I have an m/m/1 queue with arrival rates $\lambda_0=3, \lambda_1=2,\lambda_2=1$ and for all $n>2$ we have $\lambda_n=0$. With constant service rates of $\mu=2$
I need here to calculate the stationary distribution for the above system, however when doing this I run into a problem. Namely that when we try to calculate the distribution we get
$\pi_{1}=\pi_{0},$ $\pi_{2}=2\pi_{0},$ $\pi_{3}=4\pi_{0},$ $\pi_{3}=\pi_{2},$ $\pi_{4}=\pi_{3}$
so $\pi_i=\pi_{i+1}$ for all $i\geq2 \Rightarrow \pi_0 = 0$ and then that all probabilities are zero. I feel as though I am doing something wrong here but I can't put my finger on it. Any help would be greatly appreciated.
thanks
I think your calculations are correct. Given the rates that you have, $$ 0 \rightleftharpoons^3_2 1 \rightleftharpoons^2_2 2 \rightleftharpoons^1_2 3,$$ the transition rate matrix would be: $$A=\begin{bmatrix}-3 & 3 & 0 & 0\\ 2& -4 &2 &0\\ 0 & 2 & -3 &1 \\ 0 & 0 & 2 & -2\end{bmatrix}$$ The transition matrix for the Markov chain is: $$P=\begin{bmatrix}0 & 1 & 0 & 0\\ \frac{2}{4}& 0 &\frac{2}{4} &0\\ 0 & \frac{2}{3} & 0 &\frac{1}{3} \\ 0 & 0 & 1 & 0\end{bmatrix}$$ Then the equation $\pi P= \pi$ has only one (invalid) trivial solution $\pi=[\pi_0, \pi_1, \pi_2, \pi_3]=\vec{0}$. Hence, an stationary distribution does not exist.