Consider a closed queueing network where there are two queueing nodes, each with one server, FCFS queueing disciplines, and independent exponential service times. The server at node $i$ has mean service time $\frac{1}{\mu_i}$ for i = 1, 2. Assume $\mu_2 < \mu_1$. Customers leaving node 1 enter node 2 and those leaving node 2 enter node 1 in an infinite cycle. Let there be $n$ customers trapped in this closed system. I want to analyze the stationary distribution of this queueing system.
It seems to me that this system is same as the birth and death process with $n+1$ states $S_0,S_1, \ldots, S_n$ where $S_i=(\text{$i$ in queue 1, $n-i$ in queue 2})$ and the transition rates from $S_i$ to $S_{i+1}$ is $\mu_2$ and from $S_i$ to $S_{i-1}$ is $\mu_1$. Is this correct? Or do I have to somehow use Burkes theorem?