# Chapter 1

## Amdahl's Law

Before speedup
|-----------------|----------------------------| $T_{old}$
$(1-\alpha) T_{old}$ $\quad \alpha T_{old}$

After speedup
|-----------------|-------------------| $T_{new}$
$(1-\alpha) T_{old}$ $\quad (\alpha T_{old})/k$

\begin{aligned} T_{new} &= (1 - \alpha) T_{old} + (\alpha T_{old})/k \\ &= T_{old}[(1 - \alpha) + \alpha / k] \end{aligned}

So,we can compute the speedup $S = T_{old} / T_{new}$
$$S = \frac{1}{(1 - \alpha) + \alpha / k}$$
when
$$\begin{cases} \alpha = 0.6 \\ k = 3 \end{cases} \\$$
get
\begin{aligned} S &= \frac{1}{(1 - 0.6) + 0.6 / 3}\\ &=1.67 \times \end{aligned}
while
$$\begin{cases} \alpha = 0.6 \\ k \to +\infty \end{cases} \\$$
get
\begin{aligned} S_{\infty} &= \frac{1}{(1 - \alpha)} \\ &= 2.5 \times \end{aligned}
we routinely improve performance by factors of 2 or more. Such high factors can only be achieved by optimizing large parts of a system.

Last modification：April 5th, 2021 at 10:36 pm