CHAPTER 14 Waiting Lines and Queuing Theory Models TRUE/FALSE 14.1 The three parts of a queuing system are the arrivals, the queue, and the service facility. 14.2 Two characteristics of arrivals are the line length and queue discipline. 14.3 Queuing theory models can also apply to customers placing telephone calls and being placed on hold. 14.4 The only objective of queuing theory is to minimize customer dissatisfaction. 14.5 Should a customer leave a queue before being served, it is said that the customer has reneged. 14.6 Balking refers to customers who enter the queue but may become impatient and leave without completing their transactions. 14.7 Most systems use the queue discipline known as the first-in, first-out rule. 14.8 In a very complex queuing model, if all of the assumptions of the traditional models are not met, then the problem cannot be handled. 14.9 Before using exponential distributions to build queuing models, the quantitative analyst should determine if the service time data fit the distribution. 443
Transcript
CHAPTER 14Waiting Lines and Queuing Theory Models
TRUE/FALSE
14.1 The three parts of a queuing system are the arrivals, the queue, and the service facility.
14.2 Two characteristics of arrivals are the line length and queue discipline.
14.3 Queuing theory models can also apply to customers placing telephone calls and being placed on hold.
14.4 The only objective of queuing theory is to minimize customer dissatisfaction.
14.5 Should a customer leave a queue before being served, it is said that the customer has reneged.
14.6 Balking refers to customers who enter the queue but may become impatient and leave without completing their transactions.
14.7 Most systems use the queue discipline known as the first-in, first-out rule.
14.8 In a very complex queuing model, if all of the assumptions of the traditional models are not met, then the problem cannot be handled.
14.9 Before using exponential distributions to build queuing models, the quantitative analyst should determine if the service time data fit the distribution.
14.10 For practical purposes, queue length is almost always modeled with a finite queue length.
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14.11 The Greek letter is used to represent the average service rate at each channel.
14.12 For a single channel model that has Poisson arrivals and exponential service rates, the Greek letter is the utilization factor.
14.13 In a multi-channel, single-phase queuing system, the arrival will pass through at least two different service facilities.
14.14 In a multi-channel model = /( M ).
14.15 A goal of many waiting line problems is to help a firm find the ideal level of services to be offered.
14.16 Any waiting line problem can be investigated using an analytical queuing model.
14.17 One of the difficulties in waiting line analyses is that it is sometimes difficult to place a value on customer waiting time.
14.18 The goal of most waiting line problems is to identify the service level that minimizes service cost.
14.19 One of the limitations of analytical waiting line models is that they do not give information on extreme cases (e.g., maximum waiting time or maximum number in the queue).
14.20 An "infinite calling population" occurs when the likelihood of a new arrival does not depend upon the number of past arrivals.
14.21 All practical problems can be described by an "infinite" population waiting model.
14.22 On a practical note – if we are using waiting line analysis for a problem studying customers calling a telephone number for service, balking is probably not an issue.
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14.23 On a practical note– if we are using waiting line analysis to study cars passing through a single tollbooth, reneging is probably not an issue.
14.24 On a practical note – if we are studying patrons moving through checkout lines at a grocery store, and we note that these patrons sometimes move from one line to another, we should consider balking as an issue.
14.25 On a practical note – if we were to study the waiting lines in a hair salon which had only five chairs for patrons waiting, we would have to use a finite queue waiting line model.
14.26 All practical waiting line problems can be viewed as having a FIFO queue discipline.
14.27 A hospital emergency room will usually employ a FIFO queue discipline.
14.28 If we wish to study a bank, in which patrons entered the building and then, depending upon the service desired, chose one of several tellers in front of which to form a line, we would employ a set of single-channel queuing models.
14.29 On a practical note – we should probably view the checkout counters in a grocery store as a set of single channel systems.
14.30 A cafeteria, in which cold dishes are separated from hot dishes, is probably best viewed as a single-channel, single-phase system.
14.31 An emergency room might be viewed as a multi-channel, multi-phase system.
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14.32 A single highway with multiple tollbooths should be viewed as a single-channel system.
14.33 In a doctor's office, we would expect the arrival rate distribution to be Poisson distributed, and the service time distribution to be negative exponential.
