Quiz 13: Queuing Analysis
Business
Q 1Q 1
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
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Essay
Queueing system is a system that has any type of waiting lines(queues). Various models can be built from the queueing system to calculate the parameters like wait time, length of the queue, etc.
Poisson process is a counting process in which events occur on a continuous basis with mean rate ?. It is represented as { X(t); t = 0}.
a.
In a single-server waiting line model, calculate the average waiting time in the queue using the following formula:
Since a victim arrives after every 7 minutes, calculate
as follows:
Since a victim is serviced after every 4.5 minutes, calculate
as follows:
Calculate the waiting time as follows:
Thus, the average waiting time for victims will be
.
Now, when the distribution of service times is undefined,
= 9 as aforementioned and calculate
as follows:
Compute the average time using the following formula:
Here,
= expected number of victims in the queue
Now, compute
using the following formula:
Obtain the values of
and
as follows:
Thus, the average waiting time for victims when the distribution of service times is undefined will be
.
b.
Here, a multi-server model is considered with
= 9 as aforementioned and calculate
as follows:
Calculate the expected number of customers in the system using the following formula:
Here, compute
using the following formula:
Here,
Calculate
and
as shown below:
Calculate the average waiting time for a victim in the system as follows:
Calculate the average waiting time for a victim in the queue as follows:
c.
Here,
= 18 (number of victims)
Now, calculate the average waiting time for a victim in the system as follows:
Calculate the average number of victims in the queue as follows:
Calculate the average waiting time for a victim in the queue as follows:
d.
Here, s = 2 (number of servers)
Now, calculate
and
as follows:
Calculate
and
as shown below:
Now, calculate the average waiting time for a victim in the system as follows:
Calculate the average number of victims in the queue as follows:
Calculate the average waiting time for a victim in the queue as follows:
e.
This is a multiple server model with s = 5 (number of physicians).
Since a victim arrives after every 8.5 minutes, calculate
as follows:
Calculate
as follows:
Now, compute
and
as shown below:
Calculate the average waiting time for a victim in the system as follows:
Calculate the average waiting time for a victim in the queue as follows:
f.
Here,
= 23 (number of victims)
Now, calculate the average waiting time for a victim in the system as follows:
Calculate the average number of victims in the queue as follows:
Calculate the average waiting time for a victim in the queue as follows:
g.
The most useful model in analyzing the disaster situation will be model e (multi-server model with five physicians in hospital M) since it has the least waiting time for the victims in the queue and the system. Also, the length of the waiting line is the least in the model.
To use other models together to analyze the disaster situation , the pros and cons of each model needs to be adjusted so that the overall waiting time and the length of the waiting queue is minimum. This can be done by incorporating more physicians in model b or more emergency vehicles in model c.
Other type of waiting line models that might be useful in analyzing the disaster situation will be as follows:
• Jockeying - This means the movement between the queues is possible which occurs when there are more than one servers which in turn is preceded by queue.
• The service time of the victim is not exponentially distributed or undefined.
• Service is provided to the victims not in first-come first basis but on the basis of alphabetical order.
• There can also be a model in which victims can be shifted to another hospital if the queue is very long in the hospital to which the patient is first taken to.
Q 2Q 2
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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Essay
BEI airport has a problem in staffing and security checking. In the peak season it is having the problem of long waiting lines and waiting times. BEI decided to forecast the airport passenger arrival in order to find out the number of check points that are needed to avoid the rush and waiting time for 2 hour time segment. Service times are exponentially distributed with a mean of 11.6 seconds.
The operational characteristics for the above problem can be calculated using the single server model.
The probability that no customers are in the queuing system (either in the queue or being served)
Calculation of the security check points for two hour time segment-
By applying the above formulas we can now calculate the values of Po, Pw, L, Lq, W and Wq.
• 4:00 A.M to 6:00 A.M
The number of check points required is
.
• 6:00 A.M to 8:00 A.M
The number of check points required is
.
