A single server queue with Markov modulated service rates and impatient customers

We consider a single server queue in which the customers wait for service for a fixed time and leave the system if the service has not begun within that time. The customers arrive according to a Poisson process and each arriving customer brings in a certain amount of phase-type distributed work. The service rate of a server varies according to the underlying continuous time Markov process with finite states. We construct a Markov process by using the age process and then obtain the stationary distribution of the Markov process. By using the results of the stationary distribution of the Markov process, we obtain the loss probability, the waiting time distribution and the system size distribution.