Amdahl’s Law

MATH133: Unit 3 Individual Project 2B In 1967, Dr. Gene Amdahl developed Amdahl’s Law for predicting speed-up gained using multiple processors in parallel while executing a computer program. One version of Amdahl’s Law states that the speed-up (or efficiency) of using multiple parallel processors can be calculated approximately using the following rational function:

𝑆(𝑥) = 𝑥 𝑝𝑝 + (1 − 𝑝)

Here, x is the number of processors used and p is the decimal equivalent of the percent of the program that must be run sequentially.

For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places.

1. Based on the first letter of your last name, in the table below choose a percent of a program that must be run sequentially. If your choice is not a whole number percent, then use at most two decimal places. First letter of your last name. Possible values for p. A–F 5.00%–9.99% G–L 10.00%–14.99% M–R 15.00%–19.99% S–Z 20.00%–24.99%

2. Using your chosen value for p in part 1, write your version of this speed-up rational function, S(x).

3. Choose 6 x-values between 1 and 200, one of which must be 10. Calculate their corresponding values of S(x), and then complete the following table. (See part 6 for the reason why one x-value has to be 10.) To receive full credit, be sure to show all of your calculation details. Chosen x-values Calculated S(x)-values 10 Page 2 of 2

4. Graph your speed-up function using Excel or another graphing utility. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into the supplied Word Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values in part 3 above.

5. Given a computer program with your chosen value of p, the graph shows the relation between the number of parallel processors used and their efficiencies. Looking at your graph, what appears to be the maximum possible speed-up?

6. Suppose the amount of the program that must be completed sequentially could be reduced by 2% (e.g., your value of p% minus 2%). What would be the new speedup for x= 10 parallel processors?

7. Compare your speed-up in part 6 to the speed-up for x = 10 in part 3 above. What conclusions can you draw from this comparison? Explain how you arrived at this conclusion.

References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx