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
$$
\begin{equation}
\begin{cases}
\alpha = 0.6 \\
k = 3
\end{cases}
\\
\end{equation}
$$
get
$$
\begin{aligned}
S &= \frac{1}{(1 - 0.6) + 0.6 / 3}\\
&=1.67 \times
\end{aligned}
$$
while
$$
\begin{equation}
\begin{cases}
\alpha = 0.6 \\
k \to +\infty
\end{cases}
\\
\end{equation}
$$
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.