• 8:00 A.M to 10:00 AM
The number of check points required is
•
10:00 A.M to NOON
The number of check points required is
• NOON- 2:00 PM
The number of check points required is
• 2:00 P.M TO 4:00 P.M
The number of check points required is
• 4:00 P.M TO 6:00 P.M
The number of check points required is
.
• 6:00 P.M to 8:00 P.M
The number of check points required is
• 8:00 P.M to 10:00 P.M
The number of check points required is
.
The number of security checkpoints is sufficient to keep passengers move freely through the security systems without excessive delays.
Q 3Q 3
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.
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Essay
Determine the order service configuration to achieve their set goals of meeting the targets in the form of Excel Spreadsheet.
As per the stated case, the current system of achieving the company goals is not feasible and the management of NB does not think if they work 8hours, 5 days per week, they would be able to serve the customers effectively.
Calculate the second alternative that is 7 days and 8 hours service in the Excel spreadsheet.
The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls.
Insert the formulas in the following way:
The terms for 9 days service are also shown below:
Calculate the third alternative that is 5 days and 16 hours service in the Excel spreadsheet.
The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls.
Insert the formulas in the following way:
The terms for 7 days service are also shown below:
Calculate the fourth alternative that is 7 days and 16 hours service in the Excel spreadsheet.
The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls.
The terms for 5 days service are also shown below:
Among all the alternatives, second alternative would be the most feasible option for the company P. The cost incurred in meeting the targets is the least that is $22,600 among all the other alternatives.
Q 4Q 4
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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Q 5Q 5
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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Q 6Q 6
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
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Q 7Q 7
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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Q 8Q 8
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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Q 9Q 9
Provide an example of when a first-in, first-out (FIFO) rule for queue discipline would not be appropriate.
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Q 10Q 10
Under what conditions will the single-server queuing model with Poisson arrivals and undefined service times provide the same operating characteristics as the basic model with exponentially distributed service times
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Q 11Q 11
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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Q 12Q 12
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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Q 13Q 13
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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Q 14Q 14
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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Q 15Q 15
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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Q 16Q 16
The Peachtree Airport in Atlanta serves light aircraft. It has a single runway and one air traffic controller to land planes. It takes an airplane 12 minutes to land and clear the runway. Planes arrive at the airport at the rate of 4 per hour.
a. Determine the average number of planes that will stack up, waiting to land.
b. Find the average time a plane must wait in line before it can land.
c. Calculate the average time it takes a plane to clear the runway once it has notified the airport that it is in the vicinity and wants to land.
d. The FAA has a rule that an air traffic controller can, on average, land planes a maximum of 45 minutes out of every hour. There must be 15 minutes of idle time available to relieve the tension. Will this airport have to hire an extra air traffic controller
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Q 17Q 17
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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Q 18Q 18
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
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Q 19Q 19
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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Q 20Q 20
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
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Q 21Q 21
In Problem, the dean of the college of business at State University is considering the addition of a second adviser in the college advising office to serve students waiting to have their schedules approved. This new adviser could serve the same number of students per hour as the present adviser. Determine L , L q , W , and W q for this altered advising system. As a student, would you recommend adding the adviser
Problem
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
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Q 22Q 22
The Dynaco Manufacturing Company has an assembly line that feeds two drill presses. As partially completed products come off the line, they are lined up to be worked on as drill presses become available. The units arrive at the workstation (containing both presses) at the rate of 100 per hour. Each press operator can process an average of 60 units per hour. Compute L , L q , W , and W q.
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Q 23Q 23
The Acme Machine Shop has five machines that periodically break down and require service. The average time between breakdowns is 4 days, distributed according to an exponential distribution. The average time to repair a machine is 1 day, distributed according to an exponential distribution. One mechanic repairs the machines in the order in which they break down.
a. Determine the probability that the mechanic is idle.
b. Determine the mean number of machines waiting to be repaired.
c. Determine the mean time machines wait to be repaired.
d. Determine the probability that three machines are not operating (are being repaired or waiting to be repaired).
