Parallel Computation with Serial Sections Model

• In this computational model, it is assumed that a fraction $f$ of the given task (computation) is not dividable into concurrent subtasks. The remaining part $(1-f)$ is assumed to be dividable into concurrent subtasks.
• The time required to execute the task on $n$ processors is $t_p=t_s*f+(1-f)*(t_s/n)$. The speedup factor is therefore given by
  

  $$S(n)=\frac{t_s}{t_s*f+(1-f)*(t_s/n)}=\frac{n}{1+(n-1)*f}$$ (4)

  
  

• According to this equation, the potential speedup due to the use of $n$ processors is determined primarily by the fraction of code that cannot be divided.
• If the task (program) is completely serial, that is, $f=1$, then no speedup can be achieved regardless of the number of processors used.
• This principle is known as Amdahl's law. It is interesting to note that according to this law, the maximum speedup factor is given by
  
  $$lim_{n\rightarrow \infty} S(n)=\frac{1}{f}$$
  
  

• Therefore, according to Amdahl's law the improvement in performance (speed) of a parallel algorithm over a sequential one is limited not by the number of processors employed but rather by the fraction of the algorithm that cannot be parallelized.
• According to Amdahl's law, researchers were led to believe that a substantial increase in speedup factor would not be possible by using parallel architectures.
• The effect of the communication overhead on the speedup factor, given that a fraction, $f$, of the computation is not parallelizable.
  

  $$S(n)=\frac{t_s}{t_s*f+(1-f)*(t_s/n)+t_c}=\frac{n}{(n-1)*f+1+n*(t_c/t_s)}$$ (5)

  
  
  The maximum speedup factor under such conditions is given by
  
  $$lim_{n\rightarrow \infty} S(n)=lim_{n\rightarrow \infty} \frac{n}{(n-1)*f+1+n*(t_c/t_s)}=\frac{1}{f+(t_c/t_s)}$$
  
  

• The above formula indicates that the maximum speedup factor is determined not by the number of parallel processors employed but by the fraction of the computation that is not parallelized and the communication overhead.
• Recall that the efficiency is defined as the ratio between the speedup factor and the number of processors, $n$. The efficiency can be computed as:
  

  $$\begin{array}{l} E(no~communication~overhead)=\frac{1}{1+(n-... ...nication~overhead)=\frac{1}{(n-1)*f+1+n*(t_c/t_s)} \end{array}$$ (6)

  
  

• As the number of processors increases, it may become difficult to use those processors efficiently. In order to maintain a certain level of processor efficiency, there should exist a relationship between the fraction of serial computation, f, and the number of processor employed.