14.34 The M/M/1 queuing model assumes that the arrival rate does not change over time.
14.35 The analytical queuing models typically provide operating characteristics that are averages (e.g., average waiting time, average number of customers in the queue).
14.36 The analytical queuing models can be used to tell us how many people are presently waiting in line.
14.37 The quantity is the probability that one or more customers are in a single channel system.
14.38 In the multi-channel model (M/M/m), we must assume that the average service time for all channels is the same.
14.39 If we compare a single-channel system with = 15, to a multi-channel system (with 3 channels) with the service rate for the individual channel of = 5, we will find that the average wait time is less in the single-channel system.
14.40 If we compare a single-channel system with exponential service rate (=5) to a constant service time model (=5), we will find that the average wait time in the constant service time model is less than that in the probabilistic model.
14.41 As a general rule, any time that the number of people in line can be a significant portion of the total population, we should use a finite population model.
14.42 Whether or not we use the finite population queuing model depends upon the relative arrival and service rates, not just the size of the population from which the arrivals come.
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14.43 Whether or not we use the finite population queuing model depends upon the amount of space we have in which to form the queue.
14.44 If a waiting line problem is particularly complex, we may have to turn to a simulation model.
14.45 If we are using a simulation queuing model, we still have to abide by the assumption of a Poisson arrival rate, and negative exponential service rate.
14.46 Using a simulation model allows one to ignore the common assumptions required to use analytical models.
*14.47 If we are studying the arrival of automobiles at a highway toll station, we can assume an infinite calling population.
*14.48 If we are studying the need for repair of electric motors on a small assembly line, we can assume an infinite calling population.
*14.49 The difference between balking and reneging is that balking implies that the arrival never joined the queue, while reneging implies that the arrival joined the queue, but became impatient and left.
*14.50 When looking at the arrivals at the ticket counter of a movie theater, we can assume an unlimited queue.
*14.51 When looking at the arrivals at a barbershop, we must assume a finite queue.
*14.52 A bank, in which a single queue is used to move customers to several tellers, is an example of a single-channel system.
*14.53 A fast food drive-through system is an example of a multi-channel queuing system.
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*14.54 A fast food drive-through system is an example of a multi-phase queuing system.
*14.56 In a single-channel, single-phase system, reducing the service time only reduces the total amount of time spent in the system, not the time spent in the queue.
*14.57 The wait time for a single-channel system is more than twice that for a two channel system using two servers working at the same rate as the single server.
MULTIPLE CHOICE
14.58 The expected cost to the firm of having customers or objects waiting in line to be serviced is termed the
(a) expected service cost.(b) expected waiting cost.(c) total expected cost.(d) expected balking cost.(e) expected reneging cost.
14.59 Which of the following is not an assumption in common queuing mathematical models?
(a) Arrivals come from an infinite, or very large, population.(b) Arrivals are Poisson distributed.(c) Arrivals are treated on a first-in, first-out basis and do not balk or renege.(d) Service times follow the negative exponential distribution.(e) The average arrival rate is faster than the average service rate.
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14.60 Which of the following is not a key operating characteristic for a queuing system?
(a) utilization rate(b) percent idle time(c) average time spent waiting in the system and in the queue(d) average number of customers in the system and in the queue(e) none of the above
14.61 Three parts of a queuing system are
(a) the inputs, the queue, and the service facility.(b) the calling population, the queue, and the service facility.(c) the calling population, the waiting line, and the service facility.(d) All of the above are appropriate labels for the three parts of a queuing system.
14.62 Upon arriving at a convention, if a person must line up to first register at a table, then proceed to a table to gather some additional information, and then pay at another single table, this is an example of a
(a) single-channel, multi-phase system.(b) single-channel, single-phase system.(c) multi-channel, multi-phase system.(d) multi-channel, single-phase system.(e) none of the above
14.63 The utilization factor for a system is defined as
(a) the mean number of people served divided by the mean number of arrivals per time period.(b) the average time a customer spends waiting in a queue.(c) the proportion of the time the service facilities are in use.(d) the percent idle time.(e) none of the above
14.64 Which of the following is not a characteristic of the calling population and its behavior?
(a) Size is considered to be limited or unlimited.(b) Queue discipline.(c) A customer is usually patient.(d) Customers can arrive randomly.(e) none of the above
14.65 In queuing theory, the objective is to
(a) maximize productivity.(b) minimize customer dissatisfaction as measured in balking and reneging.(c) minimize the sum of the costs of waiting time and providing service.
