Consider the closed queueing network in the figure below. There are three customers who are doomed forever to cycle between node 1 and node 2. Both nodes use FCFS service and have exponential IID service times. The service times at one node are also independent of those at the other node and are independent of the customer being served. The server at node i has mean service time 1/μi, i = 1, 2. Assume to be specific that μ2 <> μ1.

(a) The system can be represented by a four-state Markov process. Draw its graphical representation and label it with the individual states and the transition rates between them.

(b) Find the steady-state probability of each state.

(c) Find the time-average rate at which customers leave node 1.

(d) Find the time-average rate at which a given customer cycles through the system.

(e) Is the Markov process reversible? Suppose that the backward Markov process is interpreted as a closed queueing network. What does a departure from node 1 in the forward process correspond to in the backward process? Can the transitions of a single customer in the forward process be associated with transitions of a single customer in the backward process?

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