Waiting Lines and Queuing Theory Models l CHAPTER 14TRUE/FALSE 14.1 The three parts of a queuing system are the arrivals, the queue, and the service facility. ANSWER: TRUE 14.2 Two characteristics of arrivals are the line length and queue discipline. ANSWER: TRUE 14.3 Queuing theory models can also apply to customers placing telephone calls and being placed on hold. ANSWER: TRUE 14.4 The only objective of queuing theory is to minimize customer dissatisfaction. ANSWER: FALSE 14.5 Should a customer leave a queue before being served, it is said that the customer has reneged. ANSWER: TRUE 14.6 Balking refers to customers who enter the queue but may become impatient and leave without completing their transactions. ANSWER: FALSE 14.7 Most systems use the queue discipline known as the first-in, first-out rule. ANSWER: TRUE 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. ANSWER: FALSE 14.9 Before using exponential distributions to build queuing models, the quantitative analyst should determine if the service time data fit the distribution. ANSWER: TRUE 14.10 For practical purposes, queue length is almost always modeled with a finite queue length. ANSWER: FALSE 187 Waiting Lines and Queuing Theory Models l CHAPTER 14 14.11 The Greek letter is used to represent the average service rate at each channel. ANSWER: FALSE 14.12 For a single channel model that has Poisson arrivals and exponential service rates, the Greek letter is the utilization factor. ANSWER: TRUE 14.13 In a multi-channel, single-phase queuing system, the arrival will pass through at least two different service facilities. ANSWER: FALSE 14.14 In a multi-channel model = /( M ). ANSWER: TRUE 14.15 A goal of many waiting line problems is to help a firm find the ideal level of services to be offered. ANSWER: TRUE 14.16 Any waiting line problem can be investigated using an analytical queuing model. ANSWER: FALSE 14.17 One of the difficulties in waiting line analyses is that it is sometimes difficult to place a value on customer waiting time. ANSWER: TRUE 14.18 The goal of most waiting line problems is to identify the service level that minimizes service cost. ANSWER: FALSE 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). ANSWER: TRUE 14.20 An "infinite calling population" occurs when the likelihood of a new arrival does not depend upon the number of past arrivals. ANSWER: TRUE 14.21 All practical problems can be described by an "infinite" population waiting model. 14.22 ANSWER: FALSE 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. ANSWER: FALSE 188 Waiting Lines and Queuing Theory Models l CHAPTER 14 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. ANSWER: TRUE 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. ANSWER: FALSE 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. ANSWER: TRUE 14.26 All practical waiting line problems can be viewed as having a FIFO queue discipline. ANSWER: FALSE 14.27 A hospital emergency room will usually employ a FIFO queue discipline. ANSWER: FALSE 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. ANSWER: TRUE 14.29 On a practical note – we should probably view the checkout counters in a grocery store as a set of single channel systems. ANSWER: TRUE 14.30 A cafeteria, in which cold dishes are separated from hot dishes, is probably best viewed as a single-channel, single-phase system. ANSWER: FALSE 14.31 An emergency room might be viewed as a multi-channel, multi-phase system. ANSWER: TRUE 189 42 Whether or not we use the finite population queuing model depends upon the relative arrival and service rates. ANSWER: FALSE 14. average waiting time. we will find that the average wait time in the constant service time model is less than that in the probabilistic model. average number of customers in the queue).41 As a general rule. 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. ANSWER: TRUE 14. and the service time distribution to be negative exponential.39 If we compare a single-channel system with = 15. ANSWER: TRUE 14.33 In a doctor's office. we would expect the arrival rate distribution to be Poisson distributed. ANSWER: TRUE 14. we should use a finite population model.37 The quantity is the probability that one or more customers are in a single channel system.g. ANSWER: FALSE 14.35 The analytical queuing models typically provide operating characteristics that are averages (e. ANSWER: TRUE 190 .. ANSWER: FALSE 14.34 The M/M/1 queuing model assumes that the arrival rate does not change over time.