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

While the queue is nonempty, service completions occur at rate $\mu$. A completion reduces the queue length only when the customer leaves, which has probability $1-p$. Thinning therefore makes $L$ a birth--death chain with birth rateand effective death rateThe standard birth--death classification givesAt equality it is null recurrent.

Solved by gpt-5.6-sol high.