2 Attachment(s)

M/M/1 queue (totally desperate)

Hi,

We have a homework to give by Wednesday, and many colleagues and I are trying to answer a question which consists of a proof...

$\displaystyle L =\sum\limits_{n=0}^{s-1} n P_{n}+L_{q}+ s \left ( 1 - \sum\limits_{n=0}^{s-1} P_{n} \right )$

L is the expected value of the number of users in the system (in the queue + being served)

$\displaystyle L_{q}$ is the expected value of the number of users in the queue (excluding those being server)

(the rest of the equation) is the expected value of the number os users being server (excluding those waiting in the queue)

So we have to proof that (the rest of the equation) is equal to $\displaystyle \frac{\lambda}{\mu}$ where $\displaystyle \lambda$ is the arrival rate (users / hour) and $\displaystyle \mu$ the service rate (users / hour) (it is supposed that $\displaystyle \frac{\lambda}{\mu} < 1$).

But how the hell am I supposed to get there ? We supposedly have a page with everything we need, but I tried everything thing I could, and nothing worked so far !

You may want to have a look at the attached files even though they're in French... you will at least understand the equations (I hope)...

20121216 01 Devoir 4 (version brouillon).pdf is what I tried (starts at page 2)

prob.pdf is the page that is supposed to help us which doesn't...