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Consider a system of two infinite queues in series, where each of the two service facilities has a single server. All service times are independent and have an exponential distribution, with a mean of 3 minutes at facility 1 and 4 minutes at facility 2. Facility 1 has a Poisson input process with a mean rate of 10 per hour.
(a) Find the steady-state distribution of the number of customers at facility 1 and then at facility 2. Then show the product form solution for the joint distribution of the number at the respective facilities.
(b) What is the probability that both servers are idle?
(c) Find the expected total number of customers in the system and the expected total waiting time (including service times) for a customer.
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