Q. Amdahl Law to measure speed up performance?

Remember that speed up factor assists us in knowing relative gain attained in shifting execution of a task from sequential computer to parallel computer and performance doesn't increase linearly with rise in number of processors. Because of above reason of saturation in 1967 Amdahl's law was derived. Amdahl's law defines that a program includes 2 types of operations that is complete sequential operations that should be done serially and complete parallel operations that can be executed on numerous processors. The declaration of Amdahl's law can be illustrated with help of following example. 

Let's discuss a problem say P that has to be solved with the help of parallel computer. In accordance with Amdahl's law there are principally two types of operations. So the problem would have a number of serial operations and a number of parallel operations. We already know that this requires T (1) amount of time to carry out a problem using a sequential machine and sequential algorithm. The time to calculate sequential operation is a fraction α (alpha) (α ≤ 1) of total execution time that is T (1) and time to calculate parallel operations is (1- α). So S (N) can be calculated as below:

Dividing by T(1)

Consider the value of α is amid 0 and 1. Now let's put a number of values of α and calculate speed up factor for rising values of number of processors.  We find that S(N) continue reduces with increase in value of α (which is number of serial operations as displayed in Figure below). 

Figure: Speed-up vs. Number of Processors

The graph in Figure clearly demonstrates that there is a bottleneck caused because of serial operations in parallel computer. Even when numbers of serial operations are more after rising number of processors speed up factor S (N) degrades. 

The sequential fraction which is α can be compared with speed up factor S (N) for a permanent value of N let's say 500. Figure below demonstrates a pictorial view of effect of Amdahl's law on speed up factor.

Figure: (n) vs. α (Graph is not to scale)