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(d) minimize the percent of idle time.(e) minimize queue length.
14.66 In queuing problems, the size of the calling population is important because
(a) it is usually easier to deal with the mathematics if the calling population is considered infinite.
(b) it is usually easier to deal with the mathematics if the calling population is considered finite.
(c) it is impossible to deal with the mathematics (except through monte carlo simulation) if the calling population is infinite.
(d) it is impossible to deal with the mathematics (except through monte carlo simulation) if the calling population is finite.
(e) none of the above
14.67 An arrival in a queue that reneges is one who
(a) after joining the queue, becomes impatient and leaves.(b) refuses to join the queue because it is too long.(c) goes through the queue, but never returns.(d) jumps from one queue to another, trying to get through as quickly as possible.(e) none of the above
14.68 A balk is an arrival in a queue who
(a) refuses to join the queue because it is too long.(b) after joining the queue, becomes impatient and leaves.(c) goes through the queue, but never returns.(d) jumps from one queue to another, trying to get through as quickly as possible.
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14.69 Queue discipline may be
(a) FIFO (first-in, first-out).(b) FIFS (first-in, first-served).(c) LIFS (last-in, first-served).(d) by assigned priority.(e) all of the above
14.70 If the arrival rate and service times are kept constant and the system is changed from a single-channel to a two-channel system, then the average time an arrival will spend in the waiting line or being serviced (W) is
(a) increased by 50 percent.(b) reduced by 50 percent.(c) exactly doubled.(d) the same.(e) none of the above
14.71 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 increase.(d) the average queue length will double and the average waiting time will double.(d) none of the above
14.72 If a queuing situation becomes extremely complex,
(a) there is always a mathematical model to solve it.(b) the only alternative is to study the real situation.(c) there are tables available for any combination of complexities.(d) computer simulation is an alternative.(e) you should make simplifying assumptions and use the mathematical procedure which most
closely approximates the system to be studied.
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14.73 Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute, what is the average number of customers in the system?
(a) 0.50(b) 0.75(c) 2.25(d) 3.00(e) none of the above
14.74 Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute, what is the average number of customers waiting in line behind the person being served?
(a) 0.50(b) 0.75(c) 2.25(d) 3.00(e) none of the above
14.75 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute, what proportion of the time is the server busy?
(a) 0.25(b) 0.50(c) 0.75(d) 2.25(e) 3.00
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14.76 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute, on average, how much time will elapse from the time a customer enters the line until he/she leaves the cafeteria?
(a) 0.25(b) 0.50(c) 0.75(d) 2.25(e) 3.00
14.77 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks. What is the average length of the line?
(a) 3.429(b) 1.929(c) 1.143(d) 0.643(e) none of the above
14.78 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks. How long does the average person spend waiting for a clerk to become available?
(a) 3.429(b) 1.929(c) 1.143(d) 0.643(e) none of the above
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14.79 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks. What proportion of the time are both clerks idle?
(a) 0.643(b) 0.250(c) 0.750(d) 0.143(e) none of the above
14.80 A finite population model differs from the other models because, with a finite population,
(a) the queue line is never empty.(b) there is a dependent relationship between the length of the queue and the arrival rate.(c) the service rate will be less than the arrival rate.(d) the average number in the system is the same as the average number in the queue.
14.81 At an automatic car wash, cars arrive randomly at a rate of 9 cars every 20 minutes. The car wash takes exactly 2 minutes (this is constant). On average, what would the length of the line be?
(a) 8.1(b) 4.05(c) 9(d) 1(e) none of the above
14.82 At an automatic car wash, cars arrive randomly at a rate of 9 cars every 20 minutes. The car wash takes exactly 2 minutes (this is constant). On average, how long would each car spend at the car wash?
(a) 0.9 minutes(b) 0.45 minutes(c) 9 minutes(d) 18 minutes(e) none of the above
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14.83 According to Table 14-1, which provides a queuing problem solution, what proportion of the time is the system totally empty?
(a) 0.111(b) 0.333(c) 0.889(d) 0.667(e) none of the above
14.84 According to Table 14-1, which provides a queuing problem solution, on average, how long does each customer spend waiting in line?
