Amdahl's Law Calculator

Amdahl's Law describes how much a task can be maximally accelerated if only part of the task is improved. By entering the proportion of the program that can run in parallel and the number of processors, the calculator gives the overall acceleration ratio, the efficiency of the processors as well as a hard upper limit that cannot be exceeded. It also supports reverse calculation: If either the proportion of parallel parts or the number of processors are left out, the corresponding values can be calculated from the already measured acceleration ratio.

What is Amdahl's Law?

The Amdahl's Law is named after computer architect Gene Amdahl who stated it in 1967. It reveals a simple but persistent fact: a program consists of a part that can be distributed to multiple processors, and another part that must be executed sequentially, step by step. Adding more processors speeds up the parallel part, but has no effect on the serial part.

Thus the serial part becomes a bottleneck. As more processors are added, the time required for parallel work approaches zero while the time required for sequential work remains constant, causing the overall speedup ratio to eventually taper off. This is an example of the law of diminishing returns expressed in a formula.

Amdahl's Law formula:

For a parallel portion P that is divided among N processors, the overall speedup ratio is:

where P is the fraction of the program that can be run in parallel during execution time and is represented by a decimal number between 0 and 1; (1 - P) is the non-parallelizable, sequential part. N is the number of processors, cores or threads involved in running the parallel portion. If we consider N as the speedup factor for the improved portion, then this formula applies to any kind of improvement, not just parallel hardware.

As n approaches infinity, the term p/n disappears, resulting in a maximum acceleration factor of:

An example calculation:

Suppose a certain task can be performed 95 percent in parallel and you have 8 processors. Then p is equal to 0.95, (1-p) is equal to 0.05, and n is equal to 8:

Eight processors give a speedup of 5.93 times, not 8 times. The parallel efficiency is the speedup factor divided by the number of processors, so 5.93 divided by 8, which is about 74 percent. Regardless of how many more processors are added, the limit to the fraction of code that can be run in parallel is 95 percent, which is 1 divided by 0.05, or 20 times. Finally, the serial execution of 5 percent of the code limits overall performance.

Efficiency and effective core count.

The speedup ratio indicates how much faster a system is while the efficiency indicates how well the hardware is being used. The efficiency is equal to the speedup ratio divided by the number of processors:

With 8 processors and a speedup of 5.93, the efficiency is about 74 percent. This means that approximately one quarter of the total processing power of serial code is consumed. You can also think of the ratio of the speedup as being the number of effective cores. A speedup of 5.93 means that your 8 physical cores are performing at a level equivalent to about 5.93 fully loaded cores. This way of thinking makes it easier to compare chips with different numbers of cores and clock speeds.

How to determine the proportion of the parallel part:

In general P is not known beforehand. The practical approach is to do measurements: first measure the execution time on a single core, then on multiple cores, and then calculate the actual speedup ratio in both cases. In this calculator you leave the parallel fraction blank, enter the measured speedup ratio and number of processors used, and the calculator will compute P. This back-calculated value is the serial fraction in Karp-Flatt style that you can use in calculations to estimate performance on other hardware.

Comparison of Amdahl's Law and Gustafson's Law:

Amdahl's Law assumes that the problem size is constant. Therefore, adding processors can only speed up a fraction of the work, and the ratio of acceleration eventually reaches a saturation point. Gustafson's Law takes an opposite approach. In practice, more computing power often solves a larger problem rather than running the same task faster. For such scalable workloads, the ratio of acceleration grows approximately linearly with the number of processors:

Both laws are correct. Amdahl's law is the right perspective for fixed tasks where faster execution is desired; Gustafson's law is the right perspective for workloads that expand to fill purchased hardware. The Gustafson switch in this calculator shows both side by side.

Important restrictions:

The Amdahl's Law gives an idealized upper bound and should therefore be considered as a ceiling rather than a promise. It assumes that the parallel part is perfectly scalable and there are no coordination costs but real parallel code has to deal with communication, synchronization and memory conflicts which cause the result to fall below the curve.

It is also assumed that the processor is the bottleneck; if memory, disk or network are the limiting factors then adding more cores will not provide significant benefits until the time predicted by the algorithm is reached. Also many programs have a fixed number of threads hard-coded and different tasks within the same program often have different scaling factors so a single value for P may only be an average.

This calculator is for planning and teaching purposes only. The actual acceleration ratio will depend on your code, hardware, and specific workload. So be sure to run measurements on your own system before making purchasing or design decisions.