# Queueing server with exponential+deterministic stages

1. Dec 20, 2012

### lahanadar

Hi, does anyone know how to model probability density function, b(x), of a system that has two parallel servers:

-The first is selected by the customer with probability $p$, it has an exponential rate of $\mu$.
-The second server is selected by the customer with probability $1-p$, it has a deterministic rate of $k$.
-When a customer chooses a server, another servers waits until the customer being served leaves.

If the second server were an exponential rate server, it would simply be an M/Er=2/1 system but now I don't know how to model this. Can anyone help me, plz?

2. Dec 22, 2012

### chiro

Hey lahanadar.

Are you allowed to use Monte-Carlo simulation to simulate the process so that you get a PDF (by stipulating a large enough number of simulations)?

3. Dec 22, 2012

### chiro

Hey lahanadar.

Are you allowed to use Monte-Carlo simulation to simulate the process so that you get a PDF (by stipulating a large enough number of simulations)?

4. Dec 22, 2012

### lahanadar

Not really, I should find it by computation. I think one way could be representing the over all server pdf by linear combination of individual servers' pdfs, while first has an exponential pdf and second unit step function. Do you think this works?

Last edited: Dec 22, 2012
5. Dec 22, 2012

### chiro

Have you constructed a Markovian system for your queues (if you can't use simulation)?

Even if you do it analytically, I would suggest you use simulation to double check your work and get into the habit of double checking things in this way for the future.

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