Chapter 7, Problem 9E

a.

Program Plan Intro

Given:

The fraction of work performed by CPU (f) = $55\%=\frac{55}{100}=0.55$

The fraction of work performed by DISK (f) = $45\%=\frac{45}{100}=0.45$

Speedup of CPU (k) = 1.4

Speedup of the DISK (k) = 2.5

To Find: Overall speedup of the CPU and DISK

Solution:

According to Amdahl’s law; $\text{Overall speedup }\left(\text{S}\right)\text{=}\frac{\text{1}}{\text{(1-f)+(}\frac{\text{f}}{\text{k}}\text{)}}$ --- (1)

Substitute the value for “f” and “k” in the above equation. So, the equation (1) becomes,

Finding overall speedup for CPU:

$\begin{array}{l}{\text{S}}_{\text{CPU}}\text{=}\frac{\text{1}}{\text{(1-0}\text{.55)+(}\frac{\text{0}\text{.55}}{\text{1}\text{.4}}\text{)}}\\ {\text{S}}_{\text{CPU}}\text{=}\frac{\text{1}}{\text{0}\text{.45+0}\text{.39}}\\ {\text{S}}_{\text{CPU}}\text{=}\frac{\text{1}}{0.84}\end{array}$

$\boxed{{\text{S}}_{\text{CPU}}\text{=1}\text{.19}}$

Convert 1.19 to percentage:

=>$1.19\times 100\%=119\%$. Subtract 100% from 119% to get the overall speedup percentage of the new system.

Overall speedup percentage of CPU = 119% - 100% = 19%

Finding overall speedup for DISK:

$\begin{array}{l}{\text{S}}_{\text{DISK}}\text{=}\frac{\text{1}}{\text{(1-0}\text{.45)+(}\frac{\text{0}\text{.45}}{\text{2}\text{.5}}\text{)}}\\ {\text{S}}_{\text{DISK}}\text{=}\frac{\text{1}}{\text{0}\text{.55+0}\text{.18}}\\ {\text{S}}_{\text{DISK}}\text{=}\frac{\text{1}}{0.73}\end{array}$

$\boxed{{\text{S}}_{\text{DISK}}\text{=1}\text{.37}}$

Convert 1.37 to percentage:

=>$1.37\times 100\%=137\%$. Subtract 100% from 137% to get the overall speedup percentage of the new system.

Overall speedup percentage of DISK = 137% - 100% = 37%

b.

Explanation of Solution

If one does not care about money, then, one would go for the upgrade of disk activity because:

From the above part (a), it is clear that the speed of the DISK activity ${\text{S}}_{\text{DISK}}\text{=37\%}$ is more when compared with

c.

Explanation of Solution

Breakeven points for the upgrade:

From the part (a):

Overall speedup for CPU is: 19%

Money needed to upgrade CPU is: $263.15

Overall speedup for DISK is: 37%

Money needed to upgrade DISK is: $216.22

The breakeven points for the upgrades are:

$\begin{array}{l}\text{Breakeven point = Money to upgrade CPU × speedup of DISK}\\ \text{Breakeven point = }\$263.15\text{ × 37}\%\\ \boxed{\text{Breakeven point = 9736}}\end{array}$