A single-server queue has Poisson arrivals at finite rate $\displaystyle \lambda > 0$ and exponential services at finite rate $\displaystyle \mu > 0$. When there are $\displaystyle n$ customers in the system an arrival joins the queue with probability $\displaystyle \frac{1}{(n+1)}$ and decides not to join with probability $\displaystyle \frac{n}{(n+1)}$. Let $\displaystyle X_t$ be the number of customers in the system at time $\displaystyle t$.

What are the transition rates for the Markov process $\displaystyle (X_t)$.

Under what condition(s) does the limiting distribution exist? What is the limiting distribution of the number of customers in the system?