Can someone show me how to solve this problem?

For a program that is 70% parallelizable, what will be the speedup relative to a uniprocessor from running it on a 4,8, and 16 way multiprocessor assuming perfect load balancing.

Thanks!

To solve this problem, we need to understand the concept of speedup and how it is calculated in the context of parallel computing.

Speedup measures the improvement in performance achieved by running a program on multiple processors compared to running it on a single processor. It is defined as the ratio of the execution time on a single processor to the execution time on multiple processors.

In this case, the program is assumed to be 70% parallelizable, which means that 70% of the program's workload can be executed in parallel. This implies that the remaining 30% of the workload cannot be parallelized and will still run sequentially on a single processor.

Assuming perfect load balancing, we can calculate the speedup using Amdahl's Law, which is given by the formula:

Speedup = 1 / [(1 - p) + (p / N)]

where p is the parallelizable portion of the workload (expressed as a decimal) and N is the number of processors.

Let's calculate the speedup for a 4-way, 8-way, and 16-way multiprocessor.

For a 4-way multiprocessor:
Parallelizable portion (p) = 0.7
Number of processors (N) = 4

Speedup = 1 / [(1 - 0.7) + (0.7 / 4)]
= 1 / [0.3 + 0.175]
= 1 / 0.475
= 2.105

Therefore, the speedup relative to a uniprocessor for a 4-way multiprocessor is approximately 2.105.

For an 8-way multiprocessor:
Parallelizable portion (p) = 0.7
Number of processors (N) = 8

Speedup = 1 / [(1 - 0.7) + (0.7 / 8)]
= 1 / [0.3 + 0.0875]
= 1 / 0.3875
= 2.58

Therefore, the speedup relative to a uniprocessor for an 8-way multiprocessor is approximately 2.58.

For a 16-way multiprocessor:
Parallelizable portion (p) = 0.7
Number of processors (N) = 16

Speedup = 1 / [(1 - 0.7) + (0.7 / 16)]
= 1 / [0.3 + 0.04375]
= 1 / 0.34375
= 2.91

Therefore, the speedup relative to a uniprocessor for a 16-way multiprocessor is approximately 2.91.

So, the speedup relative to a uniprocessor for a program that is 70% parallelizable running on a 4-way, 8-way, and 16-way multiprocessor (assuming perfect load balancing) is approximately 2.105, 2.58, and 2.91, respectively.