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

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.

The recurrent chain is positive recurrent precisely when the drift is strictly towards zero:The equality case is null recurrent.

Assume $\rho =\lambda /[\mu (1-p)]<1$. Detailed balance givesso normalization yieldsThis is a geometric distribution on ${\mathbb{Z}}_{+}$, and therefore

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