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?
A single-server queue has Poisson arrivals at finite rate and exponential services at finite rate . When there are customers in the system an arrival joins the queue with probability and decides not to join with probability . Let be the number of customers in the system at time .
What are the transition rates for the Markov process .
Under what condition(s) does the limiting distribution exist? What is the limiting distribution of the number of customers in the system?