# Introduction to Management Science Study Set 3

## Quiz 13 :Queuing Analysis

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The Petroco Service Station has one pump for regular unleaded gas, which (with an attendant) can service 10 customers per hour. Cars arrive at the regular unleaded pump at a rate of 6 per hour. Determine the average queue length, the average time a car is in the system, and the average time a car must wait. If the arrival rate increases to 12 cars per hour, what will be the effect on the average queue length
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Arrival rate
Service rate
L q = Average number of customers in the waiting line
W = Average time a customer will spend in the queue system
W q = Average waiting time in the line per customer
Given Data:
Use the following formula to calculate the average queue length:
There are on an average 0.9 cars in the queue.
Use the following formula to calculate the average time a car will spend in the total queue system:
The average waiting time in the entire system per car is 0.25 hours.
Use the following to calculate how long each car must wait:
The average waiting in line time per car is 0.15 hours.
If the arrival rate increases to 12 cars per hour, this is higher than the service rate therefore an infinite line would be formed.

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A single-server queuing system with an infinite calling population and a first-come, first-served queue discipline has the following arrival and service rates (Poisson distributed): = 16 customers per hour = 24 customers per hour Determine P 0 , P 3 , L , L q , W , W q , and U.
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Arrival rate
Service rate
P 0 = Probability that no customers in the queue
P n = Probability that there are three customers in the queue
L = Average number of customers in the queue
L q = Average number of customers in the waiting line
W = Average time a customer will spend in the queue system
W q = Average waiting time in the line per customer
U= Probability that the server was busy
Given Data:
Customers per hour
Customers per hour
Use the following formula to solve for the probability that no customers exist in the queue :
The probability that there are no customers in the queue system is 0.33.
Use the following formula to determine the probability that there are three customers in this queue:
The probability of three customers in the queue system is 0.097
Use the following formula to calculate the average number of customers in the queue system:
Use the following formula to calculate the average number of customers in the waiting line :
The average numbers of customers waiting in the line are 1.33.
Use the following formula to calculate average time a customer will spend in the queue system:
The average waiting time in the entire system per customer is 0.125 hours.
Use the following formula to calculate the average waiting time in the line per customer:
The average waiting in line time per customer is 0.083 hours.
Use the following formula to calculate the probability that the server was busy :
There is a 0.67 probability that the server was busy and the customer waited.

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The Dynaco Manufacturing Company produces a particular product in an assembly line operation. One of the machines on the line is a drill press that has a single assembly line feeding into it. A partially completed unit arrives at the press to be worked on every 7.5 minutes, on average. The machine operator can process an average of 10 parts per hour. Determine the average number of parts waiting to be worked on, the percentage of time the operator is working, and the percentage of time the machine is idle.
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Arrival rate
Service rate
L q = Average number of products in the waiting line
U = Probability that the server was busy
I = Percentage of time the machine is idle
Given Data:
Time Interval = 7.5 min
Calculate the average number of parts waiting to be worked on, using the following formula as shown below:
The average number of parts waiting to be assembled is 3.2 parts.
The percentage of time the operator is working is calculated as shown below:
Multiplying the value of 0.8 by 100 will give us 80 percent.
The operator is working for 80 percent of the time.
In order to find the percentage of time the machine is idle use the formula below:
Multiplying the value of 0.20 by 100 will give us 20 percent.
The machine is idle for 20 percent of the time.

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The ticket booth on the Tech campus is operated by one person, who is selling tickets for the annual Tech versus State football game on Saturday. The ticket seller can serve an average of 12 customers per hour; on average, 10 customers arrive to purchase tickets each hour (Poisson distributed). Determine the average time a ticket buyer must wait and the portion of time the ticket seller is busy.
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The management of Dynaco Manufacturing Company (see Problem) likes to have its operators working 90% of the time. What must the assembly line arrival rate be for the operators to be as busy as management would like Problem The Dynaco Manufacturing Company produces a particular product in an assembly line operation. One of the machines on the line is a drill press that has a single assembly line feeding into it. A partially completed unit arrives at the press to be worked on every 7.5 minutes, on average. The machine operator can process an average of 10 parts per hour. Determine the average number of parts waiting to be worked on, the percentage of time the operator is working, and the percentage of time the machine is idle.
