Measure of Performance

The Speed-Up is the ratio of sequential time $T\left(1\right)$ and the parallel time on $P$ processors $T\left(P\right)$ :(1)

$$S\left(P\right)=T\left(1\right)/T\left(P\right)$$

The efficiency is defined as:(2)

$$E\left(P\right)=S\left(P\right)/P;E\left(P\right)\le 1$$

The Amdahl’s law for multitasking is used to determine the speed-up:(3)

$$S\left(P\right)=\frac{\left(T_{Seq}+T_{Par}\right)}{\left(T_{Seq}+\raisebox{1ex}{$T_{Par}$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)}$$

Where, $T_{Par}$ and $T_{Seq}$ are the computation times respectively related to parallel and non-parallel parts. As $T_{Seq}+T_{Par}=1$ , write:(4)

$$S\left(P\right)=\frac{1}{\left(T_{Seq}+\raisebox{1ex}{$T_{Par}$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)}$$

The limit value can be obtained when the process number tends to infinite:(5)

$$S\left(\infty \right)=\frac{1}{T_{Seq}}$$

Table 1 provides the Speed-Up in function of number of processors and the rate of parallelization in the program. It can be seen that if the rate of parallelization is less 95%, the computation acceleration will not be greater than 20; however, the number of processors. This means that to obtain a good scalability of a code, at least 99% of the program must be parallel.