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Q 24Q 24
McBurger's fast-food restaurant has a drive-through window with a single server who takes orders from an intercom and also is the cashier. The window operator is assisted by other employees who prepare the orders. Customers arrive at the ordering station prior to the drive-through window every 4.5 minutes (Poisson distributed), and the service time is 2.8 minutes (exponentially distributed). Determine the average length of the waiting line and the waiting time. Discuss the quality of the service.
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Q 25Q 25
Game World, a video game arcade at Tanglewood Mall, has just installed a new virtual reality battle game. It requires exactly 2.7 minutes to play. Customers arrive, on average, every 2.9 minutes (Poisson distributed) to play the game. How long will the line of customers waiting to play the game be, and how long, on average, must a customer wait
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Q 26Q 26
Drivers who come to get their licenses at the Department of Motor Vehicles have their photograph taken by an automated machine that develops the photograph onto the license card and laminates the complete license. The machine requires a constant time of 4.5 minutes to prepare a completed license. If drivers arrive at the machine at the mean rate of 10 per hour (Poisson distributed), determine the average length of the waiting line and the average waiting time.
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Q 27Q 27
A vending machine at City Airport dispenses hot coffee, hot chocolate, or hot tea, in a constant service time of 20 seconds. Customers arrive at the vending machine at a mean rate of 60 per hour (Poisson distributed). Determine the average length of the waiting line and the average time a customer must wait.
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Q 28Q 28
Norfolk, Virginia, a major seaport on the East Coast, has a ship coal-loading facility. Coal trucks filled with coal arrive at the port facility at the mean rate of 149 per day (Poisson distributed). The facility operates 24 hours a day. The coal trucks are unloaded one at a time, on a first-come, firstserved basis, by automated mechanical equipment that empties the trucks in a constant time of 8 minutes per truck, regardless of truck size. The port authority is negotiating with a coal company for an additional 30 trucks per day. However, the coal company will not use this port facility unless the port authority can assure it that its coal trucks will not have to wait to be unloaded at the port facility for more than 12 hours per truck, on average. Can the port authority provide this assurance
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Q 29Q 29
The Bay City Police Department has eight patrol cars that are on constant call 24 hours per day. A patrol car requires repairs every 20 days, on average, according to an exponential distribution. When a patrol car is in need of repair, it is driven into the motor pool, which has a repair person on duty at all times. The average time required to repair a patrol car is 18 hours (exponentially distributed). Determine the average time a patrol car is not available for use and the average number of patrol cars out of service at any one time. Indicate whether the repair service seems adequate.
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Q 30Q 30
The Rowntown Cab Company has four cabs that are on duty during normal business hours. The cab company dispatcher receives requests for service every 8 minutes, on average, according to an exponential distribution. The average time to complete a trip is 20 minutes (exponentially distributed). Determine the average number of customers waiting for service and the average time a customer must wait for a cab.
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Essay
Q 31Q 31
In Problem, in addition to course approvals, suppose that the advisor also answers questions and provides course counseling, such that the service time can no longer be assumed to be exponentially distributed. Instead, the service time distribution is undefined (i.e., general), with a mean of 2 minutes and a standard deviation of 5 minutes. Compute L , L q , W , and W q and compare these results with those in part (a) in Problem.
Problem
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
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Essay
Q 32Q 32
The College Avenue Sub Shop is located in a small college town. Virtually all of its phone delivery orders are in the evening, between 5:00 p.m. and 1:00 a.m., so the shop has a delivery person for that time period. It receives an average of about 3 orders per hour (Poisson distributed). However, because of the different distances the delivery person must travel, the service time is undefined, with a mean of 15 minutes and a standard deviation of 6 minutes. Compute L , L q , W , and W q.
