Question

2. 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

Solution:

Let T denote the time that you’re in the bank;

let Si be the service time of the ith person in line (i = 1, 2, 3, 4);

let R be the remaining service time of the guy in service when you show up;

let S be your service time.

Obviously, the Si ’s and S are i.i.d. Exp(µ).

By the memoryless property, so is R. Therefore,

E[T] = E[R + S1 + S2 + S3 + S4 + S]

E[T] = 6/µ.