Question

Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times are all exponential with rate μ, what is the expected amount of time you will spend in the bank?

Answer 1

The expected amount of time you will spend in the bank can be calculated as follows:

Let's call T the expected service time for each customer. Then, T = 1/μ.

The expected time you spend waiting in line is given by the formula:
(number of people ahead of you * T) + T

So, in this case, the expected time you spend in the bank is:
(4 * T) + T = 5 * T = 5/μ.

Therefore, the expected amount of time you will spend in the bank is 5/μ.