Question

When a program is adapted to run on multiple processors in a multiprocessor system, the execution time on each processor is comprised of computing time and the overhead time required for locked critical sections and/or to send data from one processor to another.

Assume a program requires t = 100 s of execution time on one processor. When run p processors, each processor requires t/p s, as well as an additional 4 s of overhead, irrespective of the number of processors. Compute the per-processor execution time for 2, 4, 8, 16, 32, 64, and 128 processors. For each case, list the corresponding speedup relative to a single processor and the ratio between actual speedup versus ideal speedup (speedup if there was no overhead).

Short answer

The answer is listed below for each case:

Processor number	Execution Time	Speed-up	Speed-up without the overhead	Ratio
2	54	1.85	2	0.925
4	29	3.45	4	0.8625
8	16.5	6.06	8	0.7575
16	10.25	9.76	16	0.61
32	7.125	14.04	32	0.43875
64	5.5425	18.05	64	0.281875
128	4.78125	20.92	128	0.1634375

Step-by-step solution

Define the concept.

$$\begin{gathered} \mathrm{The}\mathrm{execution}\mathrm{time}\mathrm{of}\mathrm{the}\mathrm{program}=\left(\frac{100}{\mathrm{Number}\mathrm{of}\mathrm{the}\mathrm{processor}}+4\right) \\ \mathrm{The}\mathrm{corresponding}\mathrm{speedup}\mathrm{relative}\mathrm{to}\mathrm{single}\mathrm{processor}=\frac{100}{\mathrm{Executive}\mathrm{time}\mathrm{of}\mathrm{the}\mathrm{program}} \end{gathered}$$

$$\begin{gathered} \mathrm{The}{\mathrm{ratio}}_{\mathrm{Atcual}\mathrm{speedup}\mathrm{versus}\mathrm{ideal}\mathrm{speedup}}=\frac{\mathrm{Speedup}}{\mathrm{Number}\mathrm{of}\mathrm{the}\mathrm{processor}} \\ \mathrm{The}\mathrm{speedup}\mathrm{without}\mathrm{overhead}=\frac{100}{{\displaystyle \frac{100}{\mathrm{Number}\mathrm{of}\mathrm{the}\mathrm{processor}}}} \end{gathered}$$

Determine the calculation.

Given that, the value of “t” is 100s.

And the execution time is calculated for the processors of 2, 4, 8, 16, 32, 64, and 128.

Processor number	Execution Time	Speed-up	Speed-up without overhead	Ratio
2	$\left(\frac{100}{2}+4\right)=\left(50+4\right)=54$	$\left(\frac{100}{54}\right)=1.85$	$\left(\frac{100}{50}\right)=2$	$\left(\frac{1.85}{2}\right)=0.925$
4	$\left(\frac{100}{4}+4\right)=25+4=29$	$\left(\frac{100}{29}\right)=3.45$	$\left(\frac{100}{24}\right)=4$	$\left(\frac{3.45}{4}\right)=0.8625$
8	$\left(\frac{100}{8}+4\right)=\left(12.5+4\right)=16.5$	$\left(\frac{100}{16.5}\right)=6.06$	$\left(\frac{100}{12.5}\right)=8$	$\left(\frac{6.06}{8}\right)=0.7575$
16	$\left(\frac{100}{16}+4\right)=\left(6.25+4\right)=10.25$	$\left(\frac{100}{10.25}\right)=9.76$	$\left(\frac{100}{6.25}\right)=16$	$\left(\frac{9.76}{16}\right)=0.61$
32	$\left(\frac{100}{32}+4\right)=\left(3.125+4\right)=7.125$	$\left(\frac{100}{7.125}\right)=14.04$	$\left(\frac{100}{3.125}\right)=32$	$\left(\frac{14.04}{32}\right)=0.43875$
64	$\left(\frac{100}{64}+4\right)=\left(1.5425+4\right)=5.5425$	$\left(\frac{100}{5.5425}\right)=18.05$	$\left(\frac{100}{1.5625}\right)=64$	role="math" localid="1655127771056" $\left(\frac{18.05}{64}\right)=0.281875$
128	$\left(\frac{100}{125}+4\right)=4.78125$	$\left(\frac{100}{4.78125}\right)=20.92$	128	=0.1634375