14.85 According to Table 14-1, which provides 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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14.86 According to Table 14-2, which provides a queuing problem solution, on average, how many units are in the line?
(a) 5.455(b) 3.788(c) 1.091(d) 0.758(e) 0.833
14.87 According to Table 14-2, which provides a queuing problem solution, what proportion of the time is at least one server busy?
(a) 0.833(b) 0.758(c) 0.091(d) 0.909(e) none of the above
14.88 According to Table 14-2, which provides a queuing problem solution, there are two servers in this system. Counting each person being served and the people in line, on average, how many people would be in this system?
(a) 5.455(b) 3.788(c) 9.243(d) 10.900(e) none of the above
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14.89 According to Table 14-3, 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
14.90 According to Table 14-3, 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
14.91 According to Table 14-3, 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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14.92 According to Table 14-3, 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
14.93 The most appropriate cost to be considered in making a waiting line decision is the
(a) expected service cost.(b) expected waiting cost.(c) total expected cost.(d) expected balking cost.(e) expected reneging cost.
14.94 Which of the following is not an assumption in common queuing mathematical models?
(a) Arrivals come from an infinite, or very large, population.(b) Arrivals are Poisson distributed.(c) Arrivals are treated on a first-in, first-out basis and do not balk or renege.(d) Service rates follow the normal distribution.(e) The average service rate is faster than the average arrival rate.
14.95 The utilization factor is defined as the
(a) percent of time the system is idle.(b) average percent of time the customers wait in line.(c) average time the service system is open.(d) percent of time that a single customer is in the system.(e) none of the above
14.96 Which of the following is usually the most difficult cost to determine?
(a) service cost(b) facility cost(c) calling cost(d) waiting cost(e) none of the above
14.97 Lines at banks where customers wait to go to a teller window are usually representative of a
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(e) none of the above
14.98 A restaurant in which one must go to the maitre d' in order to be seated in one of three dining rooms is an example of a
(a) single-channel, multi-phase system.(b) single-channel, single-phase system.(c) multi-channel, multi-phase system.(d) multi-channel, single-phase system.(e) none of the above
14.99 A gasoline station which has a single pump and where the customer must enter the building to pay is an example of a
(a) single-channel, multi-phase system.(b) single-channel, single-phase system.(c) multi-channel, multi-phase system.(d) multi-channel, single-phase system.(e) none of the above
14.100 A vendor selling newspapers on a street corner is an example of a
(a) single-channel, multi-phase system.(b) single-channel, single-phase system.(c) multi-channel, multi-phase system.(d) multi-channel, single-phase system.(e) none of the above
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14.101 The utilization factor for a system tells one the
(a) mean number of people served divided by the mean number of arrivals per time period.(b) average time a customer spends waiting in a queue.(c) proportion of the time the service facilities are in use.(d) percent idle time.(e) none of the above
14.102 Which of the following is not an assumption about the queue in the waiting line models presented in this chapter?
(a) Queue length is considered to be unlimited.(b) Queue discipline is assumed to be FIFO.(c) A customer in the queue is usually patient.(d) Customers arrive to enter the queue in a random fashion.(e) none of the above
14.103 Assume that we are using a waiting line model to analyze the number of service technicians required to maintain machines in a factory. Our goal should be to
(a) maximize productivity of the technicians.(b) minimize the number of machines needing repair.(c) minimize the downtime for individual machines.(d) minimize the percent of idle time of the technicians.(e) minimize the total cost (cost of maintenance plus cost of downtime).
14.104 In queuing problems, the size of the calling population is important because
(a) we have models only for problems with infinite calling populations.(b) we have models only for problems with finite calling populations.(c) the size of the calling population determines whether or not the arrival of one customer
influences the probability of arrival of the next customer.(d) we will have to consider the amount of space available for the queue.(e) none of the above
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14.105 The behavior of jumping from one queue to another trying to get through as quickly as possible is called:
(a) balking.(b) reneging.(c) cropping.(d) blithering.(e) none of the above
14.106 The customer who arrives at a bank, sees that there is a long line, and leaves to return at another time is
(a) balking.(b) cropping.(c) reneging.(d) blithering.(e) none of the above
14.107 The term queue discipline describes the
(a) degree to which members of the queue renege.(b) sequence in which members of the queue arrived.(c) degree to which members of the queue are orderly and quiet.(d) sequence in which members of the queue are serviced.(e) all of the above
14.108 If the arrival rate and service times are kept constant and the system is changed from a two-channel system to a single-channel system, then the average time an arrival will spend in the waiting line is
(a) decreased.(b) increased.(c) exactly doubled.(d) the same as before.(e) could be any of the above depending on other parameters of the problem.
