# Math Help - Markov Process:Poisson Queue with exponential services

1. ## Markov Process:Poisson Queue with exponential services

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

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

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

2. I can't do the second two questions until I have calculated the transition rates, and I am unsure how to calculate them in this question. Can anyone please help me to understand this?