kc = 0.63.56×
10Kc = 0.63.58×
11.5Kc = 0.63.59×
13.2Kc = 0.63.61×
15.1Kc = 0.63.63×
17.2Kc = 0.63.65×
19.7Kc = 0.63.67×
22.6Kc = 0.63.69×
25.9Kc = 0.63.7×
29.6Kc = 0.63.72×
33.9Kc = 0.63.74×
38.8Kc = 0.63.76×
44.4Kc = 0.63.78×
50.9Kc = 0.63.79×
58.2Kc = 0.63.81×
66.6Kc = 0.63.83×
76.3Kc = 0.63.85×
87.3Kc = 0.63.86×
100Kc = 0.63.88×

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.

Tail at scale

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