Let's consider the situation from the start of your service until the end.
Since service times are independent identically distributed exponential random variables, the time it takes for you to be served follows an exponential distribution with a certain rate parameter λ. The sum of n independent exponential random variables with rate λ is a gamma random variable with shape parameter n and rate parameter λ.
Since there are only two tellers and you are served by one of them, there is a 1/2 chance that the other teller will finish before you. So the probability that you will be the last to leave is 1/2.
When you enter your bank, you find that there are only two tellers, both busy serving other customers, and that there are no other customers in line. Assume that the service times for you and for each of the customers being served are independent identically distributed exponential random variables. Also assume that after a service completion, the next customer in line immediately begins to be served. What is the probability that you will be the last to leave? Hint: Think of the situation at the time that you start getting served
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