Question

Consider a bank with two tellers. Three people, Anne, Betty and Carol enter the bank at almost the same time and in that order. Anne and Betty go directly into service while Carol waits or the first available teller. Suppose that the service times for two servers are exponentially distributed with mean three and six minutes (or they have rates of 20 and 10 per hour).
(a)What is the expected total amount of time for Carol to complete her businesses?
(b) What is the expected total time until the last of the three customers leaves?
(c) What is the probability for Anne, Betty, and Carol to be the last one to leave?

Answer 1

Answer / Step-by-step explanation:

To properly answer this question, we will assume some parameters. that includes us saying :

(a) Where S1 is teller 1 service time

                  S2 is teller 2 service time

Therefore,

E ( Total time carol spent completing her business)

= E ( Minimum ( S1, S2 ) ) + P (S1 > S2) . 6 + P ( S1 < S2) . 3

= 1 /1/3 + 1/6 +  1/6 / 1/3 + 1/6 . 6 + 1/3 / 1/3 + 1/6 . 3

= 2 +2 + 2 = 6

(b) E ( Minimum ( S1, S2)  + E ( Max ( S1, S2)

= 1/ 1/3 + 1/6  +3 + 6 - 1 / 1/3 + 1/6 = 9

(c) P (S1 > S2) P (S2 > S1 ) + P (S1 < S2 ) P (S1 > S2)

1 /3 . 2.3 . 2

= 4 /9