Two stage cyclic queue Given a cyclic queue of two servers of exponential service rates, if there are N customers at one server at time t, how do i start about showing that N can be modeled as a birth and death process? and then find the BD rates?? 
Totally have no idea how to begin. what are we supposed to show to prove it can be modeled as a BD process?
 A: Suppose server 1 serves at rate $\mu_1$ and server 2 at rate $\mu_2$ and that each server can only serve a single job at a time. All other jobs at the service node are waiting.
As the system is closed and cyclic, if there are $k$ customers at node 1 we know that there are $N-k$ customers at node 2. Write $X_t$ for the number of customers at node 1 at time $t$. This can take values in $\{0,1,2,3,\ldots,N\}$.
In states $\{0,1,2,3,\ldots,N-1\}$ arrivals can happen as there are jobs at node 2 to be served. These arrivals happen at rate $\mu_2$. In state $N$ no arrivals are possible as all jobs are at node 2.
In state $0$ there can be no departures/services because all the jobs are at node 2, there are no jobs at node 1 to serve. In states $\{1,2,3,\ldots,N\}$ departures are possible (as a job completes service) and happen at rate $\mu_1$. Therefore the transition rate matrix $Q$ is (zeroes in the blanks)
$$Q=\begin{pmatrix}
-\mu_2 & \mu_2 \\
\mu_1 & -\mu_1 -\mu_2 & \mu_2 \\
&\mu_1 & -\mu_1 -\mu_2 & \mu_2 \\
&&\mu_1 & -\mu_1 -\mu_2 & \mu_2 \\
&&&& \ddots \\
&&&&\mu_1 & -\mu_1 -\mu_2 & \mu_2 \\
&&&&& \mu_1 & -\mu_1
\end{pmatrix}.$$
The model $X_t$ is a finite capcity M/M/1 queue. The process is a birth-death process because the transition rate matrix is a tridiagonal matrix; it has non-zero elements only on the main diagonal, on the first diagonal above this and the first below this.