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All trucks traveling on Interstate 40 between Albuquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 200 per 8-hour day, and the station can weigh, on the average, 220 trucks per day. a. Determine the average number of trucks waiting, the average time spent waiting and being weighed at the weigh station by each truck, and the average waiting time before being weighed for each truck. b. If the truck drivers find out they must remain at the weigh station longer than 15 minutes, on average, they will start taking a different route or traveling at night, thus depriving the state of taxes. The state of New Mexico estimates that it loses $10,000 in taxes per year for each extra minute trucks must remain at the weigh station. A new set of scales would have the same service capacity as the present set of scales, and it is assumed that arriving trucks would line up equally behind the two sets of scales. It would cost$50,000 per year to operate the new scales. Should the state install the new set of scales
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Bob and Carol Packer operate a successful outdoorwear store in Vermont called Northwoods Backpackers. They stock mostly cold weather outdoor items such as hiking and backpacking clothes, gear, and accessories. They established an excellent reputation throughout New England for quality products and service. Eventually, Bob and Carol noticed that more and more of their sales were to customers who did not live in the immediate vicinity but were calling in orders on the telephone. As a result, the Packers decided to distribute a catalog and establish a phone-order service. The order department consisted of five operators working 8 hours per day from 10:00 a.m. to 6:00 p.m., Monday through Friday. For a few years the mail-order service was only moderately successful; the Packers just about broke even on their investment. However, during the holiday season of the third year of the catalog-order service, they were overwhelmed with phone orders. Although they made a substantial profit, they were concerned about the large number of lost sales they estimated they had incurred. Based on information provided by the telephone company regarding call volume and complaints from customers, the Packers estimated that they lost sales of approximately $100,000. Also, they felt they had lost a substantial number of old and potentially new customers because of the poor service of the catalog order department. Prior to the next holiday season, the Packers explored several alternatives for improving the catalog-order service. The current system includes the five original operators with computer terminals who work 8-hour days, 5 days per week. The Packers hired a consultant to study this system, and she reported that the time for an operator to take a customer order is exponentially distributed, with a mean of 3.6 minutes. Calls are expected to arrive at the telephone center during the 6-week holiday season, according to a Poisson distribution, with a mean rate of 175 calls per hour. When all operators are busy, callers are put on hold, listening to music until an operator can answer. Waiting calls are answered on a FIFO basis. Based on her experience with other catalog telephoneorder operations and data from Northwoods Backpackers, the consultant has determined that if Northwoods Backpackers can reduce customer call-waiting time to approximately 0.5 minute or less, the company will save$135,000 in lost sales during the coming holiday season. Therefore, the Packers have adopted this level of call service as their goal. However, in addition to simply avoiding lost sales, the Packers believe it is important to reduce waiting time in order to maintain their reputation for good customer service. Thus, they would like for about 70% of their callers to receive immediate service. The Packers can maintain the same number of workstations and computer terminals they currently have and increase their service to 16 hours per day with two operator shifts running from 8:00 a.m. to midnight. The Packers believe when customers become aware of their extended hours, the calls will spread out uniformly, resulting in a new call average arrival rate of 87.5 calls per hour (still Poisson distributed). This schedule change would cost Northwoods Backpackers approximately $11,500 for the 6-week holiday season. Another alternative for reducing customer waiting times is to offer weekend service. However, the Packers believe that if they offer weekend service, it must coincide with whatever service they offer during the week. In other words, if they have phone-order service 8 hours per day during the week, they must have the same service during the weekend; the same is true with 16-hours-per-day service. They feel that if weekend hours differ from weekday hours, it will confuse customers. If 8-hour service is offered 7 days per week, the new call arrival rate will be reduced to 125 calls per hour, at a cost of$3,600. If they offer 16-hour service, the mean call arrival rate will be reduced to 62.5 calls per hour, at a cost of $7,200. Still another possibility is to add more operator stations. Each station includes a desk, an operator, a phone, and a computer terminal. An additional station that is in operation 5 days per week, 8 hours per day, will cost$2,900 for the holiday season. For a 16-hour day, the cost per new station is $4,700. For 7-day service, the cost of an additional station for 8-hours-per-day service is$3,800; for 16-hours-per-day service, the cost is $6,300. The facility Northwoods Backpackers uses to house its operators can accommodate a maximum of 10 stations. Additional operators in excess of 10 would require the Packers to lease, remodel, and wire a new facility, which is a capital expenditure they do not want to undertake this holiday season. Alternatively, the Packers do not want to reduce their current number of operator stations. Determine what order service configuration the Packers should use to achieve their goals and explain your recommendation. Essay Answer: Tags Choose question tag In the Fast Shop Market example in this chapter, Alternative II was to add a new checkout counter at the market. This alternative was analyzed using the single-server model. Why was the multiple-server model not used Essay Answer: Tags Choose question tag Two area hospitals have jointly initiated several planning projects to determine how effectively their emergency facilities can handle disaster-related situations at nearby Tech University. These disasters could be weather related (such as a tornado), a fire, accidents (such as a gas main explosion or