An airline phone-reservation call center is staffed with 3 agents. An incoming call for reservations is handled by an agent if one is available, otherwise the caller is put on hold. The system can put a maximum of 2 callers on hold, any additional callers get a busy signal and are lost. Calls arrive according to P P(), and service times are exponentially distributed with rate .

a. How do you model this system as a CTMC? Construct the rate matrix.

b. Suppose the call center starts to operate at 8am everyday with all agents idle. The customer calls arrive at a rate of 2 per hour, and the mean service time is 15 minutes. What is the expected number of busy agents at 3pm? Show how you would compute in detail, but do not compute.

c. What is the probability that the system is empty in the long-run? Compute by solving the balance and the normalizing equations.

d. Given that there is one customer in the system at time 0, compute the expected time until the system becomes empty.

e. Suppose each busy agent costs $20 per hour and each customer in the system results in a revenue of $25 per hour spent in the system. Compute the long-run average profit per hour.