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

Setting the mean equation's derivative to zero givesWhen the actual current occupancy is $n=25$, the next admission has exponential distribution rate five and the next discharge has rate $n\delta =5$. The memoryless property makes these independent competing clocks conditional on the current occupancy. ThereforeThe rate comparison uses the actual stated occupancy. Knowing only the mean occupancy would not justify replacing the random $X$ inside $X\delta /(\alpha +X\delta )$ by its mean.