a building collapse), or acts of terrorism. One of these projects has focused on the transport of disaster victims from the Tech campus to the two hospitals in the area, Montgomery Regional and Radford Memorial. When a disaster occurs at Tech, emergency vehicles are dispatched from Tech police, local EMT units, hospitals, and local county and city police departments. Victims are brought to a staging area near the disaster scene and wait for transport to one of the two area hospitals. Aspects of the project analysis include the waiting times victims might experience at the disaster scene for emergency vehicles to transport them to the hospital, and waiting times for treatment once victims arrive at the hospital. The project team is analyzing various waiting line models, as follows. (Unless stated otherwise, arrivals are Poisson distributed, and service times are exponentially distributed.) a. First, consider a single-server waiting line model in which the available emergency vehicles are considered to be the server. Assume that victims arrive at the staging area ready to be transported to a hospital on average every 7 minutes and that emergency vehicles are plentiful and available to pick up and transport victims every 4.5 minutes. Compute the average waiting time for victims. Next, assume that the distribution of service times is undefined, with a mean of 4.5 minutes and a standard deviation of 5 minutes. Compute the average waiting time for the victims. b. Next, consider a multiple-server model in which there are eight emergency vehicles available for transporting victims to the hospitals, and the mean time required for a vehicle to pick up and transport a victim to a hospital is 20 minutes. (Assume the same arrival rate as in part a.) Compute the average waiting line, the average waiting time for a victim, and the average time in the system for a victim (waiting and being transported). c. For the multiple-server model in part b, now assume that there are a finite number of victims, 18. Determine the average waiting line, the average waiting time, and the average time in the system. (Note that a finite calling population model with multiple servers will require the use of the QM for Windows software.) d. From the two hospitals' perspectives, consider a multiple- server model in which the two hospitals are the servers. The emergency vehicles at the disaster scene constitute a single waiting line, and each driver calls ahead to see which hospital is most likely to admit the victim first, and travels to that hospital. Vehicles arrive at a hospital every 8.5 minutes, on average, and the average service time for the emergency staff to admit and treat a victim is 12 minutes. Determine the average waiting line for victims, the average waiting time, and the average time in the system. e. Next, consider a single hospital, Montgomery Regional, which in an emergency disaster situation has five physicians with supporting staff available. Victims arrive at the hospital on average every 8.5 minutes. It takes an emergency room team, on average, 21 minutes to treat a victim. Determine the average waiting line, the average waiting time, and the average time in the system. f. For the multiple-server model in part e, now assume that there are a finite number of victims, 23. Determine the average waiting line, the average waiting time, and the average time in the system. (Note that a finite calling population model with multiple servers will require the use of the QM for Windows software.) g. Which of these waiting line models do you think would be the most useful in analyzing a disaster situation How do you think some, or all, of the models might be used together to analyze a disaster situation What other type(s) of waiting line model(s) do you think might be useful in analyzing a disaster situation Essay Answer: Tags Choose question tag During registration at State University every semester, students in the college of business must have their courses approved by the college adviser. It takes the adviser an average of 2 minutes to approve each schedule, and students arrive at the adviser's office at the rate of 28 per hour. a. Compute L , L q , W , W q , and U. b. The dean of the college has received a number of complaints from students about the length of time they must wait to have their schedules approved. The dean feels that waiting 10.00 minutes to get a schedule approved is not unreasonable. Each assistant the dean assigns to the adviser's office will reduce the average time required to approve a schedule by 0.25 minute, down to a minimum time of 1.00 minute to approve a schedule. How many assistants should the dean assign to the adviser Essay Answer: Tags Choose question tag In Problem, suppose passing truck drivers look to see how many trucks are waiting to be weighed at the weigh station. If they see four or more trucks in line, they will pass by the station and risk being caught and ticketed. What is the probability that a truck will pass by the station Problem All trucks traveling on Interstate 40 between Albuquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 200 per 8-hour day, and the station can weigh, on the average, 220 trucks per day. a. Determine the average number of trucks waiting, the average time spent waiting and being weighed at the weigh station by each truck, and the average waiting time before being weighed for each truck. b. If the truck drivers find out they must remain at the weigh station longer than 15 minutes, on average, they will start taking a different route or traveling at night, thus depriving the state of taxes. The state of New Mexico estimates that it loses$10,000 in taxes per year for each extra minute trucks must remain at the weigh station. A new set of scales would have the same service capacity as the present set of scales, and it is assumed that arriving trucks would line up equally behind the two sets of scales. It would cost $50,000 per year to operate the new scales. Should the state install the new set of scales Essay Answer: Tags Choose question tag The copy center at the college of business at State University has become an increasingly contentious item among the college administrators. The department heads have complained to the associate dean about the long lines and waiting times for their secretaries at the copy center. They claim that it is a waste of scarce resources for the secretaries to stand in line talking when they could be doing more productive work in the office. Alternatively, Handford Burris, the associate dean, says the limited operating budget will not allow the college to purchase a new copier, or several copiers, to relieve the problem. This standoff