Past exam of the options II course of the University of Oxford / 2009 / M / M5 / b / ii / Solution

For $j\ge 1$, the only incoming transition is from $j-1$, so a stationary distribution must satisfyNormalization gives ${\pi }_0{e}^{\lambda }=1$, henceThe zero-state balance also holds: ${\sum }_i{\pi }_i(1-\lambda /(i+1))=1-{\sum }_{j\ge 1}{\pi }_j={\pi }_0$. Thus the Poisson distribution is indeed stationary, including when $\lambda =1$. Irreducibility gives uniqueness.