# Amdahl's Law: Definition, Formula & Examples

Lesson Transcript

Shadi has a Ph.D. in Computer Science and more than 20 years experience in industry and higher education.

If you want to improve a system which is composed of different components, how would you decide which component to choose to get the best overall improvement? Amdahl's law gives you the means to make this decision.

## What Is Amdahl's Law?

Imagine a situation where three friends, Sam, Jack, and Harry, are going to a party. There are two conditions to be able to get into the party hall; first, they need to go separately, and second, all of them need to be present at the door to be able to get in. Now, assume that Sam is coming in his car, Jack is using a motorcycle, and Harry is coming by foot. In this case, Harry is considered a bottleneck in the performance of the whole system. In other words, no matter how fast Sam and Jack can reach the party hall, all need to wait for Harry to arrive to be able to attend the party. This means that in order to accelerate the overall process, you need to concentrate on improving the performance of Harry much more than the other two.

This is precisely what Amdahl's law is about. Amdahl's law relates the performance improvement of a system with the parts that didn't perform well. This considerably affects the design of computing systems. While manufacturers make enormous efforts on improving the performance of processors, input/output devices like memory and storage devices are still too slow compared to the processors. This means that the overall speed improvement is limited by the low speed of input/output devices. In other words, at a certain point, manufacturers will need to pay more attention in improving IO speed instead of processor's speed.

### Calculating Performance Improvement Using Amdahl's Law

Mathematically, Amdahl's law is formulated as follows:

Where

Smax is the maximum possible improvement of the system

p is the part that can be improved. In other words, (1 - p) is the part of the system that cannot be improved.

s is the performance improvement factor of p after applying the enhancements.

Let us now look at how the maximum improvement can be calculated using Amdahl's law. For a given system, if the part that can be improved is 25% of the overall system and its performance can be doubled, then:

Let us now suppose that for a different system, the part that can be improved is 75% of the overall system and its performance can be doubled, then:

Comparing the two values, we can see that the more important the part that cannot be improved, the less the benefit of the improvement.

Note, that if the whole system can benefit from the improvement, Smax will have a value of 2, which is the maximum possible performance improvement.

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