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March 28, 2017

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Hi folks, I'm having trouble with figuring out the intensity matrix for a particular birth/death process.

Customers arrive in pairs in a Poisson stream with intensity lambda. There is waiting room for one customer. Service time is exponentially distributed with parameter mu. If the server is busy and the waiting room is empty when a pair arrives, one person stays and the other person leaves.

So the state space is {0,1,2} corresponding to the number of customers in the shop. I've figured out some entries of the intensity matrix (could be wrong, but hope not), it's the entries on the the second row (state 1) that I'm having trouble with.

| -lambda 0 lambda |
| mu ? ? |
| 0 mu -mu |

As for the stationary distribution, I have no idea what to do there.
Any help would be greatly appreciated.

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