 Solution

The stationarity equation at an interior state collects arrivals from its two neighbors and itself:This rearranges to the required recurrence. At zero, stationarity gives ${\pi }_0=(1-a){\pi }_0+b{\pi }_1$, hence $\boxed{b{\pi }_1=a{\pi }_0}$.

The recurrence says the flux $a{\pi }_{i-1}-b{\pi }_i$ equals $a{\pi }_i-b{\pi }_{i+1}$. The boundary equation makes that flux zero, so ${\pi }_i=(a/b{)}^i{\pi }_0$. Since $0\le a/b<1$, normalization givesWhen $a=0$ this means ${\pi }_0=1$ and ${\pi }_i=0$ for $i>0$. The probability assumptions also require $a,b\ge 0$, $a+b\le 1$, as implicit in the transition rules.