A Line is a Line, or is it?

Stan Teal Mary Riehle Jean Glore

Project Revision

Deon Hickey John Roberts

Class: Pre-calculus

Materials: Data sheets (included) and Random number generator.

Goals: This project offers students the opportunity to explore the use of simulation to model a probability situation.

Time Required: Two days. Have the students work on the project on Friday and present their reports in class on Monday.

Background: This project is designed for a student in Algebra or Mathematical Analysis who has a background in probability and random number generators.

Setting: There are two fast food restaurants, and each has two check out lanes open. In McBurgers, a customer comes in and chooses the shortest line for service. In Handy Andy's, the customers wait in one line for the first available cashier. Which system gives faster service?

Problem: In choosing a model for this problem, the teacher should emphasize that it is necessary to make certain assumptions. Questions such as how many customers there are, when they come in, and how much time their service will take must be considered.

Before developing the following model with your students, have them consider ways in which they might approach the problem. Try to work with their ideas in developing the approach presented below. If given some time, most students will come up with the idea of Atrying it out@ or Arole playing@ and helping them add detail to this kind of an approach in their own words is more valuable than simply showing them the finished model. The exercises below can help you guide the discussion.

This problem can be modeled by using a random number generator to indicate the length of time it takes to serve each customer. The data we have indicates the time it takes to serve the customers as defined by the following chart.

 Minutes to serve a customer Percent of this type of customer 1 10 % 2 20 % 3 40 % 4 20 % 5 10 %

A spinner can be used to generate random numbers to simulate the minutes each customer needs for service.

Suppose we also assume that one customer comes in each minute for twelve minutes. Spin the spinner twelve times to simulate each customer=s waiting time. The results of the twelve spins should be recorded on both data charts in the columns marked "Customer, number, service minutes". The customer number and the minute they arrive should be the same number. The following abbreviation would indicate customer number five with four minutes to be served: "c5,4".

On the McBurger chart, it is necessary to determine which line a person chooses. Assume they always choose the shorter line. If both lines are of equal length, flip a coin. Heads indicates they go to Line 1, tails to Line 2.

Complete the information in the chart. Assume that exactly twelve customers come in and each stays until their service is complete.

If you complete the chart correctly, the last column should contain the number of customers waiting or being served during that particular minute. Just count the customers in each column to find the total waiting time for your twelve customers. Divide this by the number of customers served to find the average waiting time per customer.

Spin the spinner twelve more times and repeat the above steps for the previous charts. Find the average waiting times for each restaurant. Decide which system works best.

EXERCISES TO GUIDE DISCUSSION:

1) Make a list of the assumptions discussed in the problem.

2) What other assumptions did this model make?

3) What can you conclude from your experiment with this model? Does this seem like a reasonable conclusion? Would you be justified in using only your data to encourage a change in the system?

4) What are the weaknesses and strengths of this model for the problem?

5) Combine the data from the entire class and observe the results. Would it be more reasonable to draw a conclusion based on these results than only one of your results? Why?

After compiling your data, each group must make an oral presentation to the class involving all members of the group.

Following the oral presentations, each person should write a summary of what was learned in this problem and any conclusions they make from the problem.

Sample Solution: An example of each chart is attached.

Evaluation:

The project evaluation should include the following: Neatness (20%), Correct calculations and use of random generator (20%); Exercise answers (20%); Oral presentation (20%); Follow-up assumptions (20%).

Extension(s):

This project could be adjusted to a different context (airline emergency exits, escape routes from public buildings).

Combine the data from the entire class and observe the results. Would it be more reasonable to draw a conclusion based on these results than only one result?

What would happen to the model if you change the distribution of service time? Suppose more of the people coming in required four minutes of time for service. Make a new spinner to reflect changes in the percent of people requiring various service times. Run the experiment using this spinner.

Can you find another model for this problem based on different assumptions? For example, suppose more than one person can come in each minute. Write assumptions, describe the approach you used, and find a solution using your model.

Teacher Notes:

You might explore the fast food restaurants in your area to see what method they use to serve customers. If one way is the "best", why does any business do differently?

Students should work in groups of three of four. Each group will need a copy of the problem, spinner, and charts. They will also need a paper clip to use with the spinner. (Use a pencil point to hold the paper clip at the center of the spinner and spin.)

Graphing calculators can be used to generate random numbers instead of a spinner. A random number, N, generated between 0 and 1 can be used as follows to determine time of service:

0 < N < .1 ===> 1minute

.1 < N < .3 ===> 2minutes

.3 < N < .7 ===> 3minutes

.7 < N < .9 ===> 4minutes

.9 < N < 1 ===> 5minutes

1) Assumptions: One customer comes in each minute for twelve minutes; customers choose the shorter line; if lines are the same length they will choose their line at random; exactly twelve customers come in and all stay until they are served.

2) Additional assumptions might be: Only one customer will come in each minute; each customer places only one order; customers do not stand and wait for their order while the cashier begins to help another customer. Answers will vary.

