7.16 Consider an M/M/2 queueing system with hysteresis. Specifically, the system operates as follows. Customers arrive according to a Poisson process with rate λ customers per second. There are two identical servers, each of which serves at the rate of μ customers per second, but as long as the number of customers in the system is less than eight, only one server is busy serving them. When the number of customers exceeds eight, the second server is brought in, and the two will continue to serve until the number of customers in the system decreases to four when the server that has just completed a service is retired and only one server is allowed in the system.

a. Give the state transition rate diagram of the process.

b. What is the probability that both servers are idle?

c. What is the probability that exactly one server is busy?

d. What is the expected waiting time in the system?