Title
Waiting In Line (Eric Knuth)
Goals
(1) Students will investigate basic concepts involved in queuing theory.
(2) Students will represent wait times using recursive expressions.
Abstract
Everyone has experienced the frustration of waiting in long lines. Scheduling any type of appointment or setting the timing sequence for assembly lines requires an understanding of what is known as queuing theory. The amount of time spent waiting is dependent on several variables: arrival time, service time, and the number of people who are waiting to be served. This activity examines how these variables effect the length of time one has to wait.
Problem Statement
As the school registrar, part of your job entails making the registration process as painless as possible for the students. One aspect of the registration process that influences the length of the time required to register is the amount of time students spend waiting in line. To get a better idea of the factors that influence the wait time, you have decided to model a few different situations.
Instructor Suggestions
(1) Describe each situation to the students. You may find it helpful to demonstrate how to model the situation using a table. Another option might be to model a simple situation--your students might play the role of registering students.
Situation A
Assume that no student is waiting in line. Each student needs two minutes to pick up his/her registration materials and students are scheduled to arrive every three minutes. Answer: No wait time. When the service time is less than the arrival time, there will be no waiting.
Situation B
Assume that no student is waiting in line. Each student needs two minutes to pick up his/her registration materials and students are scheduled to arrive each minute.
Answer: In this case the wait time keeps getting longer with each student. When the service time is greater than the arrival time, there is always a wait.
Situation C
Assume that there are four students waiting in line as registration begins and that each new student arrives three minutes apart. Also assume that the service time is still two minutes.
Answer: Average wait is 3.7 minutes. After the ninth arriving student the registrar will be caught up.
Situation D
Assume that five students are waiting and each transaction takes three minutes with arrivals four minutes apart.
Answer: Average wait is 7.1 minutes. After the sixteenth arriving student the registrar will be caught up.
Instructor Suggestions (cont.)
Sample analysis Situation C
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note: b1 represents first student already waiting
s1 represents first student arriving
At student number thirteen, the wait time is down to zero.
Average wait time is 48 ÷ 13 = 3.7 minutes.
Arrival time = (n - 1) * 3, where n = number of arriving student
Beginning service time = 6 + 2n, where n = number of arriving student
Wait time will be zero when arrival time equals beginning service time:
(n - 1) * 3 = 6 + 2n
(2) Separate students into small groups to investigate each situation. You may chose to have each group investigate only one or two of the situations.
(3) Each group should present its results to the class.
(4) Discuss students' solutions in terms of the registration process and the ways in which it could be improved.
Materials
worksheet describing problem situations
Time
Problem introduction (5 minutes), group work (25 minutes), small group presentation (15 minutes), class discussion (15 minutes)
Mathematics Concepts
Discrete Mathematics Concepts
recursion, graphs
Related Mathematics Concepts
algebraic expressions, modeling
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections (within mathematics and across disciplines), Algebra, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into (1) an Algebra class as students begin to use equations for solving word problems; (2) any mathematics class in which students use modeling in solving problem situations.
Further Investigation
(1) A receptionist in a dentist's office schedules patients every thirty minutes beginning at 9:00 am. Each appointment takes 25 minutes. If the fifth patient is twenty minutes late, when will the dentist catch up (assume the dentist doesn't take a break)? When will the dentist catch up if she takes a 3 minute rest between patients?
(2) A conveyor belt carries jars of salsa past two machines. The first machine tightens the caps and the second places a label on each jar. The second machine needs to shut down for 18 minutes in order to refill it with labels. The label machine takes 30 seconds to apply each label. When the machine begins again at 2:00 PM, there are thirty jars waiting. Jars come from the first machine every 3 seconds. When will there be no wait for jars to pass through the second machine?
Variations/Comments
References/Resources
Davis, Gretchen. (1993). First come, first served. Communicator, 18(3), California Mathematics Council.
Queues (1987), Janson Publications, Inc.