Chapter 1, Problem 1.9.1E

Number of processors	Arithmetic instructions	L/S instructions	Branch instructions	Execution Time (Seconds)	Speedup
1	2.56 × 109	1.28 × 109	2.56 × 109	9.6	1
2	1.83 × 109	9.14 × 108	2.56 × 109	7.04	1.36
4	9.12 × 108	4.57 × 108	2.56 × 109	3.84	1.83
8	4.57 × 108	2.29 × 108	2.56 × 109	2.24	1.71

The arithmetic instructions are parallelized to run over multiple cores as follows

Arithmatic instructions = Number of instructions0.7 × Number of processors

Substitute, “2.56 × 109” for “number of instructions” and “2” for “number of processors” in the above formula

Arithmatic instructions(2) = 2.56 × 1090.7 × 2= 1.83 × 109

Thus, the number of arithmetic instructions with 2 processors is 1.83 × 109.

Substitute, “2.56 × 109” for “number of instructions” and “4” for “number of processors” in the above formula

Arithmatic instructions(4) = 2.56 × 1090.7 × 4= 9.12 × 108

Thus, the number of arithmetic instructions with 4 processors is 9.12 × 108.

Substitute, “2.56 × 109” for “number of instructions” and “8” for “number of processors” in the above formula

Arithmatic instructions(8) = 2.56 × 1090.7 × 8= 4.57 × 108

Thus, the number of arithmetic instructions with 8 processors is 4.57 × 108.

The L/S instructions are parallelized to run over multiple cores as follows

L/S instructions = Number of instructions0.7 × Number of processors

Substitute, “1.28 × 109” for “number of instructions” and “2” for “number of processors” in the above formula

L/S instructions(2) = 1.28 × 1090.7 × 2= 9.14 × 108

Thus, the number of L/S instructions with 2 processors is 9.14 × 108.

Substitute, “1.28 × 109” for “number of instructions” and “4” for “number of processors” in the above formula

L/S instructions(4) = 1.28 × 1090.7 × 4= 4.57 × 108

Thus, the number of L/S instructions with 4 processors is 4.57 × 108.

Substitute, “1.28 × 109” for “number of instructions” and “8” for “number of processors” in the above formula

L/S instructions(8) = 1.28 × 1090.7 × 8= 2.29 × 108

Thus, the number of L/S instructions with 8 processors is 2.29 × 108.

The total execution time of the program can be calculated using the following formula

Execution time = CPI × Number of instructionsClock rate

Substitute, “1” for “CPI of arithmetic instructions”, “12” for “CPI of L/S instructions”, “5” for “CPI of branch instructions”,, “2.56 × 109” for “number of arithmetic instructions”, “1.28 × 109” for “number of L/S instructions” “0.25 × 109”for “number of branch instructions” and “2 × 109” for “clock rate”

Execution time(1) = (1× 2.56 × 109) + (12× 1.28 × 109) + (5× 0.25 × 109)2 × 109= 9.6

Thus, the total execution time of the program with 1 processor is “9.6seconds”.

Substitute, “1” for “CPI of arithmetic instructions”, “12” for “CPI of L/S instructions”, “5” for “CPI of branch instructions”,, “2.56 × 1090.7 × 2” for “number of arithmetic instructions”, “1.28 × 1090.7 × 2” for “number of L/S instructions” “0.25 × 109”for “number of branch instructions”, “and “2 × 109” for “clock rate”

Execution time(2) = (1× 2.56 × 1090.7 × 2) + (12× 1.28 × 1090.7 × 2) + (0.25 × 109)2 × 109= 7.04

Thus, the total execution time of the program with 2 processors is “7.04seconds”.

Substitute, “1” for “CPI of arithmetic instructions”, “12” for “CPI of L/S instructions”, “5” for “CPI of branch instructions”,, “2.56 × 1090.7 × 4” for “number of arithmetic instructions”, “1.28 × 1090.7 × 4” for “number of L/S instructions” “0.25 × 109”for “number of branch instructions”, “and “2 × 109” for “clock rate”

Execution time(4) = (1× 2.56 × 1090.7 × 4) + (12× 1.28 × 1090.7 × 4) + (0.25 × 109)2 × 109= 3.84

Thus, the total execution time of the program with 4 processors is “3.84seconds”.

Substitute, “1” for “CPI of arithmetic instructions”, “12” for “CPI of L/S instructions”, “5” for “CPI of branch instructions”,, “2.56 × 1090.7 × 8” for “number of arithmetic instructions”, “1.28 × 1090.7 × 8” for “number of L/S instructions” “0.25 × 109”for “number of branch instructions”, “and “2 × 109” for “clock rate”

Execution time(8) = (1× 2.56 × 1090.7 × 8) + (12× 1.28 × 1090.7 × 8) + (0.25 × 109)2 × 109= 2.24

Thus, the total execution time of the program with 8 processors is “2.24seconds”.

The relative speedup of the processor can be calculated using the following formula

Speedup = Execution time(required)Execution time(current)

Substitute, “9.6” for “execution time of required processor” and “7.04” for “execution time of current processor”

Speedup(2-1) = 9.67.04= 1.36

Therefore, the relative speedup of 2 processors with a single processor is “1.36”.

Substitute, “9.6” for “execution time of required processor” and “3.84” for “execution time of current processor”

Speedup(4-1) = 9.63.84= 2.5

Therefore, the relative speedup of 4 processors with a single processor is “2.5”.

Substitute, “9.6” for “execution time of required processor” and “2.24” for “execution time of current processor”

Speedup(8-1) = 9.62.24= 4.28

Therefore, the relative speedup of 8 processors with a single processor is “4.28”.