Consider a pre-emptive resume last come first serve (LCFS) queueing system with two classes of customers. Type A customer arrivals are Poisson with rate A and Type B customer arrivals are Poisson with...

Consider a pre-emptive resume last come first serve (LCFS) queueing system with two classes of customers. Type A customer arrivals are Poisson with rate A and Type B customer arrivals are Poisson with rate B. The service time for type A customers is exponential with rate µA and that for type B is exponential with rate µB. Each service time is independent of all other service times and of all arrival epochs. Define the “state” of the system at time t by the string of customer types in the system at t, in order of arrival. Thus state AB means that the system contains two customers, one of type A and the other of type B; the type B customer arrived later, so is in service. The set of possible states arising from transitions out of AB is as follows: ABA if another type A arrives. ABB if another type B arrives. A if the customer in service (B) departs. Note that whenever a customer completes service, the next most recently arrived resumes service, so the state changes by eliminating the final element in the string.