Exponential expected service time question
A customer enters a store where he is serviced by clerk 1, followed by clerk 2, finally by clerk 3. Service time by each of the clerks is an exponentially distributed variable with , where . Assume you enter the the store and there is a single customer being serviced by clerk 3.
a) Find the probability that clerk 3 will still be busy when you move to clerk 2.
b) Find the probability that clerk 3 will remain busy when you move to see him.
c) Find the total time in the store
a) For this I'm thinking that for clerk 3 to be busy when you finish with clerk 1. Therefore the probability that clerk 3 is still busy when you move to clerk 2 would be . But this doesn't seem correct.
b) You're done with clerk 1 and 2 and you have to wait to see clerk 3. Thus the expected time of clerk 3 would have to be greater then that of clerk 1 and 2 combined so you would have which would imply
c) For this one I'm assuming it would be the total service time + total waiting time (assuming clerk 3 is still busy when you go see him)
so I would have :