At my local post office there is a central queue served by three assistants. From past experience, I know that the service times of the assistants are exponentially distributed with means 12 minutes, 2 minutes and 3 minutes.
One day a customer enters the post office to discover that all three assistants are busy, but nobody else is waiting to be served.
(i) Find the distribution of the time she has to wait before she can move forward for service, and hence show that her expected waiting time is 40 seconds.
Calculate the probability that she has to wait more than three minutes for an
assistant to be free to serve her.
I know that the combination of three exponential processes with rates r1,
r2, r3 is a single exponential process with a rate r1 + r2 + r3.
How can I do that?
(ii) Three minutes after entering the post office she is still waiting. Another
customer enters the post office and stands behind her in the queue. (So now
there are the same three people being served as there were at the start, and
two people standing in the queue.) State the distribution of this latest> arrival?s waiting time before his service commences, and hence find the mean
and standard deviation of his waiting time.
I think I will have to condition on the relative probabilities of
the first person in line being served by each server. Is that right??? HOW???