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Essay
Q 33Q 33
The Riverton Post Office has four stations for service. Customers line up in single file for service on a FIFO basis. The mean arrival rate is 40 per hour, Poisson distributed, and the mean service time per server is 4 minutes, exponentially distributed. Compute the operating characteristics for this operation. Indicate whether the operation appears to be satisfactory in terms of the following: (a) postal workers' (servers') idle time; (b) customer waiting time and/or the number waiting for service; and (c) the percentage of the time a customer can walk in and get served without waiting at all.
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Essay
Q 34Q 34
In Problem, the Dynaco Company has found that if more than three units (average) are waiting to be processed at any one workstation, then too much money is being tied up in work-inprocess inventory (i.e., units waiting to be processed). The company estimates that (on average) each unit waiting to be processed costs $50 per day. Alternatively, operating a third press would cost $150 per day. Should the company operate a third press at this workstation
Problem
The Dynaco Manufacturing Company has an assembly line that feeds two drill presses. As partially completed products come off the line, they are lined up to be worked on as drill presses become available. The units arrive at the workstation (containing both presses) at the rate of 100 per hour. Each press operator can process an average of 60 units per hour. Compute L , L q , W , and W q.
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Essay
Q 35Q 35
Cakes baked by the Freshfood Bakery are transported from the ovens to be packaged by one of three wrappers. Each wrapper can wrap an average of 200 cakes per hour. The cakes are brought to the wrappers at the rate of 500 per hour. If a cake sits longer than 5 minutes before being wrapped, it will not be fresh enough to meet the bakery's quality control standards. Does the bakery need to hire another wrapper
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Essay
Q 36Q 36
The associate dean of the college of business at Tech is determining which of two copiers he should lease for the college's administrative suite. A regular copier leases for $8 per hour, and it takes an employee an average of 6 minutes (exponentially distributed) to complete a copying job. A deluxe, high-speed copier leases for $16 per hour, and it requires an average of 3 minutes to complete a copying job. Employees arrive at the copying machine at a rate of 7 per hour (Poisson distributed), and an employee's time is valued at $10 per hour. Determine which copier the college should lease.
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Essay
Q 37Q 37
The Quick Wash 24-hour Laundromat has 16 washing machines. A machine breaks down every 20 days (exponentially distributed). The repair service with which the Laundromat contracts takes an average of 1 day to repair a machine (exponentially distributed). A washing machine averages $5 per hour in revenue. The Laundromat is considering a new repair service that guarantees repairs in 0.50 day, but it charges $10 more per hour than the current repair service. Should the Laundromat switch to the new repair service
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Essay
Q 38Q 38
The Regency Hotel has enough space at its entrance for six taxicabs to line up, wait for guests, and then load passengers. Cabs arrive at the hotel every 8 minutes; if a taxi drives by the hotel and the line is full, it must drive on. Hotel guests require a taxi every 5 minutes, on average. It takes a cab driver an average of 3.5 minutes to load passengers and luggage and leave the hotel (exponentially distributed).
a. What is the average time a cab must wait for a fare
b. What is the probability that the line will be full when a cab drives by, causing it to drive on
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Essay
Q 39Q 39
The local Burger Doodle fast-food restaurant has a drive-through window. Customers in cars arrive at the window at the rate of 10 per hour (Poisson distributed). It requires an average of 4 minutes (exponentially distributed) to take and fill an order. The restaurant chain has a service goal of an average waiting time of 3 minutes.
