Suppose that customers arrive at a bank with 6 tellers according to a Poisson process with rate $\lambda$.
Customers go to each teller with probability 1/6 (i.e. there are 6 independent M/M/1 queues), and suppose that each teller serves customers with rate $\mu$ > $\lambda$/6 (i.e. a stationary distribution exists).
Now, suppose that customer A enters the bank immediately after the system becomes stationary, and that customer B is the very next customer to enter the bank after customer A.
Let $T_A$ be the time at which customer A exits the bank (i.e. finishes being served) and let $T_B$ be the time at which customer B exits the bank.
Define $D := T_B - T_A$.
Find
(a) $\mathbb{E}(D$ | B joins the same queue as A$)$
(b) $\mathbb{E}(D)$