# Queuing thoeory

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Determine the Waiting Line Model and Solve Below.

6. The Bijou Theater in Hermosa Beach, California, shows vintage movies. 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. (Because of these added services, many customers don't get in until after the feature started.)

a. What is the average customer waiting time in the system?

b. What would be the effect on system waiting time of having a second ticket taker doing nothing but validations and card punching, thereby cutting the average service time to 20 seconds?

c. Would system waiting time be less than you found in b if a second window was opened with each server doing all three tasks?

8. A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn follow a Poisson distribution at the rate of three per minute. In serving themselves, customers take about 15 seconds, exponentially distributed.

a. How many customers would you expect to see on the average at the coffee urn?

b. How long would you expect it to take to get a cup of coffee?

c. What percentage of time is the urn being used?

d. What is the probability that three or more people are in the cafeteria?

e. If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 15 seconds, how does this change your answers to a and b?

13. Benny and Barber owns a one-chair shop. At barber college, they told Benny that his customers would exhibit a Poisson arrival distribution and that he would provide an exponential service distribution. His market survey data indicate that customers arrive at a rate of two per hour. It will take Benny an average o 20 minutes to give a haircut. Based on these figures, find the following:

a. The average number of customers waiting

b. The average time a customer waits

c. The average time a customer is in the shop.

d. The average utilization of Benny's time.

15. Customers enter the camera department of a store at the average rate of six per hour. The department is staffed by one employee, who takes an average of six minute to serve each arrival. Assume this is a simple Poisson arrival exponentially distributed service time situation.

a. As a casual observer, how many people would you expect to spend in the camera department (excluding the clerk)? How long would a customer expect to spend in the camera department (total time)?

b. What is the utilization of the clerk?

c. What is the probability that there are more than two people in the camera department (excluding the clerk)?

d. Another clerk has been hired for the camera department who also takes an average of six minutes to serve each arrival. How long would a customer expect to spend in the department now?

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