**Exploring Modeling**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk__

1. The exit to a parking lot is controlled by a traffic light. If there is a line of cars at the signal inside the lot, how many cars will be able to exit during the green phase of the light?

Things to think about:

a. What data might need to be collected to answer this question?

b. Is there any difference between this question and what would go into asking how many cars stopped for a red light would be able to clear a red light on a street that has one lane of traffic in each direction?

2. A busy north-south street in an otherwise residential neighborhood has a cross street (east-west) with no traffic signal. The nearest traffic light north of the intersection is 6 blocks north and the nearest traffic light south of the street is 7 blocks away. The cross street has a moderate amount of traffic as well as considerable pedestrian traffic because the street ends, in the westerly direction, at a commuter railroad station. Analyze if it would be a good idea to place a light at this intersection.

3. What considerations might go into deciding the relative lengths of the green, yellow, and red phases of a traffic signal at different kinds of intersections?

4. Discuss the issues involved in deciding whether a new urban expressway project should have two lanes or three lanes of traffic in each direction.

6. Two identical machines can process tasks which take the following times (in seconds) to complete:

55, 45, 84, 101, 121, 78, 78, 99, 112, 57, 78, 56, 96, 45, 94, 111, 66, 66, 66, 55, 87, 89, 111, 69, 96, 78, 87, 77, 64, 58, 61, 83, 83, 59, 55, 77, 87, 96, 66, 77, 52, 84,76

i. What is the earliest time that one can complete the job consisting of these tasks?

ii. Resolve this problem if a. 3 identical machines are available to do the work and b. 4 identical machines are available to do the work.

iii. What is the minimum number of machines that would be necessary to finish all of these tasks in a. 4 minutes b. 5 minutes c. 8 minutes?

iv. Develop a framework for attaching general versions of problems of these two kinds.

v. Can you cite situations in the "real world" where being able to solve questions of this type arise?

7. Operations research (OR) (sometimes called management science) is the branch of mathematics which concerns itself with making the operations of businesses and governments work more efficiently. Many operations research problems appear in newspapers on a daily basis. For example, organizing the return of American troops from Iraq is an operations research problem, as is the deployment of police to ensure public safety during the running of the NYC or Boston marathons.

a. Find examples of operations research problems that are implicit in stories that you see in the newspaper or hear about on television.

b. Find examples of OR problems which involve:

i. arithmetic

ii. geometry

iii. functions

iv. statistics

v. probability