The Bijou Theater in Bellevue shows vintage movies.For this problem, the system can be considered the purchasing of the ticket. Customers arrive at the theater line at the rate of 100 per hour. The ticket seller averages 30 seconds per customer, which includes placing validation stamps on customers' parking lot receipts and punching their frequent watcher cards as well as selling tickets.Because of these added services, many customers don't get in until after the feature has started. Assume that the arrivals follow a Poisson arrival distribution and the service times are exponentially distributed. Once a customer arrives at the theater line to purchase a ticket, the customer is considered to enter the system. There is an infinite population and an infinite queue.

a) What is the average number of customers in the "system" either waiting to purchase a ticket or purchasing a ticket?

b) What is the average time a customer stands in line waiting to purchase?

c) What is the probability that there are at least two waiting in line to buy a ticket when you arrive to purchase your ticket?

d) What would be the effect on the total time it takes for the customer to enter the theater by having a second ticket taker doing nothing but validations and card punching, thereby cutting the average service time to 20 seconds per customer?This ticket taker does not directly wait on any customers but only supports the ticket seller.

e) Suppose the cost of the first ticket taker is $18 per hour while the cost of the second ticket taker is only $9 per hour. Also, suppose the cost of both waiting and buying the tickets (cost of good will and lost customers) is valued at $6 per hour per customer. What is the cost of each option (the single ticket taker and the two ticket taker options) that the theater is considering?