probability

Problem #1 [It] points) A SingleChannel Queue A small cop}r center contains a single cop}r machine located behind a counter. Customers come to the counter and hand an attendant behind the counter me originals. The attendant takes the originals, makes the copies and hands the originals along with the copies back to the customerandthen this same attendant writes up the bill, presents it tothe customer and accepts payment. Suppose that the time for malcing the copies {cop}-r time} is exponentially distributed with a mean of 3 minutes and suppose that the time to write up the bill, present it to the customer and accept payment [checkout time) is also exponentially distributed with a mean of minutes. Suppose also that on the average one customer arrives into the copy center ever}r 12 minutes according to a Poisson process. a.) {4 points} Identify the type of queue involved in this problem and compute the value of the steadystate parameter p, showing that p =: 1. You may assume that the copy time and checkout time dishibutions are independent. b.) {3 points) Compute in the long term, the average number of customers waiting in line. c.) {3 points} Compute in the long term the average time {in minutes} a customer spends at the cop}: center