Q. For this problem the system can be considered the purchasing of the ticket. Customers arrive at the theatre line at the rate of 100 per hour. The ticket seller averages 30 seconds per customer, which comprises 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. There is an infinite population and an infinite queue.

a) Illustrate what is the average time it takes for the customer to enter the theatre measured from the time the customer arrives at the theatre line to purchase a ticket?

b) Illustrate what is the average number of customers waiting in line to purchase a ticket?

c) Illustrate what is the probability that there is at least one other person waiting in line to buy a ticket?

d) Illustrate what would be the effect on the total time it takes for the customer to enter the theatre 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) Assume the cost of the first ticket taker is $20 per hour while the cost of the second ticket taker is only $10 per hour. Also, assume the cost of both waiting and buying the tickets (cost of good will and lost customers) is valued at $6 per hour per customer. Illustrate what is the cost of each option (the single ticket taker and the two ticket taker options) that the theatre is considering?