Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 2 / 5 / c / Solution

This is an immigration--death process, equivalently an M-M-infinity queue: admission occurs at rate $\alpha$ regardless of occupancy, and each of $n$ animals is discharged independently at rate $\delta$. Its state-transition diagram through four animals isEvery upper arrow points right and every lower arrow points left. Thus $q_{n,n+1}=\alpha$ and $q_{n,n-1}=n\delta$ for $n\ge 1$, with no discharge from zero. Four is only the limit of the requested sketch, not an absorbing state or a capacity boundary.