I've been stuck on this question for hours, and could use some help :)
"An M/M/1/2 queue has service rate $\mu$ and arrival rate of either $\lambda_1$ or $\lambda_2$. The rate can change only when the server is idle (state 0), whereupon it changes as a Markov process with instantaneous transition rate $\alpha$ from $\lambda_1$ to $\lambda_2$ and $\beta$ from $\lambda_2$ to $\lambda_1$."
1) Define the state space, draw its state-transition diagram: I have 6 states {$0_{\lambda_1}$, $0_{\lambda_2}$, $1_{\lambda_1}$, $1_{\lambda_2}$, $2_{\lambda_1}$, $2_{\lambda_2}$} (so the system can start at state $0_{\lambda_1}$ or $0_{\lambda_2}$, with an arrival rate $\lambda_1$ or $\lambda_2$, which it then keeps until it comes back to one of the $0$'s and switches arrival rates).
2) Show that the queue is equivalent to a standard M/M/1/5 queue and specify its service and arrival rate functions?
I can see why it 'looks' like an M/M/1/5 (as it has 6 states), but I can't figure out what M/M/1/5 (alternatively, I may have completely mis-interpreted the question...).
EDIT (how I think I've solved it)
My initial M/M/1/2 queue had the following transitions:
- $0_{\lambda_1} \rightarrow 1_{\lambda_1}: (1-\alpha)\lambda_1$
- $0_{\lambda_2} \rightarrow 1_{\lambda_2}: (1-\beta)\lambda_2$
and
- $0_{\lambda_1} \rightarrow 1_{\lambda_2}: \alpha\lambda_1$
- $0_{\lambda_2} \rightarrow 1_{\lambda_1}: \beta\lambda_2$
I don't know why I assumed that the rate changes would occur at the same time as an arrival, and this is how I got stuck. If instead, we consider the rate changes ($\lambda_1 \rightarrow \lambda_2$ and $\lambda_2 \rightarrow \lambda_1$) as occurring 'on their own', then we need to replace transitions 3 and 4 above with the following:
- $0_{\lambda_1} \rightarrow 0_{\lambda_2}: \alpha\lambda_1$
- $0_{\lambda_2} \rightarrow 0_{\lambda_1}: \beta\lambda_2$
Then it reduces to a standard M/M/1/5 (that you can get by relabelling the states in the range [0..5])
Let me know if this isn't clear enough! (I don't have time to draw the Chains in TikZ which would be better, exams start on Monday).