Please note that this all is considered as 1 problem.

11. (24 points. - Suggested statement reading time = 2 min). This problem does not require the use of Excel. However, the use of Excel is recommended for the calculations. A university bookstore has a single waiting line with a single server. Customers (mainly students) arrive at the bookstore at a rate of 21.5 customers per hour, and are served at a rate of 31.1 customers per hour. We assume Poisson arrivals and exponential service times. (a) (3 points - Suggested completion time = 2 min) Is this queuing system stable? justify you answer. (b) (2 points - Suggested completion time = 2 min) Determine the probability that 20 customers or less arrive to the bookstore in one hour (Hint: you may need to use Excel here, for convenience!). Round the probability value to 3 decimal points if needed. (c) (2 points - Suggested completion time = 2 min) Determine the probability that a cus- tomer is served in less than 2 minutes. Round the probability value to 3 decimal points if needed. (d) (3 points - Suggested completion time = 2 min) Determine the probability that no customers are present in the shop. Round the probability value to 3 decimal points if needed. (e) (3 points - Suggested completion time = 2 min) Determine the probability that exactly two customers are in the shop. Round the probability value to 3 decimal points if needed. (f) (2 points - Suggested completion time = 2 min) Determine the percentage of time the server is working (i.e. the utilization of the server). Round the percent value to 1 decimal point if needed. (g) (3 points - Suggested completion time = 3 min) Determine the average number of customers waiting for service. Round the value to 3 decimal points if needed. (h) (3 points - Suggested completion time = 3 min) Determine the average waiting time (in hours). Round the value to 3 decimal points if needed. (i) (3 points - Suggested completion time = 5 min) In order to decrease the customer waiting time, the bookstore manager considers hiring a part-time (PT) worker to help the cashier. While the new system would remain an M/M1 waiting line, the addition of this PT worker, who would assist the students in packing their purchases, would result in an estimated increase of 5 customers per hour in the service rate. The cost of hiring the PT worker would be 1500 Dirhams per week. The bookstore manager estimates that there would be gains of 500 Dirhams per week in business for every minute of waiting time saved. Should the bookstore manager hire the PT worker? Justify your answer by showing your calculations