CHAPTER 14 Waiting Lines and Queuing Theory Models

14.81 According to the information provided in Table 14-3, which presents a queuing problem solution what is the utilization rate of the service facility?

(a) 0.111

(b) 0.889

(c) 0.222

(d) 0.722

(e) 0.667

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14.82 Little’s Flow Equations are transferable to a production environment. Which of the following would be a proper interpretation of Little’s Flow Equations?

(a) Flow Rate = Inventory × Flow Time

(b) Flow Time = Inventory × Flow Rate

(c) Inventory = Flow Rate × Flow Time

(d) Time to Take an Order = Flow Rate × Flow Time

(e) Flow Rate = Time to Take an Order × Flow Time

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14.83 If everything else remains constant, including the mean arrival rate and service rate, except that the service time becomes constant instead of exponential,

(a) the average queue length will be halved.

(b) the average waiting time will be doubled.

(c) the average queue length will be doubled.

(d) There is not enough information to know what will happen to the queue length and waiting time.

(e) none of the above

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14.84 At an automatic car wash, cars arrive randomly at a rate of 7 cars every 30 minutes. The car wash takes exactly 4 minutes (this is constant). On average, what would the length of the line be?

(a) 8.171

(b) 7.467

(c) 6.53

(d) 0.467

(e) none of the above

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14.85 At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant). On average, how long would each car spend at the car wash?

(a) 28 minutes

(b) 32 minutes

(c) 17 minutes

(d) 24 minutes

(e) none of the above

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14.86 At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant). On average, how long would each driver have to wait before receiving service?

(a) 28 minutes

(b) 32 minutes

(c) 17 minutes

(d) 24 minutes

(e) none of the above

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14.87 At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant). On average, how many customers would be at the car wash (waiting in line or being serviced)?

(a) 8.17

(b) 7.46

(c) 6.53

(d) 0.46

(e) none of the above

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14.88 A(An) ___________ state is the normal operating condition of the queuing system.

(a) primary

(b) transient

(c) NOC

(d) balanced

(e) steady

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14.89 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food joint takes exactly 2 minutes (this is constant). The average total time in the system is

(a) 5.4 minutes.

(b) 6.0 minutes.

(c) 8.0 minutes.

(d) 2.5 minutes.

(e) none of the above

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Table 14-4
M/D/1
Mean Arrival Rate:	3 occurrences per minute
Constant Service Rate:	4 occurrences per minute
Queue Statistics:
Mean Number of Units in the System:	1.875
Mean Number of Units in the Queue:	1.125
Mean Time in the System:	0.625 minutes
Mean Time in the Queue:	0.375 minutes
Service Facility Utilization Factor:	0.750
Probability of No Units in System:	0.250

14.90 According to the information provided in Table 14-4, which presents a queuing problem solution for a queuing problem with a constant service rate, on average, how much time is spent waiting in line?

(a) 1.875 minutes

(b) 1.125 minutes

(c) 0.625 minutes

(d) 0.375 minutes

(e) none of the above

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14.91 According to the information provided in Table 14-4, which presents a queuing problem solution for a queuing problem with a constant service rate, on average, how many customers are in the system?

(a) 1.875

(b) 1.125

(c) 0.625

(d) 0.375

(e) none of the above

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14.92 According to the information provided in Table 14-4, which presents a queuing problem solution for a queuing problem with a constant service rate, on average, how many customers arrive per time period?

(a) 3

(b) 4

(c) 1.875

(d) 1.125

(e) none of the above

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14.93 According to Table 14-4, which presents a queuing problem with a constant service rate, on average, how many minutes does a customer spend in the service facility?

