Amdahl's Law

In computer architecture, Amdahl's law (or Amdahl's argument) gives the theoretical speedup in latency of the execution of a task at fixed workload that can be expected of a system whose resources are improved. It is named after computer scientist Gene Amdahl, and was presented at the AFIPS Spring Joint Computer Conference in 1967.

Amdahl's law can be formulated the following way:

where

Furthermore,

shows that the theoretical speedup of the execution of the whole task increases with the improvement of the resources of the system and that regardless of the magnitude of the improvement, the theoretical speedup is always limited by the part of the task that cannot benefit from the improvement.

Amdahl's law is often used in parallel computing to predict the theoretical speedup when using multiple processors. For example, if a program needs 20 hours using a single processor core, and a particular part of the program which takes one hour to execute cannot be parallelized, while the remaining 19 hours (p = 0.95) of execution time can be parallelized, then regardless of how many processors are devoted to a parallelized execution of this program, the minimum execution time cannot be less than that critical one hour. Hence, the theoretical speedup is limited to at most 20 times (1/(1 − p) = 20). For this reason parallel computing with many processors is useful only for very parallelizable programs.

A task executed by a system whose resources are improved compared to an initial similar system can be split up into two parts:

Example. — A computer program that processes files from disk. A part of that program may scan the directory of the disk and create a list of files internally in memory. After that, another part of the program passes each file to a separate thread for processing. The part that scans the directory and creates the file list cannot be sped up on a parallel computer, but the part that processes the files can.

The execution time of the whole task before the improvement of the resources of the system is denoted ${\displaystyle T}$. It includes the execution time of the part that would not benefit from the improvement of the resources and the execution time of the one that would benefit from it. The fraction of the execution time of the task that would benefit from the improvement of the resources is denoted by ${\displaystyle p}$. The one concerning the part that would not benefit from it is therefore ${\displaystyle 1-p}$ . Then

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