Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 28L / c / iv / Solution

In the stable regime, $L$ is exactly the queue-length process of an $M/M/1$ queue with arrival rate $\lambda$ and effective service rate $\mu (1-p)$. By Burke theorem, when this queue is in equilibrium its external departure process is a Poisson process of rate