= Solution
For $j\ge1$ the only entrance is from $j-1$, so stationarity requires $\pi_j=\pi_{j-1}\lambda/j$. <Induction> gives $\pi_j=\pi_0\lambda^j/j!$, and normalization using the exponential series gives $\pi_0=e^{-\lambda}$. Hence
$$
\boxed{\pi_j=e^{-\lambda}\frac{\lambda^j}{j!},\qquad j=0,1,\ldots.}
$$
The remaining zero-state equation also holds: $\sum_i\pi_i\lambda/(i+1)=e^{-\lambda}(e^\lambda-1)=1-\pi_0$, so $\sum_i\pi_i(1-\lambda/(i+1))=\pi_0$. Thus this normalized <Poisson distribution>[Poisson law] really is stationary, not just a solution of the positive-state equations. Irreducibility makes the stationary probability law unique.
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