# Amdhal’s Law

Gene Amdahl, one of the early pioneers in computing, made a simple but insightful observation about the effectiveness of improving the performance of one part of a system. This observation has come to be known as Amdahl’s law.

The main idea is that when we speed up one part of a system, the effect on the overall system performance depends on both how significant this part was and how much it sped up. Consider a system in which executing some application requires time T_o (T_old). Suppose some part of the system requires a fraction α of this time, and that we improve its performance by a factor of k. That is, the component originally required time α T_o, and it now requires time (αT_o)/k. The overall execution time would thus be:

$$T_n = (1− \alpha)T_o + (αT_)/k$$ $$= T_o[(1− \alpha) + \alpha / k]$$

From this, we can compute the speedup $$S = T_o/T_n$$ as $$S=\frac{1}{(1− α) + α/k}$$

As an example, consider the case where a part of the system that initially consumed 60% of the time (α = 0.6) is sped up by a factor of 3 (k = 3). Then we get a speedup of 1/[0.4 + 0.6/3]= 1.67×. Even though we made a substantial improvement to a major part of the system, our net speedup was significantly less than the speedup for the one part. This is the major insight of Amdahl’s law — to significantly speed up the entire system, we must improve the speed of a very large fraction of the overall system.