Consider a 3-stage flow shop with unlimited storage space that can be represented by three infinite queues in series, where each of the three manufacturing stages has a single operator (server). All processing (service) times are independent and have an exponential distribution, with a mean of 3 min./operation at stage 1, a mean of 3 min./operation also at stage 2, and a mean of 4 min./operation at stage 3. Stage 1 has a Poisson input process with a mean of 10 jobs per hour.

(a) (14 points) Find the steady-state distribution of the number of the number of jobs at each stage. Then, show the product form distribution for the joint distribution of the number of jobs at the respective stages.

(b) (10 points) What is the probability the all three servers are idle?

(c) (14 points) Find the expected total number of jobs in the flow shop and the expected total waiting time (including service times) for a job in the flow shop.