Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / a

Each arrival by time $t$ remains with probability $e^{-\mu (t-s)}$. Poisson thinning gives $X_t\sim \mathrm{Poisson}((\lambda /\mu )(1-e^{-\mu t}))$.

The invariant law is $\mathrm{Poisson}(\lambda /\mu )$, also obtained from detailed balance ${\pi }_k\lambda ={\pi }_{k+1}(k+1)\mu$.

The time-dependent Poisson mean tends to $\lambda /\mu$, so each mass converges directly to $e^{-\lambda /\mu }(\lambda /\mu {)}^k/k!={\pi }_k$.