a. Will the current system meet the restaurant's service goal
b. If the restaurant is not meeting its service goal, it can add a second drive-through window that will reduce the service time per customer to 2.5 minutes. Will the additional window enable the restaurant to meet its service goal
c. During the 2-hour lunch period, the arrival rate of drive-in customers increases to 20 per hour. Will the two-window system be able to achieve the restaurant's service goal during this rush period
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Essay
Q 40Q 40
From 3:00 p.m. to 8:00 p.m. the local K-Star Supermarket has a steady stream of customers. Customers finish shopping and arrive at the checkout area at a rate of 70 per hour (Poisson distributed). It is assumed that when customers arrive at the cash registers, they will divide themselves relatively evenly so that all the checkout lines contain an equal number. The average checkout time at a register is 7 minutes (exponentially distributed). The store manager's service goal is for customers to be out of the store within 12 minutes (on average) after they complete their shopping and arrive at a cash register. How many cash registers must the store have open to achieve the manager's service goal
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Essay
Q 41Q 41
Customers arrive to check in at the exclusive and expensive Regency Hotel's lobby at a rate of 40 per hour (Poisson distributed). The hotel normally has three clerks available at the desk to check in guests. The average time for a clerk to check in a guest is 4 minutes (exponentially distributed). Clerks at the Regency are paid $24 per hour, and the hotel assigns a goodwill cost of $2 per minute for the time a guest must wait in line. Determine whether the present check-in system is cost-effective. If it is not, recommend what hotel management should do.
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Essay
Q 42Q 42
The Riverview Clinic has two general practitioners who see patients daily. An average of six patients arrive at the clinic per hour. Each doctor spends an average of 15 minutes with a patient. The patients wait in a waiting area until one of the two doctors is able to see them. However, because patients typically do not feel well when they come to the clinic, the doctors do not believe it is good practice to have a patient wait longer than an average of 15 minutes. Should this clinic add a third doctor, and, if so, will this alleviate the waiting problem
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Essay
Q 43Q 43
The Footrite Shoe Company is going to open a new branch at a mall, and company managers are attempting to determine how many salespeople to hire. Based on an analysis of mall traffic, the company estimates that customers will arrive at the store at a rate of 10 per hour, and from past experience at its other branches, the company knows that salespeople can serve an average of 6 customers per hour. How many salespeople should the company hire to uphold a company policy that on average the probability of a customer having to wait for service be no more than.30
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Essay
Q 44Q 44
When customers arrive at Gilley's Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers, the store's staff knows that customers arrive at a rate of 40 per hour on summer days between 3:00 p.m. and 10:00 p.m., and a server can serve 15 customers per hour on average. Gilley's wants to make sure that customers wait no longer than 10 minutes for service. Gilley's is contemplating keeping three servers behind the ice cream counter during the peak summer hours. Will this number be adequate to meet the waiting time policy
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Essay
Q 45Q 45
Moore's television repair service receives an average of six TV sets per 8-hour day to be repaired. The service manager would like to be able to tell customers that they can expect 1-day service. What average repair time per set will the repair shop have to achieve to provide 1-day service, on average (Assume that the arrival rate is Poisson distributed and repair times are exponentially distributed.)
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Essay
Q 46Q 46
In Problem, suppose that Moore's television repair service cannot accommodate more than 30 TV sets at a time. What is the probability that the number of TV sets on hand (under repair and waiting for service) will exceed the shop capacity
Problem
Moore's television repair service receives an average of six TV sets per 8-hour day to be repaired. The service manager would like to be able to tell customers that they can expect 1-day service. What average repair time per set will the repair shop have to achieve to provide 1-day service, on average (Assume that the arrival rate is Poisson distributed and repair times are exponentially distributed.)
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Essay
Q 47Q 47
Maggie Attaberry is a nurse on the evening shift from 10:00 p.m. to 6:00 a.m. at Community Hospital. She has 15 patients for whom she is responsible in her area. She averages two calls from each of her patients every evening, on average (Poisson distributed), and she must spend an average of 10 minutes (negative exponential distribution) with each patient who calls. Nurse Attaberry has indicated to her shift supervisor that, although she has not kept records, she believes her patients must wait about 10 minutes, on average, for her to respond, and she has requested that her supervisor assign a second nurse to her area. The supervisor believes 10 minutes is too long for a patient to wait, but she does not want her nurses to be idle more than 40% of the time. Determine what the supervisor should do.
