Each entering customer must be served first by server $1$ss$,$ then by server $2,$ and finally by server $3.$ The

amount of time it takes to be served by server i is an exponential random variable with rate $\backslash$mu i $,$ i $=1,2,3.$

Suppose you enter the system when it contains a single customer who is being served by server $3.$

$(10$ p$)$ Find the probability that server $3$ will still be busy when you move over to server $2.$

$(10$ p$)$ Find the probability that server $3$ will still be busy when you move over to server $3.$

$(15$ p$)$ Find the expected amount of time that you spend in the system. $($Whenever you encounter a busy

server, you must wait for the service in progress to end before you can enter service.$)$

$(15$ p$)$ Suppose that you enter the system when it contains a single customer who is being served by server

$2.$ Find the expected amount of time that you spend in the