Consider a preemptive resume LCFS queueing system with two classes of customers. Type A customer arrivals are Poisson with rate λA  and Type B customer arrivals are Poisson with rate λB. The service time for type A customers is exponen- tial 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.

(a) Draw a graph for the states of the process, showing all states with two or fewer customers and a couple of states with three customers (label the empty state as E). Draw an arrow, labeled by the rate, for each state transition. Explain why these are states in a Markov process.

(b) Is this process reversible? Explain. Assume positive recurrence. Hint: If there is a transition from one state S to another state S∗, how is the number of transitions from S to S∗ related to the number from S∗ to S?

(c) Characterize the process of type A departures from the system (i.e., are they Poisson, do they form a renewal process, at what rate do they depart etc.?)

(d) Express the steady-state probability Pr{A} of state A in terms of the probability of the empty state Pr{E}. Find the probability Pr{AB} and the probability Pr{ABBA} in terms of Pr{E}. Use the notation ρA = λA/μA and ρB = λB/μB.

(e) Let Qn be the probability of n customers in the system, as a function of Q0:

(f) Pr{E}. Show that Qn = (1 - ρ)ρn where ρ = ρA + ρB.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.