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Essay
Q 48Q 48
The Escargot is a small French restaurant with six waiters and waitresses. The average service time at the restaurant for a table (of any size) is 85 minutes (Poisson distributed). The restaurant does not take reservations, and parties arrive for dinner (and stay and wait) every 18 minutes (negative exponential distribution). The restaurant owner is concerned that a lengthy waiting time might hurt its business in the long run. What are the current waiting time and queue length for the restaurant Discuss the business implications of the current waiting time and any actions the restaurant owner might take.
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Essay
Q 49Q 49
Hudson Valley Books is a small, independent publisher of fiction and nonfiction books. Each week the publisher receives an average of seven unsolicited manuscripts to review (Poisson distributed). The publisher has 12 freelance reviewers in the area who read and evaluate manuscripts. It takes a reviewer an average of 10 days (exponentially distributed) to read a manuscript and write a brief synopsis. (Reviewers work on their own, 7 days a week.) Determine how long the publisher must wait, on average, to receive a reviewer's manuscript evaluation, how many manuscripts are waiting to be reviewed, and how busy the reviewers are.
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Essay
Q 50Q 50
Amanda Fall is starting up a new house painting business, Fall Colors. She has been advertising in the local newspaper for several months. Based on inquiries and informal surveys of the local housing market, she anticipates that she will get painting jobs at the rate of 4 per week (Poisson distributed). Amanda has also determined that it will take a four-person team of painters an average of 0.7 week (exponentially distributed) for a typical painting job.
a. Determine the number of teams of painters Amanda needs to hire so that customers will have to wait no longer than 2 weeks to get their houses painted.
b. If the average price for a painting job is $1,700 and Amanda pays a team of painters $500 per week, will she make any money
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Essay
Q 51Q 51
Partially completed products arrive at a workstation in a manufacturing operation at a mean rate of 40 per hour (Poisson distributed). The processing time at the workstation averages 1.2 minutes per unit (exponentially distributed). The manufacturing company estimates that each unit of work-in-process inventory at the workstation costs $31 per day (on average). However, the company can add extra employees and reduce the processing time to 0.90 minute per unit, at a cost of $52 per day. Determine whether the company should continue the present operation or add extra employees.
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Essay
Q 52Q 52
The Atlantic Coast Shipping Company has a warehouse terminal in Spartanburg, South Carolina. The capacity of each terminal dock is three trucks. As trucks enter the terminal, the drivers receive numbers; and when one of the three dock spaces becomes available, the truck with the lowest number enters the vacant dock. Truck arrivals are Poisson distributed, and the unloading and loading times (service times) are exponentially distributed. The average arrival rate at the terminal is five trucks per hour, and the average service rate per dock is two trucks per hour (30 minutes per truck).
a. Compute L , L q , W , and W q.
b. The management of the shipping company is considering adding extra employees and equipment to improve the average service time per terminal dock to 25 minutes per truck. It would cost the company $18,000 per year to achieve this improved service. Management estimates that it will increase its profit by $750 per year for each minute it is able to reduce a truck's waiting time. Determine whether management should make the investment.
c. Now suppose that the managers of the shipping company have decided that truck waiting time from part (a) is excessive and they want to reduce the waiting time. They have determined that there are two alternatives available for reducing the waiting time. They can add a fourth dock, or they can add extra employees and equipment at the existing docks, which will reduce the average service time per location from the original 30 minutes per truck to 23 minutes per truck. The costs of these alternatives are approximately equal. Management desires to implement the alternative that reduces waiting time by the greatest amount. Which alternative should be selected (Computer solution is suggested.)