(a) 0.375 minutes

(b) 4 minutes

(c) 0.625 minutes

(d) 0.25 minutes

(e) none of the above

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Table 14-5
M/D/1
Mean Arrival Rate:	5 occurrences per minute
Constant Service Rate:	7 occurrences per minute
Queue Statistics:
Mean Number of Units in the System:	1.607
Mean Number of Units in the Queue:	0.893
Mean Time in the System:	0.321 minutes
Mean Time in the Queue:	0.179 minutes
Service Facility Utilization Factor:	0.714

14.94 According to the information provided in Table 14-5, which presents the solution for a queuing problem with a constant service rate, on average, how much time is spent waiting in line?

(a) 1.607 minutes

(b) 0.714 minutes

(c) 0.179 minutes

(d) 0.893 minutes

(e) none of the above

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14.95 According to the information provided in Table 14-5, which presents the solution for a queuing problem with a constant service rate, on average, how many customers are in the system?

(a) 0.893

(b) 0.714

(c) 1.607

(d) 0.375

(e) none of the above

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14.96 According to the information provided in Table 14-5, which presents a queuing problem solution for a queuing problem with a constant service rate, on average, how many customers arrive per time period?

(a) 5

(b) 7

(c) 1.607

(d) 0.893

(e) none of the above

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14.97 According to the information provided in Table 14-5, which presents the solution for a queuing problem with a constant service rate, on average, how many minutes does a customer spend in the system?

(a) 0.893 minutes

(b) 0.321 minutes

(c) 0.714 minutes

(d) 1.607 minutes

(e) none of the above

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14.98 According to the information provided in Table 14-5, which presents the solution for a queuing problem with a constant service rate, what percentage of available service time is actually used?

(a) 0.217

(b) 0.643

(c) 0.321

(d) 0.179

(e) none of the above

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14.99 According to the information provided in Table 14-5, which presents the solution for a queuing problem with a constant service rate, the probability that the server is idle is __________

(a) 0.217.

(b) 0.643.

(c) 0.286.

(d) 0.714.

(e) none of the above.

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14.100 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the probability the system is empty?

(a) 1.1500

(b) 1.1658

(c) .8578

(d) .8696

(e) none of the above

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14.101 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the average number of fax machines in the queue?

(a) 1.1658 fax machines

(b) 2.9904 fax machines

(c) .1563 fax machines

(d) .0142 fax machines

(e) none of the above

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14.102 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the average number of fax machines in the system?

(a) .0142 fax machines

(b) .1563 fax machines

(c) .0249 fax machines

(d) .2749 fax machines

(e) none of the above

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14.103 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the average waiting time in the queue?

(a) .0142 hours

(b) .1563 hours

(c) .0249 hours

(d) .2749 hours

(e) none of the above

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14.104 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the average time spent in the system?

a) .0142 hours

b) .1563 hours

c) .0249 hours

d) .2749 hours

e) none of the above

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14.105 The school of business has 3 fax machines. The toner in each machine needs to be changed after about 5 hours of use. There is one unit secretary who is responsible for the fax machine maintenance. It takes him 15 minutes to replace the toner cartridge. What is the probability that 2 fax machines need toner at the same time?

a).8576

b).1286

c).0129

d).1415

e)none of the above

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PROBLEMS

14.106 A new shopping mall is considering setting up an information desk manned by one employee. Based upon information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 15 per hour. It takes an average of two minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed.

(a) Find the probability that the employee is idle.

(b) Find the proportion of time that the employee is busy.

(c) Find the average number of people receiving and waiting to receive information.

(d) Find the average number of people waiting in line to get information.

(e) Find the average time a person seeking information spends at the desk.

(f) Find the expected time a person spends waiting in line to have his question answered.

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14.107 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 6 minutes, according to a Poisson distribution. Also assume that Sam's vaccinating times are exponentially distributed. Find the:

(a) probability that Sam is idle.

(b) proportion of time that Sam is busy.

(c) average number of dogs receiving or waiting to be vaccinated.

(d) average number of dogs waiting to be vaccinated.

(e) average time a dog waits before getting vaccinated.

(f) average amount (mean) of time a dog spends between waiting in line and getting vaccinated.