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14.109 If everything else remains constant, including the mean arrival rate and service rate, except that the service time becomes constant instead of exponential, the
(a) average waiting time will be decreased.(b) average queue length will be increased.(c) average number of customers in the system will be increased.(d) none of the above(e) (a), (b), & (c)
14.110 Assume that we wish to study the performance of checkout stations in a large grocery store. Some of the checkouts are reserved for those customers with fewer than twelve items; other checkouts are reserved for those customers from the bakery, deli, or pharmacy; and still other checkouts are open to all customers. We should employ
(a) a multi-channel, multi-phase queuing model.(b) a number of single-channel, single-phase models.(c) two separate multi-channel, single-phase models.(d) simplifying assumptions to make the problem fit one or another of the analytical models.(e) a simulation model.
14.111 If we want to know the maximum number of customers who will be waiting to buy tickets to a movie in a theater where there are three servers selling tickets, we should employ a
(a) single-channel, single-phase model.(b) multi-channel, single-phase model.(c) multi-channel, multi-phase model.(d) single-channel, multi-phase model.(e) none of the above
14.112 The most likely queue discipline to be followed in a hospital emergency room is
(a) FIFO (first in, first out)(b) LIFO (last in, first out)(c) FILO (first in, last out)(d) WCF (worst case first)(e) none of the above
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14.113 Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, what is the average number of customers in the system?
(a) 0.43(b) 1.67(c) 0.57(d) 1.33(e) none of the above
14.114 Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, what is the average number of customers waiting in line behind the person being served?
(a) 0.76(b) 0.19(c) 1.33(d) 1.67(e) none of the above
14.115 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, what proportion of the time is the server busy?
(a) 0.43(b) 0.57(c) 0.75(d) 0.25(e) 0.33
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14.116 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first served basis. The arrival rate follows a Poisson distribution, while service times follow an exponential distribution. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, on average, how much time will elapse from the time a customer enters the line until he/she leaves the cafeteria?
14.117 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who each work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is seven per minute and the average service rate is four per minute for each of the two clerks. What is the average length of the line?
(a) 3.429(b) 4.932(c) 5.717(d) 7.467(e) none of the above
14.118 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who each work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is seven per minute and the average service rate is four per minute for each of the two clerks. How long does the average person spend waiting for a clerk to become available?
(a) 0.067(b) 0.817(c) 1.067(d) 0.875(e) none of the above
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14.119 A post office has a single line for customers to use while waiting for the next available postal clerk. There are two postal clerks who each work at the same rate. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is seven per minute and the average service rate is four per minute for each of the two clerks. What proportion of the time are both clerks idle?
(a) 0.875(b) 0.067(c) 0.750(d) 0.817(e) none of the above
14.120 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) 6.533(d) 0.467(e) none of the above
14.121 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
14.122 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
14.123 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) 8.171(b) 7.467
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(c) 6.533(d) 0.467(e) none of the above
14.124 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). The utilization factor for this system is
(a) 0.467(b) 0.533(c) 1.000(d) 0.933(e) none of the above
14.125 At your automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant). At the moment, you have space for 7 cars in the waiting area. You should consider
(a) adding additional parking spaces.(b) reducing the price you charge for washing the car.(c) modifying the system to speed up service and reduce waiting time.(d) adding additional, optional service features.(e) none of the above
14.126 According to the information provided in Table 14-4, on average, how many units are in the line?
(a) 0.643(b) 2.191(c) 2.307(d) 0.217(e) 0.905
14.127 According to the information provided in Table 14-4, what proportion of the time is at least one server busy?
(a) 0.643(b) 0.905(c) 0.783(d) 0.091(e) none of the above
14.128 Using the information provided in Table 14-4: Counting each person being served and the people in line, on average, how many people would be in this system?
