Ensait - Waiting lines

What could the be the next coming examination
duration 2h00

All documents are allowed

Some Guidelines

1. The first quality that is expected from an engineer is the ability to submit clearly her/his findings. All listings, computations, charts and other "printer outputs" can in no way replace a statement of conclusions, coined in an accurate and scientific language.
2. Doing cut and paste with scissors and glue is probably the fastest way to incorporate the right part of all your printer outputs at the right place of your work.
3. Check the printers at the beginning of the evaluation, and print each piece as soon as possible. At exactly the specified time, printers will be disconnected.
4. On any printed document, the FAMILY_NAME/Given_name of the student must appear (especially in the title of the figures).

Use the program waits.sce. Be sure to use the most recent version (26). It can be found as usual and a certified copy is http://campus2.univ-lille2.fr/claroline/document/goto/index.php?url=/waits.sce&cidReq=N_ISC_WAITS. When launched, it prints
-waits26.sce----------------+your_working_directory
Obviously, you have to give the right value of this directory (line 10). To launch the menu chooser, you have to key "menu" in the Scilab Console.

1 "petit essai"

Change what is to be changed in lines 136..158 in order that the result of the small test (item "petit essai" in the menu) begins by : ======= arv in 13..19, srv in 14..26 == 
======= starting with seed =ddmmyyyy
where dd/mm/yyyy is your birthday.

1. Print the result of this program
2. Draw the corresponding chronogram (and explain how you are proceeding)

2 Some computations

1. Let $X$ be an uniform variable, $X\in\left[5..15\right]$. Its pdf is $f\left(x\right)=1/10$ if $5\leq x\leq15$ and $0$ otherwise. Compute (and explain) $\int_{5}^{15}f\left(x\right)\mathrm{d}x$, $\int_{5}^{15}x f\left(x\right)\mathrm{d}x$, $\int_{5}^{15}x^{2} f\left(x\right)\mathrm{d}x$, $E\left(x\right)$, $var\left(x\right)$.
2. Let $X$ be an exponential variable, with parameter $\lambda$. Its pdf is $f\left(x\right)=\lambda \exp\left(-\lambda x\right)$. Compute (and explain) $\int_{0}^{\infty}f\left(x\right)\mathrm{d}x$, $\int_{0}^{\infty}x f\left(x\right)\mathrm{d}x$, $\int_{0}^{\infty}x^{2} f\left(x\right)\mathrm{d}x$, $E\left(x\right)$, $var\left(x\right)$.

3 M/Ga/1 waiting line

Use 'système M/Ga/5' with parameters $policy=1(rand), nb\_eve=5000, nb\_bat=40$. The inter-arrivals $X$ are i.i.distributed according to an exponential law, and $E\left(X\right)=marv$, var $\left(X\right)=varv$. The service times $Y$ are i.i.distributed according to a Gamma distribution such that $E\left(Y\right)=msrv$, var $\left(Y\right)=vsrv$.

1. What are the values of parameters $a, b$ of the Gamma distribution ? What are $\lambda, \mu, \rho$ and what are their values ?
2. Print and comment graph 10.
3. Print graph 9. Read the frequencies on the histogram. Compute the mean value of the quantity plotted. What can be said about this value ?
4. Print graph 5. Explain what is a remaining service. What is the number displayed on the title bar ? What is its meaning ? What can be said about the obtained experimental value ?
5. Explain why graphs 1 and 2 aren't very different.
6. At the end of the execution, seven blocks of two lines are send to the console by the program. Print these lines (don't scramble these lines with a proportional font !). Comment them. Use 'système M/Ga/5' with parameters $policy=3(size),\, nb\_eve=5000,\, nb\_bat=40$. The inter-arrivals $X$ are i.i.distributed according to an exponential law, and $E\left(X\right)=marv$, var $\left(X\right)=varv$.
7. Here again, the execution ends by seven blocks of two lines that are send to the console. Print these lines. What has changed, what hasn't changed ? Explain.
8. Print and comment graph 6.