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Essay
Q 53Q 53
The Waterfall Buffet in the lower level of the National Art Gallery serves food cafeteria-style daily to visitors and employees. The buffet is self-service. From 7:00 a.m. to 9:00 a.m. customers arrive at the buffet at a rate of 10 per minute; from 9:00 a.m. to noon, at 4 per minute; from noon to 2:00 p.m., at 14 per minute; and from 2:00 p.m. to closing at 5:00 p.m., at 8 per minute. All the customers take about the same amount of time to serve themselves from the buffet. Once a customer goes through the buffet, it takes 0.4 minute to pay the cashier. The gallery does not want a customer to have to wait longer than 4 minutes to pay. How many cashiers should be working at each of the four times during the day
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Essay
Q 54Q 54
The Clip Joint is a hairstyling salon at University Mall. Four stylists are always available to serve customers on a first-come, first-served basis. Customers arrive at an average rate of 5 per hour, and the stylists spend an average of 35 minutes on each customer.
a. Determine the average number of customers in the salon, the average time a customer must wait, and the average number waiting to be served.
b. The salon manager is considering adding a fifth stylist. Would this have a significant impact on waiting time
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Essay
Q 55Q 55
Ships arrive at the port of Savannah at the rate of 40 per 30-day month (Poisson distributed), generally with a full cargo of containers. It takes an average of 12 hours (exponentially distributed) to unload and load an average size container ship. Currently the port has one general use terminal with sufficient quay cranes for loading and unloading.
a. Determine the average number of ships waiting for a vacant terminal berth, and the waiting time for a ship.
b. Ship traffic at the port is estimated to increase to 60 ships per month next year. Will the port be able to handle the additional traffic If not, how many terminals will the port need to handle the additional traffic
c. If the ship traffic at the port increases to 80 ships per month in the year after next, and using the number of terminals determined in part b, how will that affect ship waiting times
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Essay
Q 56Q 56
The Delacroix Inn in Alexandria is a small, exclusive hotel with 20 rooms. Guests can call housekeeping from 8:00 a.m. to midnight for any of their service needs. Housekeeping keeps one person on duty during this time to respond to guest calls. Each room averages 0.7 call per day to housekeeping (Poisson distributed), and a guest request requires an average of 30 minutes (exponentially distributed) for the staff person to respond to and take care of. Determine the portion of time the staff person is busy and how long a guest must wait for his or her request to be addressed. Does the housekeeping system seem adequate
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Essay
Q 57Q 57
Jim Carter builds custom furniture, primarily cabinets, bookcases, small tables, and chairs. He works on only one piece of furniture for a customer at a time. It takes him an average of 5 weeks (exponentially distributed) to build a piece of furniture. An average of 14 customers approach Jim to order pieces of furniture each year (Poisson distributed); however, Jim will take only a maximum of 8 advance orders. Determine the average time a customer must wait to receive a furniture order once it is placed and how busy Jim is. What is the probability that a customer will be able to place an order with Jim
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Essay
Q 58Q 58
Judith Lewis is a doctoral student at State University, and she also works full-time as an academic tutor for 10 scholarshipped student athletes. She took the job, hoping it would leave her free time between tutoring to devote to her own studies. An athlete visits her for tutoring an average of every 16 hours, and she spends an average of 1.5 hours (exponentially distributed) with him or her. She is able to tutor only one athlete at a time, and athletes study while they are waiting.
a. Determine how long a player must wait to see her and the percentage of time Judith is busy. Does the job seem to meet Judith's expectations, and does the system seem adequate to meet the athletes' needs
b. If the results in (a) indicate that the tutoring arrangement is ineffective, suggest an adjustment that could improve it for both the athletes and Judith.
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Essay
Q 59Q 59
TSA security agents at the security gate at the Tri-Cities Regional Airport are able to check passengers and process them through the security gate at a rate of 50 per hour (Poisson distributed). Passengers arrive at the gate throughout the day at a rate of 40 per hour (Poisson distributed). Determine the average waiting time and waiting line.
