Sojourn time of birth and death process

[stats_noob]

Hello, I'm having trouble working out the sojourn times and the lecture notes are no help. So far all I know is that they are independently and identically distributed. Any help would be appreciated.

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.

I believe this is a birth and death process with states S=(0,1,2), what I need is the sojourn times of these states.

Again, any help would be greatly appreciated

[BGM]

I try.

I assume the state number corresponding to the total number of customers in the waiting room
and service room.

S(0) ~ Exp(λ)
The only way to leave this state is that a new pair of customer arrive (jump to state 2)
So you will need to wait Exp(λ) time for them, which is a property of Possion process.

S(1) ~ Exp(λ+μ)
The way to leave this state is that either finish the service with this customer
(jump to state 0) or wait a new customer arrive (jump to state 2)
So the the waiting time is the minimum of the waiting time of these two events
and you can show that the minimum of these two exponential time ~ Exp(λ+μ)

S(2) ~ Exp(μ)
The only way to leave this state is to finish the serive with the current customer
and start the service with the customer in the waiting room (jump to state 1)

P.S. The pdf of Exp(λ) is f(s) = λe^(-λs), s > 0, λ > 0

[stats_noob]

Wow thanks! Yeah the state number corresponds to the number of customers, guess I should've specified that.