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Original heme by orderedlist (CC-BY-SA)


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Amdahl's Law

$$ T_{p,n} = T_s (1-p) + \frac{T_s p}{n} \\ \to \frac{T_{p,n}}{T_s} = 1 - p + \frac{p}{n} \\ \to \frac{T_s}{T_{p,n}} = \frac{1}{ 1 - p + \frac{p}{n} } \\ \to F_{p,n} = \frac{1}{ 1 - p + \frac{p}{n} } $$

For a completely serial task ($p=0$):

$$ F_{p,n} = \frac{1}{ 1 - 0 + \frac{0}{n} } = 1 $$

For a completely parallel task ($p=1$):

$$ F_{p,n} = \frac{1}{ 1 - 1 + \frac{1}{n} } = \frac{1}{ \frac{1}{n} } = n $$

When $n >> 1$:

$$ F_{p,n} = \frac{1}{ 1 - p + \frac{p}{n} } = \frac{n}{ n - n p + p } \\ \approx \frac{n}{n -n p} = \frac{1}{1-p} $$

2018-09-03