Four weeks later Pumeza visits the same branch as in exercise 2.43 . Now the service times at tellers 1 and 2 are again independent, but exponentially distributed with respective parameters \(\lambda_{1}=0.4\left[\mathrm{~min}^{-1}\right]\) and \(\lambda_{2}=0.2\left[\mathrm{~min}^{-1}\right]\).

(1) When Pumeza enters the branch, both tellers are occupied and no customer is waiting. What is the mean time Pumeza spends in the branch till the end of her service?

(2) When Pumeza enters the branch, both tellers are occupied, and another customer is waiting for service. What is the mean time Pumeza spends in the branch till the end of her service? (Pumeza does not get preferential service.)

Data from exercise 2.43

A small branch of a bank has the two tellers 1 and 2 . The service times at these tellers are independent and exponentially distributed with parameter \(\lambda=0.4\left[\mathrm{~min}^{-1}\right]\). When Pumeza arrives, the tellers are occupied by a customer each. So she has to wait. Teller 1 is the first to become free, and the service of Pumeza starts immediately. What is the probability that the service of Pumeza is finished sooner than the service of the customer at teller 2 ?