$(35$ points$)$ Consider a manufacturing system where items arrive from the outside

according to a Poisson process with rate $5$ per hour. This system consists of three

facilities. Each arriving item is first processed at Facility $1,$ then at Facility $2$ and, finally,

at Facility $3.$ There are $2$ servers in facility $1$ and also in facility $2.$ Facility $3$ has only one

server. The service times are all exponentially distributed with rates $4$ per hour, $3$ per

hour, and $6$ per hour in Facility $1,2$ and $3,$ respectively. After the tasks are performed in

Facility $1$ and Facility $2,$ the quality of the product is checked in Facility $3.$ Previous data

shows that $10\%$ of the checked products are defective and, hence, must be reworked. This

implies that a checked product is moved to Facility $1$ with probability $0\mathrm{.}1$ and is moved

out to the warehouse with probability $0\mathrm{.}9.$

a$)(10$ points$)$ Considering the models covered in the lectures, what kind of a queueing

model is this? How many items are processed on average at each facility?

b$)(10$ points$)$ What is the probability that the whole manufacturing system is empty?

c$)(10$ points$)$ What is the expected total number of items in the whole system?

d$)(5$ points$)$ What is the expected total waiting time in the whole system?