Assignment:
Case Study 1: Queuing Theory
"I hope this goes better than last time," thought Craig Rooney as he thought about having to walk into the city council's chambers next week. Craig is the assistant chief of police in Newport, VA, and, each September, he has to provide the city council with a report on the effectiveness of the city's police force. This report immediately precedes the council's discussion of the police department's budget. So Craig often feels like a tightrope artist trying to find the right balance in his presentation to both convince the council that the department is being run well and also persuade them to increase the department's budget for new officers.
The city of Newport has a total of 19 police officers assigned to 5 precincts. Currently, precinct A has 3 officers assigned to it while the others each have 4 officers. One of the town council's primary concerns each year is the amount of time it takes for an officer to begin responding when a 911 emergency call is received. Unfortunately, the city's information system does not track this data exactly, but it does keep track of the number of calls received in each precinct each hour and the amount of time that elapses between when an officer first begins responding to a call and the time he or she reports being available again to respond to other calls (this is also known as the service time for each call).
A student intern from a local university worked for Craig last summer and collected data shown in the file named Call Data.xls that accompanies this book. One of the sheets in this workbook (named Calls Per Hour) shows the number of 911 calls received during 500 randomly chosen hours of operation in each precinct. Another sheet (named Service Times) shows the services time required for each of these calls.
The student intern also set up a worksheet based on what you learnt to calculate operating characteristics of an M/M/s queue for each of the Newport's five precincts. Unfortunately, the student interim had to return to school before finishing this project. But Craig believes with a little work, he can use the data collected to figure out appropriate arrival and service rates for each precinct and complete the analysis. More importantly, he feels sure the queuing model will allow him to quickly answer many of the questions he expects the city council to ask.
1. What are the arrival rate of 911 calls and the service rates for each precinct?
2. Does the arrival rate of calls for each precinct appear to follow a Poisson distribution?
3. Does the service rate for each precinct appear to follow an exponential distribution?
4. Using an M/M/s queue, on average, how many minutes must a 911 caller in each precinct wait before a police officer begins responding?
5. Suppose Craig wants to redistribute officers among precincts so as to reduce the maximum amount of time callers in any one precinct have to wait for a police response. What should he do, and what impact would this have?
6. How many additional police officers would Newport have to hire in order for the average response time in each precinct to be less than two minutes?
Case Study 2: Decision Making Techniques
Success or failure as a farmer depends in large part of the uncertainties of the weather during the growing seasons. Consider the following quote from a recent news article:
.. Ln a summer plagued by drought and heat, many Southern crops are withering in the fields, taking farmers' profits down with them. Some farmers are fighting to break even. But others have had to give up hope that this year's crop will survive to harvest 'Farmers must decide if hey're going to continue to nurture that crop or give up and plow it under,' said George Shumaker, an Extension Service economist with the University of Georgia College of Agricultural and Environmental Sciences. Making that decision takes courage and careful calculation."
Assume that you are a fanner facing the decision of whether or not to plow under your crops. Suppose you have already invested $50 per acre in seed, water, fertilizer, and labor. You estimate it will require another $15 per acre to produce and harvest a marketable crop. If the weather remains favorable, you estimate your crop will bring a market price of $26 per acre.
However, if the weather becomes unfavorable, you estimate your crop will bring a market price of $12 per acre. Currently, the weather forecasters are predicting favorable weather conditions with a probability of 0.70. The owner of the farm next to yours (who is growing the same product and has made the same $50 per acre investment) has just decided to plow his fields under because the additional $15 per acre to produce a marketable crop would just be "throwing good money after bad."
1. Develop a decision tree for your decision problem.
2. What is the EMV of harvesting and bringing the crop to market?
3. Would you bring this crop to market or plow it under like your neighbor?
4. By how much would the probability of favorable weather have to change before your answer to the question in part c would change?
5. By how much would the $15 per acre cost of bringing the crop to market have to change before your answer to the question in part c would change?
6. What other factors might you want to consider in making this decision?