3) See if the students realize that using only one simulation will not give them adequate information on which to make a prediction.

4) For example, a weakness might be assuming that one person comes in at a time. Another might be that the selection of waiting times is not broad enough. A strength might be that the same assumptions were used for both restaurants. Also, the coin flip to determine which line to choose reflects the real situation. Answers to this question will vary.

5) Students should realize that the class average is a better prediction than any individual result.

A Line is a Line, or is it?

Student Sheet

The Problem:

There are two highly competitive fast food restaurants in our town. Some people like one more than the other, and some feel just the opposite. One source of disagreement is about which has the faster service. The restaurants both have two employees taking orders, but at McBurger's the customer chooses what appears to be the shortest (fastest) line of the two lines available. At Handy Andy's everyone gets into a single line and steps forward to the first available employee, to place an order. Determine if this factor could be responsible for one restaurant being faster than the other.

Think of different ways you could determine if the factor described above is responsible for any difference in Awaiting experiences@. Discuss your ideas with other students. After sharing your ideas, develop a plan for exploring the issue of whether Aa line is a line@. What could you do to determine if the two different ways of serving customers makes any difference?

Assumptions:

Information Needed:

Plan:

Handy Andy's

 Number of Minutes Customer Number, Service Minutes Line One Customer Served Line Two Customer Served Customers Waiting Total Customer Waiting Minutes 1 2 3 4 5 6 7 8 9 10 11 12

Total Minutes Waited __________

Number of Customers Served __________

Average Waiting Time __________

McBurgers's

 Number of Minutes Customer Number, Service Minutes C u s Line Served t o m e r s One Waiting C u s Line Served t o m e r s Two Waiting Total Customer Waiting Minutes 1 2 3 4 5 6 7 8 9 10 11 12

Total Minutes Waited __________

Number of Customers Served __________

Average Waiting Time __________

Handy Andy's

Sample Solution

 Number of Minutes Customer Number, Service Minutes Line One Customer Served Line Two Customer Served Customers Waiting Total Customer Waiting Minutes 1 c1,4 c1,4 1 2 c2,5 c1,4 c2,5 2 3 c3,2 c1,4 c2,5 c3,2 3 4 c4,3 c1,4 c2,5 c3,2 c4,3 4 5 c5,3 c3,2 c2,5 c4,3 c5,3 4 6 c6,3 c3,2 c2,5 c4,3 c5,3 c6,3 5 7 c7,3 c4,3 c5,3 c6,3 c7,3 4 8 c8,4 c4,3 c5,3 c6,3 c7,3 c8,4 5 9 c9,2 c4,3 c5,3 c6,3 c7,3 c8,4 c9,2 6 10 c10,2 c6,3 c7,3 c8,4 c9,2 c10,2 5 11 c11,1 c6,3 c7,3 c8,4 c9,2 c10,2 c11,1 6 12 c12,2 c6,3 c7,3 c8,4 c9,2 c10,2 c11,1 c12,2 7 c8,4 c9,2 c10,2 c11,1 c12,2 5 c8,4 c9,2 c10,2 c11,1 c12,2 5 c8,4 c10,2 c11,1 c12,2 4 c8,4 c10,2 c11,1 c12,2 4 c11,1 c12,2 2 c12,2 1

Total Minutes Waited ___73_____

Number of Customers Served ___12_____

Average Waiting Time ___6.08___

McBurgers's

Sample Solution

 Number of Minutes Customer Number, Service Minutes C u s Line Served t o m e r s One Waiting C u s Line Served t o m e r s Two Waiting Total Customer Waiting Minutes 1 c1,4 c1,4 1 2 c2,5 c1,4 c2,5 2 3 c3,2 c1,4 c3,2 c2,5 3 4 c4,3 c1,4 c3,2 c2,5 c4,3 4 5 c5,3 c3,2 c5,3 c2,5 c4,3 4 6 c6,3 c3,2 c5,3 c2,5 c4,3 c6,3 5 7 c7,3 c5,3 c7,3 c4,3 c6,3 4 8 c8,4 c5,3 c7,3 c8,4 c4,3 c6,3 5 9 c9,2 c5,3 c7,3 c8,4 c4,3 c6,3 c9,2 6 10 c10,2 c7,3 c8,4 c10,2 c6,3 c9,2 5 11 c11,1 c7,3 c8,4 c10,2 c6,3 c9,2 c11,1 6 12 c12,2 c7,3 c8,4 c10,2 c12,2 c6,3 c9,2 c11,1 6 c8,4 c10,2 c12,2 c9,2 c11,1 5 c8,4 c10,2 c12,2 c9,2 c11,1 5 c8,4 c10,2 c12,2 c11,1 4 c8,4 c10,2 c12,2 3 c10,2 c12,2 2 c10,2 c12,2 2 c12,2 1 c12,2 1

Total Minutes Waited ___74_____

Number of Customers Served ___12_____

Average Waiting Time ___6.17___

Funded in part by the National Science Foundation and Indiana University 1995