Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 25I / a / Solution

The stationary balance equations at level zero are $(\lambda +\alpha ){\pi }_{0C}=\beta {\pi }_{0W}$ and $\beta {\pi }_{0W}=\alpha {\pi }_{0C}+\mu {\pi }_{1W}$. At every $i\ge 1$ they areThe zero-level equations give ${\pi }_{0C}=\beta {\pi }_{0W}/(\lambda +\alpha )$ and ${\pi }_{1W}=\beta \lambda {\pi }_{0W}/[\mu (\lambda +\alpha )]$. Substituting the latter in the level-one $C$ equation givesSolve the $W$ equation for ${\pi }_{i+1,W}$, then substitute it into the next $C$ equation. This yields the required row-vector recurrence with