Chapter 11, Problem 2E

Explanation of Solution

Verification of the ratio being consistent when compared with the other system:

• System performance is considered as one of main factor of a processor.
• It is used to determine the speed a problem can be solved.
• It is also used to determine the factors such as number of problems that can be allocated at particular amount of time and also the number of problems that can be handled by the processor.
• The relative performance between the two systems is measured and expected by the run time of the program of the individual.

Given:

The information about the system A and System c are shown in the below table:

Program	Execution time System A(sec)	Execution time System B(sec)	Execution time System C(sec)
V	50	100	500
W	200	400	600
X	250	500	500
Y	400	800	800
Z	5000	4100	3500

Consider there are n programs and each programs are considered to have their own runtime on each systems.

The geometric mean of the one system’s runtime is obtained by normalizing it with the another system.

The process of normalization is carried out by taking the products of the ratio of the run time and by taking the nth root of the product.

The below tables illustrates the how the system B and system C is being normalized with that of the system A.

Program	Execution time System A(sec)	Normalized to A	Execution time System B(sec)	Normalized to B	Execution time System C(sec)	Normalized to C
V	50	1	100	0.5	500	0.1
W	200	1	400	0.5	600	0.33
X	250	1	500	0.5	500	0.5
Y	400	1	800	0.6	800	0.5
Z	5000	1	4100	1.22	3500	1.43

Calculating geometic Mean:

The formula to calculate the geometric mean:

$\text{G=}{\left(\text{x1×x2×x3×}\mathrm{...}\text{×xn}\right)}^{\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$\text{n}$}\right.}$

System A:

calculate the geometric mean for the system A:

$\begin{array}{c}\text{G=}{\left(\text{1×1×1×1×1}\right)}^{\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$\text{5}$}\right.}\\ =1\end{array}$

System B:

calculate the geometric mean for the system B:

$\begin{array}{c}\text{G=}{(\text{0}}^{}\end{array}$