Chapter 1 A Tour of Computer Systems

1.9.1 Amdahl's Law

Consider a system in which executing some application
requires time T_old.Suppose some part of the system
requires a fraction x of this time,and that we improve
its performance by a factor of k,That is,the component
originally required time xT_old,and it now requires time
(xT_old)/k,The overall execution time would thus be:

T_new = (1-x)T_old + (xT_old)/k
= T_old[(1-x) + x/k]

From this,we can compute the speedup S = T_old / T_new:

S = 1 / [(1 - x) + x/k]
As an example, consider the case where a part of the
system that initially consumed 60% of the time(x=0.6) is
sped up by a factor of 3(k = 3). Then we get a speedup
of 1/[0.4 + 0.6/3] = 1.67x.Even though we made a subs-
tantial improvement to a major part of the system,our
net speedup was significantly less than the speed up the
entire system,we mustimprove the speed of a very large
fraction of the overall system.