The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold a ticket at the rate of nine per minute. For the theater's normal offerings of older movies, customers arrive at the rate of 4.5 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.

(a)

What is the average number of customers waiting in line?

customer(s)

(b)

What is the average time (in minutes) a customer spends in the waiting line? (Round your answer to four decimal places.)

minutes

(c)

What is the average number of customers in the system?

customer(s)

(d)

What is a customer's average time (in minutes) in the system? (Round your answer to four decimal places.)

minutes

(e)

What is the probability that someone will be buying tickets when an arrival occurs?

The Grand has booked the Star Wars trilogy and expects more customers. From conversations with other theater owners, it estimates that the arrival rate will increase to 13 per minute. Output is supplied for two- and three-cashier systems as shown in the following table.

Number of Servers	2	3
Arrival Rate	13	13
Service Rate	9	9
Probability of No Units in System	0.1613	0.2244
Average Waiting Time	0.1211	0.0155
Average Time in System	0.2323	0.1266
Average Number Waiting	1.5749	0.2018
Average Number in System	3.0194	1.6463
Probability of Waiting	0.6057	0.2174
Probability of 14 in System	0.0034	0.0000

(f)

The Grand has space for 13 customers to wait indoors to buy tickets. Which system will be better?

With three servers, the chance that some customers in the system will need to wait outdoors is much lower, so this system will be better.

With two servers, the chance that some customers in the system will need to wait outdoors is much lower, so this system will be better. With three servers, the average number in the system is much higher, so this system will be better.

With two servers, the chance that a customer will need to wait in line is much lower, so this system will be better.

Both systems' operating characteristics are too close to one another to determine if one has a clear advantage over the other.

(g)

Do you think it is more sensible for the Grand to continue the one-cashier system?

The service rate is faster than the arrival rate, so the one-cashier system won't work.

No, although the one-cashier system would be much cheaper in this case, the two- and three-cashier systems have much better operating characteristics.

Yes, a one-cashier system has such a low service cost that it will offset the higher waiting cost.Yes, the operating characteristics for the two- and three-cashier systems are not much better than the one-cashier system.

The arrival rate is faster than the service rate, so the one-cashier system won't work.