24. Amdahl or Gustafson? Consider the following example. A physical process involving a flat area 100 units by 100 units is being simulated. Processing of the units can take place in parallel. However, there is a border region 10 units wide that goes round the square where parallel processing cannot take place because of relationships with the external world. In other words, parallel processing can take place in the inner square but not in the 10? unit wide strip/border round it.

a. If parallel processing is used, what speedup is possible?
b. Suppose that the system is scaled to 200 by 200 units. The border region remains at 10 units. What
speedup can parallel processing now achieve?

a. If the border is 10 units wide all the way round, the center measures 80 × 80 = 6,400 units. The border is
100 × 10 + 100 × 10 + 80 × 10 + 80 × 10 = 3,600. The serial to parallel ratio (border to center) is 3,600 to
6,400. The time taken is (3,600 + 6,400/n)T, where T is the time to process one unit and n is the number of
parallel processors. In the limit where n reduces the parallel part of the problem to zero, the irreducible
time is 3,600T. The highest speedup is 10,000T/3,600T = 2.78.

b. Suppose the system is scaled to twice the linear size. The parallelizable square is 180 × 180 units (32,400
units). The border is now 200 × 10 + 200 × 10 + 180 × 10 + 180 × 10 = 7,600. In this case the time taken is
7,600T + (32,400/n)T. If n is chosen to make the parallel term negligible, the irreducible time is 7,600T and
the speedup 40,000/7,600 = 5.26.
This demonstrates that the scaling the problem can achieve a greater speedup. In both cases Amdahl’s law
remains, but the fraction of the serial and parallel portions have changed with the scaling.