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14.108 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 4 minutes, according to a Poisson distribution. Also assume that Sam's vaccinating times are exponentially distributed. Find the:

(a) probability that Sam is idle.

(b) proportion of time that Sam is busy.

(c) average number of dogs receiving or waiting to be vaccinated.

(d) average number of dogs waiting to be vaccinated.

(e) average time a dog waits before getting vaccinated.

(f) average amount (mean) of time a dog spends between waiting in line and getting vaccinated.

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14.109 A dry cleaner has a single drive-thru window for customers. The arrival rate of cars follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is 16 per hour and the average service time is three minutes.

(a) What is the average number of cars in the line?

(b) What is the average time spent waiting to get to the service window?

(c) What percentage of time is the dry cleaner’s drive-thru window idle?

(d) What is the probability there are more than 2 cars at the drive-thru window?

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14.110 A dry cleaner has a single drive-thru window for customers. The arrival rate of cars follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is 16 per hour and the average service time is three minutes. If the dry cleaner wants to accommodate (have enough room for) all of the waiting cars at least 96 percent of the time, how many car-lengths should they make the driveway leading to the window?

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14.111 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 6 minutes, according to a Poisson distribution. Also assume that Sam's vaccinating times are exponentially distributed. Sam would like to have each waiting dog placed in a holding pen. If Sam wants to be certain he has enough cages to accommodate all dogs at least 90 percent of the time, how many cages should he prepare?

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14.112 A new shopping mall is considering setting up an information desk operated by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 4 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed.

(a) Find the proportion of the time that the employees are busy.

(b) Find the average number of people waiting in line to get some information.

(c) Find the expected time a person spends just waiting in line to have his question answered.

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14.113 Cars arrive at a parking lot entrance at the rate of 20 per hour. The average time to get a ticket and proceed to a parking space is two minutes. There are two lot attendants at the current time. The Poisson and exponential distribution appear to be relevant in this situation.

(a) What is the probability that an approaching auto must wait?

(b) What is the average waiting time?

(c) What is the average number of autos waiting to enter the garage?

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14.114 Bank Boston has a branch at Bryant College. The branch is busiest at the beginning of the college year when freshmen and transfer students open accounts. This year, freshmen arrived at the office at a rate of 40 per day (8-hour day). On average, it takes the Bank Boston staff person about ten minutes to process each account application. The bank is considering having one or two tellers. Each teller is paid $12 per hour and the cost of waiting in line is assumed to be $8 per hour.

(a) What is the total daily waiting cost for the single teller model?

(b) What is the total daily waiting cost for the two-teller model?

(c) What is the total daily service cost for the single teller model?

(d) What is the total daily service cost for the two-teller model?

(e) Which model is preferred?

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14.115 At the start of football season, the ticket office gets busy the day before the first game. Customers arrive at the rate of four every ten minutes. A ticket seller can service a customer in four minutes. Traditionally, there are two ticket sellers working. The university is considering an automated ticket machine similar to the airlines’ e-ticket system. The automated ticket machine can service a customer in 2 minutes.

(a) What is the average length of the queue for the in-person model?

(b) What is the average length of the queue for the automated system model?

(c) What is the average time in the system for the in-person model?

(d) What is the average time in the system for the automated system model?

(e) Assume the ticket sellers earn $8 per hour and the machine costs $20 per hour (amortized over 5 years). The wait time is only $4 per hour because students are patient. What is the total cost of each model?

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14.116 Consider a single-server queuing system with Poisson arrivals of 10 units per hour and a constant service time of 2 minutes per unit. How long will the customer waiting time be in seconds, on average?

SHORT ANSWER/ESSAY

14.117 With regard to queuing theory, define what is meant by balking.

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14.118 With regard to queuing theory, define what is meant by reneging.

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14.119 What is meant by a single-channel queuing system?

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14.120 What is meant by a multichannel queuing system?

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14.121 What is meant by a single-phase system?

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14.122 What is meant by a multiphase system?

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