(a) 0.905(b) 2.191
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(c) 6.037(d) 14.609(e) none of the above
14.129 According to the information provided in Table 14-4, what is the average time spent by a person in this system?
(a) 0.905 minutes(b) 2.191 minutes(c) 6.037 minutes(d) 14.609 minutes(e) none of the above
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14.130 According to the information provided in Table 14-4, what percentage of the total available service time is being used?
(a) 90.5%(b) 21.7%(c) 64.3%(d) could be any of the above; depends upon other factors.(e) none of the above
14.131 According to Table 14-5, which presents the solution for a queuing problem with a constant service rate, on the 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
14.132 According to Table 14-5, which presents the solution for a queuing problem with a constant service rate, on the 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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14.133 According to 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
14.134 According to 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
14.135 According to 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
14.136 According to 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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*14.137 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food joint takes an average of 2 minutes to serve each arrival. The utilization factor for this system is
(a) 0.467(b) 0.547(c) 0.800(d) 0.133(e) none of the above
*14.138 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food joint has restructured their serving system so that service takes exactly 2 minutes (this is constant) per arrival. The utilization factor for this system is
(a) 0.800(b) 0.723(c) 1.000(d) 0.854(e) none of the above
*14.139 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 wait time for arrivals is
(a) 5.4 minutes(b) 6.0 minutes(c) 8.0 minutes(d) 2.5 minutes(e) none of the above
*14.140 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. Service times are random (exponential) and average 2 minutes per arrival. The average time in the queue for each arrival is
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*14.141 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exponentially distributed with an average of 15 minutes. The average customer waits in line
(a) 3.0 minutes(b) 2.5 minutes(c) 3.5 minutes(d) 4.0 minutes(e) none of the above
*14.142 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exponentially distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has decided to open a second work bay, i.e., make the shop into a two-channel system. Under this new scheme, the average customer will wait in line
(a) 9.6 minutes(b) 2.5 minutes(c) 24.6 minutes(d) 2.1 minutes(e) none of the above
*14.143 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exponentially distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has decided to open a second work bay, i.e., make the shop into a two-channel system. Under this new scheme, the total time an average customer spends in the system will be
*14.144 Cars approach a set of toll booths at the rate of 75 cars per hour. There are five toll booths, with an average service time of 0.5 minutes. For the two miles before a car reaches a toll booth, the highway is five lanes wide. Under heavy traffic conditions, a car commits to a specific toll lane nearly a mile before it reaches the toll booth. On average, how long is the line in front of a specific toll booth?
*14.145 Cars approach a set of toll booths at the rate of 75 cars per hour. There are five toll booths, with an average service time of 0.5 minutes. For the two miles before a car reaches a toll booth, the highway is five lanes wide. Under light traffic conditions, a car does not have to commit to a specific toll lane until actually approaching the toll booths. On average, how long is the line in front of a specific toll booth?
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(a) 0.0179 cars(b) 0.0001 cars(c) 0.0100 cars(d) 0.5000 cars(e) none of the above
*14.146 Cars appear to approach a local Burger Basket Restaurant at the rate of 20 per hour. Average service rate averages 22 per hour. Gale Johnson, the owner, has become concerned about the waiting time under the current configuration. She is considering enlarging the facility by constructing a second takeout window. How does the waiting time with two windows change from that with only a single window?
(a) Waiting time drops from 27.3 minutes with one window to 0.7 minutes with two.(b) Waiting time drops from 32.1 minutes with one window to 21.3 minutes with two.(c) There is no change.(d) Waiting time drops from 20.3 minutes with one window to 6.5 minutes with two.(e) none of the above
*14.147 Customers arrive at the local PharmCal gas station at the rate of 40 per hour. What service rate is necessary to keep the average wait time less than 5 minutes?
(a) 45 per hour(b) 47 per hour(c) 49 per hour(d) 50 per hour(e) none of the above
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PROBLEMS
14.148 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 the time that the employee is busy.(c) Find the average number of people receiving and waiting to receive some information.(d) Find the average number of people waiting in line to get some information.(e) Find the average time a person seeking information spends at the desk.(f) Find the expected time a person spends just waiting in line to have his question answered.
14.149 A new shopping mall is considering setting up an information desk manned by two employees. Based upon 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 four 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.
