[CST-2] CSM 98.9.3?

[Nathan Dimmock]

On Sun, May 27, 2001 , garan wrote:
> Anyone got a agood answer I can look at to the proof part and the rest, I
> am rubbish.

I'm not sure if this is a "good" answer, but here's a rough stab at it:

(I assume that the question means "prove that the departure process is
exponentially distributed" not "prove that the departure process is the same
exponential distribution as the arrival process" as I don't think that is
true).

There are two cases:
1) There are <m jobs in the system. In this case, a job arriving does not
have to queue, and is serviced immediately. Obviously the departure
distribution will be the same as the service time distribution (i.e.
exponential).
2) There are >=m jobs in the system. A job entering the system joins the
queue. Its queueing time is the sum of the service times of all the jobs in
front of it in the queue. These services times are exponentially distributed
with mean (1/u). Now, by some A Level maths I've forgotten, the sum of
N exponential distributions is another exponential distribution, so the
queuing time is exponentially distributed, as is the service time of our
newly arrived job. Therefore, the total residence time of the job will be an
exponential distribution => the departure rate will be exponentially
distributed.

With all this summing of distributions (anyone know for sure if you can do
this?) it's impossible for it to have the same parameters as the arrival
distribution in all cases, hence my interpretation of the question.

HTH
-- 
Nathan                 Jesus College, Cambridge, CB5 8BL
                       http://www-jcsu.jesus.cam.ac.uk/~ned21/
Coz this is the White Room, my asylum - And you are welcome.
Good heavens! I don't know how you got here, did you sneak inside my head?
[Alisha's Attic - White Room]