This is in preparation for an exam I have coming up.
We have two $\cdot / M / \infty$ queues with external arrivals occurring according to a Poisson Process of rate $\lambda$. Service occurs with rate $\mu$.
We are told that if an arrival comes to find $n_1$ customers in Queue 1, and $n_2$ customers in Queue 2, it will join Queue $j$ with probability $\frac{n_j + 1}{n_1 + n_2 + 2}$.
I am asked to show the Markov Process $N(t) = (N_1(t), N_2(t))$ has invariant distribution $$\pi(n_1,n_2) = c\frac{\rho^{n_1 + n_2}}{(n_1 + n_2 + 1)!}$$ where $c > 0$ is some suitable constant and $\rho = \frac{\lambda}{\mu}$.
So, I first consider the transition rates. Let $\underline{n} = (n_1, n_2)$ and $\underline{e_1} = (1,0), \underline{e_2} = (0,1)$.
Then, $q(\underline{n},\underline{n} + \underline{e_j}) = \lambda\frac{n_j + 1}{n_1 + n_2 + 2}$,
and $q(\underline{n},\underline{n} - \underline{e_j}) = \mu$.
Now usually, I know that an $M/M/\infty$ queue has invariant distribution which takes the form of a Poisson distribution with parameter $\rho$, and that $\pi(n_1,n_2) = \pi_1(n_1)\pi_2(n_2)$, however due to the balking that isn't the case here.
Any guidance would be appreciated, much thanks.
Since you are only asked to verify that the equilibrium distribution is as given, you can just write down the balance equations for each state $(n_1,n_2)$ and plug in the equilibrium probabilities and check that the equations are indeed satisfied. You must also check if the equilibrium probabilities can be normalized, i.e. check if
\begin{equation} \sum_{n_1,n_2 \ge 0} \pi(n_1,n_2) < \infty. \end{equation}