Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 27K (Applied Probability)

The queue length is a birth-death process with birth rate $\lambda r^n$ in state $n$ and death rate $\mu \min (n,2)$. Its reversible weights satisfyFor $0<r<1$, the factor $r^{n(n-1)/2}$ makes their sum finite for every $\lambda ,\mu >0$. For $r=1$, the tail is geometric with ratio $\lambda /(2\mu )$, so a stationary distribution exists exactly when $\lambda <2\mu$.

On states $0,\dots ,N$, births have rate $1$ below $N$ and deaths have rate $\mu$ above $0$. For $\mu >0$ this finite continuous-time Markov chain is irreducible, so its unique stationary law isFor $\mu =1$ this is uniform. If the parameter value $\mu =0$ is admitted, state $N$ is absorbing and the unique stationary law is instead the point mass at $N$.

In equilibrium the chain is reversible by detailed balance. Reverse time. Forward departures become reverse accepted arrivals; reversibility identifies the reversed chain with the original one, whose attempted arrivals form a Poisson process. The accepted points are exactly those occurring while the state is below $N$. The standard quasi-reversibility argument therefore makes the equilibrium departure process Poisson, with rate equal to the mean departure rate