14.150 Due to a recent increase in business, a secretary in a certain law firm is now having to type 20 letters a day on average. It takes her approximately 20 minutes to type each letter. Assuming the secretary works 8 hours a day:
(a) What is the secretary's utilization rate?(b) What is the average waiting time before the secretary types a letter?(c) What is the average number of letters waiting to be typed?(d) What is the probability that the secretary has more than 5 letters to type?
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14.151 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can "shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every six minutes, according to a Poisson distribution. Also assume that Sam's shooting times are exponentially distributed. Find the:
(a) probability that Sam is idle.(b) proportion of the time that Sam is busy.(c) average number of dogs being shot or waiting to be shot.(d) average number of dogs waiting to be shot.(e) average time a dog waits before getting shot.(f) average amount (mean) of time a dog spends between waiting in line and getting shot.
14.152 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can "shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every four minutes, according to a Poisson distribution. Also assume that Sam's shooting times are exponentially distributed. Find the:
(a) probability that Sam is idle.(b) proportion of the time that Sam is busy.(c) average number of dogs being shot or waiting to be shot.(d) average number of dogs waiting to be shot.(e) average time a dog waits before getting shot.(f) average amount (mean) of time a dog spends between waiting in line and getting shot.
14.153 Calls arrive at the hotel switchboard at a rate of two per minute. The average time to handle each of these is 15 seconds. There is only one switchboard operator at the current time. The Poisson and exponential distribution appear to be relevant in this situation.
(a) What is the probability that the operator is busy?(b) What is the average time that a call must wait before reaching the operator?(c) What is the average number of calls waiting to be answered?
14.154 At the start of football season, the ticket office gets very busy the day before the first game. Customers arrive at the rate of four every ten minutes, and the average time to transact business is two minutes.
(a) What is the average number of people in line?(b) What is the average time that a person would spend in the ticket office?(c) What proportion of time is the server busy?
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14.155 At the start of football season, the ticket office gets very busy the day before the first game. Customers arrive at the rate of four every ten minutes, and the average time to transact business is one minute. There are two servers in the ticket office, each of whom operate at the same rate of speed.
(a) What is the average number of people in line?(b) What is the average time that a person would spend in the ticket office?(c) What proportion of time is the server busy?
14.156 A post office has a single drive-in window for customers to use. The arrival rate of cars follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is 20 per hour and the average service time is two 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 the time is the postal clerk idle?
14.157 A company has six computers that are used to run an automated manufacturing facility. Each of these runs an average of 90 minutes without requiring any attention from the technician. Each time the technician is required to adjust a computer, an average of 15 minutes (following an exponential distribution) is required to fix the problem.
(a) On average, how many computers are waiting for service? (b) On average, how long is a computer out of service?(c) What is the average waiting time in the queue to be serviced?
14.158 The new Providence shopping mall is considering setting up an information desk manned by one employee. Because of the complex design of the mall, it is expected that people will arrive at the desk at about twice the rate for most malls. The expected rate is 25 per hour. It will also take longer to answer their questions – approximately four minutes per person on average.
(a) Find the probability that the employee is idle.(b) Find the proportion of the time that the employee is busy.(c) Find the average number of people receiving and waiting to receive some information.(d) Find the average number of people waiting in line to get some information.(e) Find the average time a person seeking information spends waiting and at the desk.(f) Find the expected time a person spends just waiting in line to have his question answered.
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14.159 The new Providence shopping mall is considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately four minutes for the Information Desk employee to help the average person. By utilizing a new map and special guide signs, the
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manager believes that the required service time can be reduced to an average of two minutes per person. Assuming that he implements the new map and guide signs:
(a) Find the probability that the Information Desk employee is idle.(b) Find the proportion of the time that the Information Desk employee is busy.(c) Find the average number of people receiving and waiting to receive some information.(d) Find the average number of people waiting in line to get some information.(e) Find the average time a person seeking information spends waiting and at the desk.(f) Find the expected time a person spends just waiting in line to have his question answered.