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Essay
Q 60Q 60
Tech has a student population of 30,000 and is located in a small college town in Virginia. DirectCast Cable TV has a small service staff or three installation trucks and technicians that is sufficient to handle service calls and installations for almost the entire year. However, for the month-long period right before and during the beginning of fall semester in August, when all the students return, the cable TV service is overwhelmed. During the normal year DirectCast averages about 15 service requests per day (Poisson distributed), and service calls average about 50 minutes (exponentially distributed) during their 8-hour workday. However, during the month before school starts, the demand for service calls triples, and almost all of these are requests for installations, so the average service time increases to 80 minutes. During normal demand periods, DirectCast guarantees service within a 24-hour period.
a. What will the cable service and customer waiting time be like during the month before the semester starts
b. It is possible for DirectCast to "borrow" trucks and crews from other company offices. How many extra trucks would it need to provide its normal service guarantee
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Essay
Q 61Q 61
In Problem, as an alternative to borrowing trucks and crews from other offices (which can be costly), DirectCast is considering extending the normal workday to 16 hours and paying its own technicians overtime. Would this satisfy demand to the extent that additional trucks and crews would not have to be borrowed from other offices Discuss the financial information DirectCast would likely have to consider in making these decisions.
Problem
Tech has a student population of 30,000 and is located in a small college town in Virginia. DirectCast Cable TV has a small service staff or three installation trucks and technicians that is sufficient to handle service calls and installations for almost the entire year. However, for the month-long period right before and during the beginning of fall semester in August, when all the students return, the cable TV service is overwhelmed. During the normal year DirectCast averages about 15 service requests per day (Poisson distributed), and service calls average about 50 minutes (exponentially distributed) during their 8-hour workday. However, during the month before school starts, the demand for service calls triples, and almost all of these are requests for installations, so the average service time increases to 80 minutes. During normal demand periods, DirectCast guarantees service within a 24-hour period.
a. What will the cable service and customer waiting time be like during the month before the semester starts
b. It is possible for DirectCast to "borrow" trucks and crews from other company offices. How many extra trucks would it need to provide its normal service guarantee
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Essay
Q 62Q 62
In Problem, when the students come back for fall semester, the Sub Shop experiences a large increase in demand for about a month before it begins to level off again, such that the number of phone orders increases to an average of 8 per hour (Poisson distributed). If the shop now assumes that service times are exponentially distributed, with a mean of 20 minutes, how many delivery people will it need to provide a reasonable service level
Problem
The College Avenue Sub Shop is located in a small college town. Virtually all of its phone delivery orders are in the evening, between 5:00 p.m. and 1:00 a.m., so the shop has a delivery person for that time period. It receives an average of about 3 orders per hour (Poisson distributed). However, because of the different distances the delivery person must travel, the service time is undefined, with a mean of 15 minutes and a standard deviation of 6 minutes. Compute L , L q , W , and W q.
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Essay
Q 63Q 63
In Problem, the passenger traffic arriving at the airport security gate varies significantly during the day. At times close to flight takeoffs, the traffic is very heavy, while at other times there is little or no passenger traffic through the security gate. At the times leading up to flight takeoffs, passengers arrive at a rate of 110 per hour (Poisson distributed). Suggest a waiting line system that can accommodate this level of passenger traffic.
Problem
TSA security agents at the security gate at the Tri-Cities Regional Airport are able to check passengers and process them through the security gate at a rate of 50 per hour (Poisson distributed). Passengers arrive at the gate throughout the day at a rate of 40 per hour (Poisson distributed). Determine the average waiting time and waiting line.
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Essay
Q 64Q 64
The inland port at Front Royal, Virginia, is a transportation hub that primarily transfers shipping containers from trucks coming from eastern seaports to rail cars for shipment to inland destinations. Containers arrive at the inland ports at a rate of 8 per hour. It takes 28 minutes (exponentially distributed) for a crane to unload a container from a truck, place it on a flatbed railcar, and secure it. Suggest a waiting line system that will effectively handle this level of container traffic at the inland port.
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Essay
Q 65Q 65
The Old Colony theme park has a new ride, the Double Cyclone. The ride holds 30 people in double roller coaster cars, and it takes 3.8 minutes to complete the ride circuit. It also takes the ride attendants another 3 minutes (with virtually no variation) to load and unload passengers. Passengers arrive at the ride during peak park hours at a rate of 4 per minute (Poisson distributed). Determine the length of the waiting line for the ride.
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Essay