Post-AP Calculus Investigations Queuing Theory: The Calculus of Lines

The purpose of this project is for students to see the real power of calculus to describe and explain the world around them. Waiting in line is an experience we all face if not every day, at least every week of our lives. Lines at the grocery store, in traffic, at the school cafeteria, and during times of emergency, for food, water, and shelter are a common experience for everyone.

This project has five main components:

An Introduction to Probability Density Functions
Properties of the Exponential Probability Density Function
Crossing a Street Using Calculus
Quick-Lube Queuing Simulation (Netlogo Instructions)
The Calculus of Lines

The first two components are a prerequisite for the others and combined into a single section. The Introduction to Probability Density Functions gives a short overview of probability density functions (pdf’s) and some simple examples. The area under a curve is used to compute the probability of an event given a pdf. The Properties of the Exponential Probability Density Function examines specific properties, such as the mean, median, and standard deviation of the exponential distribution and the famous “memoryless” property of the distribution. Techniques of integration are vital in this section.

The third component extends and applies the exponential distribution to the problem of waiting for a break in a line of cars when attempting to cross a busy street. Under what conditions should a crosswalk with a light be installed. In addition to integration, students will learn to use derivatives to sum infinite series.

The final two components go together. The lead the students to investigate the expected length of a line at an oil change station with a single service bay. The simulation will illustrate the growth and decay of the line and allow students to investigate how changing the arrival and service rates affects the expected length of the line and the resulting waiting time. Students will need to download the free, cross-platform agent-based modeling software Netlogo to run the simulation.

The final component develops a model of the probabilities of the line being n cars long. The definition of derivative coupled with some algebra result in a system of differential equations that can be approximated using Euler’s method. Instead of the approximation, students will do a steady-state analysis that is common for this kind of application. We again use differentiation to find the sum of an infinite series and use the sum to predict the average line length and expected wait time in line.

We generally try to give at least two weeks for projects like this one. Student material and teacher material, most with solutions, are provided so teachers can see what content is involved in each section.

Teacher Materials

Introduction and Properties of Exponential Probability Density Functions (components 1 & 2)

Crossing the Street

NetLogo Instructions and Queuing Model Simulation

Queuing Theory

Solutions

Solutions are password protected. Please contact Christine Belledin or Dan Teague for the password.

View Solutions

If you choose to have your students work through this investigation and find errors or have questions, please e-mail one of us at: "belledin@ncssm.edu” or "teague@ncssm.edu." One of us (or our colleagues) will get back to you with support.