Question

A program is divided into 3 blocks that are being compiled on 3 parallel computers. Each block takes an Exponential amount of time, 5 minutes on the average, independently of other blocks. The program is completed when all the blocks are compiled. Compute the expected time it takes the program to be compiled.

Answer 1

Answer:

The answer is "15 minutes".

Explanation:

The time of compilation for one block with all three computers would be denoted by the $Y_1,Y_2 and Y_3$.

Of all them.

$Y_i \sim Exp(\frac{1}{5}) \ for \ j = \ 1,\ 2,\ and\ 3$ and the average:

$=\frac{1}{\lambda } \\ \\=5$

Let,

$Z=Y_1+Y_2+Y_3, \\\\ then \ Z \sim \ Gamma ( \alpha =3, \ \ \lambda =\frac{1}{5})$

since independent of the total sum and the gamma distribution of parameters, $\alpha = \ n \ \ and \ \lambda$ are accompanied by exponential variables.  

Now the predicted value of Z can be found.

$\bold{Formula:} \\\\ \to E(Z)=\frac{\alpha}{\lambda }$

$=\frac{3}{\frac{1}{5}}\\\\= 3 \times 5\\\\= 15$

Therefore, it takes 15 minutes for the program to be compiled.