14.160 The new Providence shopping mall has been considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person. By utilizing a new map and special guide signs, he believes that the required service time can be reduced to an average of two minutes per patron. The manager has also noticed that the people seeking help at the information desk may come from one of two groups: (a) mall patrons, and (b) mall staff, or delivery persons. Therefore, the manager has decided that he wants to consider another option: establishing two information desks – one desk to help mall patrons, the other to help staff and delivery persons. The manager believes that he can expect patrons to arrive at the rate of 20 per hour, and mall staff or delivery persons to arrive at the rate of 5 per hour. It is likely to take an average of two minutes to answer the questions of a patron, and an average of ten minutes to answer those of a staff or delivery person. Assuming that the two-desk concept is implemented,
(a) find the probability that both Information Desk employees are idle.(b) determine whether the patron or the staff person is likely to have the longer wait.
14.161 The new Providence shopping mall has been considering setting up an information desk manned by one employee. The layout for this mall is quite complex, leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person. He has now come to realize that employing only a single person at the information desk would lead to a very lengthy line – theoretically, an infinite line! He has decided, therefore, to employ two staff members at the information desk.
(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.(d) If the manager has a goal that, for the average patron, the time spent having one's question
answered is less than half the time spent waiting, has he met that goal?(e) Assuming that the manager also has the goal that the combined idle time for the two
workers does not exceed 45 minutes in an 8 hour day has he met that goal?
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(f) What would the arrival rate have to be for the manager to meet the 45 minute idle time goal?
14.162 Bank Boston now has a branch at Bryant College. The branch is always busiest at the beginning of the college year when freshmen and transfer students arrive on campus and 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.
(a) What is the staff person's utilization rate?(b) What is the average time a student has to wait before getting his application processed?(c) What is the probability that there are more than three students waiting in line?(d) The office has a total of five chairs for students, four for waiting, and one at the service
desk. Is it likely that any students will have to stand?
14.163 Sam the Vet is running a rabies vaccination clinic for cats at the local grade school. Sam can "shoot" a cat every four minutes. It is estimated that the cats will arrive independently and randomly throughout the day at a rate of one cat every five minutes, according to a Poisson distribution. Also assume that Sam's shooting times are exponentially distributed.
(a) What is the probability that a cat will have to wait?(b) On the average, how many cats will be in the waiting room?(c) If a cat has to wait more than 20 minutes, it will become obnoxious. Is this likely to
present a serious problem?
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14.164 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can "shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every six minutes, according to a Poisson distribution. Also assume that Sam's shooting times are exponentially distributed.
Sam would like to have each waiting dog placed in a holding pen during the waiting period. If Sam wants to be certain to have enough cages to accommodate all dogs at least 90 percent of the time, how many cages should he prepare?
14.165 Cars arrive at the entrance to a parking lot at the rate of 20 per hour. The average time to get a ticket and proceed to a parking space is two minutes. There is only one lot attendant 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?
14.166 At the start of football season, the ticket office gets very busy the day before the first game. Customers arrive at the rate of four every ten minutes. Would the customer be better off if the stadium employed a single ticket seller who could service a customer in two minutes, or two ticket sellers, each of whom could service a customer in three minutes?
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14.167 At the start of ballet season, the ticket office gets very busy the day before the first performance. Customers arrive at the rate of four every fifteen minutes, and the average time to transact business is 6 minutes. There are two servers in the ticket office, both of whom operate at the same speed.
(a) What is the average number of people in line?(b) What is the average time that a person would spend in the ticket office?(c) What proportion of time is at least one server busy?
14.168 A post office has a single drive-in window for customers to use. The arrival rate of cars follows a Poisson distribution, while the service time follows an exponential distribution. The average arrival rate is 20 per hour and the average service time is two minutes.
If the post office wants to accommodate all of the waiting cars at least 95 percent of the time, how many car-lengths should they make the driveway leading to the window?
14.169 A company has six computers that are used to run an automated manufacturing facility. Each of these runs an average of 90 minutes without requiring any attention from the technician. Each time the technician is required to adjust a computer, an average of 12 minutes (following an exponential distribution) is required to fix the problem.
How many spares should the technician keep on hand if she wishes to be 90 percent certain that she will have a working machine to swap for a defect before repairing the defective machine? Assume that it takes only three minutes to swap the machines.
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SHORT ANSWER/ESSAY
14.170 With regard to queue theory, define what is meant by balking.
14.171 With regard to queue theory, define what is meant by reneging.
14.172 How is FIFO used in describing a queuing theory problem?
14.173 List three key operating characteristics of a queuing system.
14.174 What is meant by a single-channel queuing system?
14.175 What is meant by a multi-channel queuing system?