has been going on for several years. To make her case for improved copying facilities, Lauren Moore, the department head for management science, assigned students a class project to gather some information about the copy center. The students were to record the arrivals at the center and the length of time it took to do a copy job once the secretary actually reached a copy machine. In addition, the students were to describe how the copy center system worked. When the students completed the project, they turned in a report to Professor Moore. The report described the copy center as containing two machines. When secretaries arrived for a copy job, they joined a queue, which looked more like milling around to the students. But they acknowledged that the secretaries knew when it was their turn, and, in effect, they formed a single queue for the first available copy machine. Also, because copy jobs are assigned tasks, secretaries always stayed to do the job, no matter how long the line was or how long they had to wait. They never left the queue. From the data the students gathered, Professor Moore is able to determine that secretaries arrive every 8 minutes for a copy job and that the arrival rate is Poisson distributed. Furthermore, she was able to determine that the average time it takes to complete a job is 12 minutes and that this is exponentially distributed. Using her own personnel records and some data from the university's personnel office, Dr. Moore determines that a secretary's average salary is$8.50 per hour. From her academic calendar she adds up the actual days in the year when the college and departmental offices are open and finds there are 247. However, as she adds up working days, it occurs to her that during the summer months, the workload is much lighter, and the copy center also probably gets less traffic. The summer includes about 70 days, during which she expects the copy center traffic to be about half of what it is during the normal year, but she speculates that the average time of a copying job remains about the same. Professor Moore next calls a local office supply firm to check the prices on copiers. A new copier of the type in the copy center now would cost $36,000. It would also require$8,000 per year for maintenance and would have a normal useful life of 6 years. Do you think Dr. Moore will be able to convince the associate dean that a new copier machine will be cost-effective
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a. Is the following statement true or false "The single-phase, single-channel model with Poisson arrivals and undefined service times will always have larger (i.e., greater) operating characteristic values (i.e., W , W q , L , L q ,) than the same model with exponentially distributed service times." Explain your answer. b. Is the following statement true or false "The single-phase, single-channel model with Poisson arrivals and constant service times will always have smaller (i.e., lower) operating characteristic values (i.e., W , W q , L , L q ) than the same model with exponentially distributed service times." Explain your answer.
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For each of the following queuing systems, indicate whether it is a single- or multiple-server model, the queue discipline, and whether its calling population is infinite or finite. a. Hair salon b. Bank c. Laundromat d. Doctor's office e. Adviser's office f. Airport runway g. Service station h. Copy center i. Team trainer j. Web site
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Provide an example of when a first-in, first-out (FIFO) rule for queue discipline would not be appropriate.
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Why do waiting lines form at a service facility even though there may be more than enough service capacity to meet normal demand in the long run
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The First American Bank of Rapid City has one outside drive-up teller. It takes the teller an average of 4 minutes to serve a bank customer. Customers arrive at the drive-up window at a rate of 12 per hour. The bank operations officer is currently analyzing the possibility of adding a second drive-up window, at an annual cost of $20,000. It is assumed that arriving cars would be equally divided between both windows. The operations officer estimates that each minute's reduction in customer waiting time would increase the bank's revenue by$2,000 annually. Should the second drive-up window be installed
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In the "Forecasting Airport Passenger Arrivals" case problem in Chapter, the objective is to develop a forecasting model to predict daily airline passenger arrivals for 2-hour time segments from 4:00 a.m. to 10:00 p.m. for July at Berry International Airport (BEI). Such a forecasting model is necessary in order to determine how many security gates will be needed at the South concourse during each of the daily time segments for any day in July, the airport's busiest travel month. Use the forecast developed in the Chapter case to perform this type of waiting line analysis to determine how many security checkpoints are needed during each time segment. Assume that as passengers arrive at the South concourse security gate they join a single line to have their boarding pass and identification checked at one of several stations. When passengers leave these stations they again form a single line and are approximately equally distributed among the security checkpoints by security personnel before going through the various detection machines. For July the airport plans to staff six security checkpoints for each 2-hour time segment from 4:00 a.m. to 6:00 p.m., and then three checkpoints from 6:00 p.m. to 8:00 p.m., and two checkpoints from 8:00 p.m. to 10:00 p.m. Assume that the arrival rate at this point in the security system is Poisson distributed, with the forecasted passenger arrivals developed in the Chapter case as the mean arrival rate. Further, assume that service times are exponentially distributed with a mean of 11.6 seconds. Determine whether the number of security checkpoints the airport plans to use for each 2-hour time segment is sufficient to keep passengers moving freely through the security system without excessive delays. If the current number of checkpoints is not sufficient, what is the likely result If the planned system is not likely to be sufficient, determine the number of checkpoints that would be needed for each 2-hour segment in order for passengers to move quickly through the security checkpoints without excessive waiting times.
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