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Queuing :M/M/s model

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The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold its ticket at the rate of 6 per minute. For the theater's normal offerings of older movies, customers arrive at the rate of 3 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. (7 points)

a. What is the average number of customers waiting in line?
b. What is the average time a customer spends in the waiting line?
c. What is the average number of customers in the system?
d. What is a customer's average time in the system?
e. What is the probability that someone will be buying tickets when an arrival occurs?

The Grand has booked the Stars Wars Trilogy and expects more customers. From conversations with other theater owners, it estimates that the arrival rate will increase to 10 per minute. Output is supplied for a two-cashier and a three-cashier system.Number of Channels 2 3
Arrival Rate 10 10
Service Rate 6 6
Probability of No Units in System 0.0909 0.1727
Average Waiting Time 0.3788 0.0375
Average Time in System 0.5455 0.2041
Average Number Waiting 3.7879 0.3747
Average Number in System 5.4545 2.0414
Probability of Waiting 0.7576 0.2998
Probability of 11 in System 0.0245 less than .0088

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

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

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