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.32 A single highway with multiple tollbooths should be viewed as a single-channel system. ANSWER: FALSE 14.40 If we compare a single-channel system with exponential service rate (=5) to a constant service time model (=5). ANSWER: TRUE 14. ANSWER: TRUE 14. we must assume that the average service time for all channels is the same. not just the size of the population from which the arrivals come.36 The analytical queuing models can be used to tell us how many people are presently waiting in line.38 In the multi-channel model (M/M/m). ANSWER: TRUE 14. any time that the number of people in line can be a significant portion of the total population. while reneging implies that the arrival joined the queue.51 When looking at the arrivals at a barbershop. and negative exponential service rate. ANSWER: FALSE *14. we may have to turn to a simulation model. ANSWER: TRUE 14. we still have to abide by the assumption of a Poisson arrival rate.46 Using a simulation model allows one to ignore the common assumptions required to use analytical models. but became impatient and left. ANSWR: FALSE 14. ANSWER: TRUE *14. is an example of a single-channel system.47 If we are studying the arrival of automobiles at a highway toll station.Waiting Lines and Queuing Theory Models l CHAPTER 14 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. ANSWER: TRUE *14. ANSWER: TRUE *14. ANSWER: TRUE *14.52 A bank. ANSWER: FALSE 191 . in which a single queue is used to move customers to several tellers. ANSWER: TRUE *14.53 A fast food drive-through system is an example of a multi-channel queuing system. we can assume an infinite calling population. we can assume an unlimited queue. ANSWER: FALSE 14.44 If a waiting line problem is particularly complex.49 The difference between balking and reneging is that balking implies that the arrival never joined the queue.45 If we are using a simulation queuing model. ANSWER: FALSE *14. we can assume an infinite calling population. we must assume a finite queue.48 If we are studying the need for repair of electric motors on a small assembly line.50 When looking at the arrivals at the ticket counter of a movie theater. reducing the service time only reduces the total amount of time spent in the system. Service times follow the negative exponential distribution. ANSWER: TRUE MULTIPLE CHOICE 14.54 A fast food drive-through system is an example of a multi-phase queuing system. population. Arrivals are treated on a first-in.Waiting Lines and Queuing Theory Models l CHAPTER 14 *14.58 The expected cost to the firm of having customers or objects waiting in line to be serviced is termed the (a) (b) (c) (d) (e) expected service cost.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. ANSWER: b 14.59 Which of the following is not an assumption in common queuing mathematical models? (a) (b) (c) (d) (e) Arrivals come from an infinite. ANSWER: e 192 . first-out basis and do not balk or renege.56 In a single-channel. or very large. single-phase system. ANSWER: TRUE *14. ANSWER: FALSE *14. Arrivals are Poisson distributed. The average arrival rate is faster than the average service rate. total expected cost. expected balking cost. expected waiting cost. expected reneging cost. not the time spent in the queue. the queue. Customers can arrive randomly. if a person must line up to first register at a table. multi-phase system.64 Which of the following is not a characteristic of the calling population and its behavior? (a) (b) (c) (d) (e) 14. then proceed to a table to gather some additional information. A customer is usually patient.65 Size is considered to be limited or unlimited. the objective is to (a) maximize productivity. single-channel. the calling population. and the service facility. the percent idle time.63 The utilization factor for a system is defined as (a) (b) (c) (d) (e) the mean number of people served divided by the mean number of arrivals per time period.62 Upon arriving at a convention. single-phase system. All of the above are appropriate labels for the three parts of a queuing system.61 Three parts of a queuing system are (a) (b) (c) (d) the inputs. 193 . the proportion of the time the service facilities are in use. Queue discipline. the waiting line. multi-channel. none of the above ANSWER: a 14. the calling population. single-phase system. and the service facility. multi-phase system. multi-channel. the average time a customer spends waiting in a queue. this is an example of a (a) (b) (c) (d) (e) single-channel. and then pay at another single table. none of the above ANSWER: c 14.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. none of the above ANSWER: b In queuing theory. and the service facility. the queue. ANSWER: d 14.60 Which of the following is not a key operating characteristic for a queuing system? (a) (b) (c) (d) (e) utilization rate percent idle time average time spent waiting in the system and in the queue average number of customers in the system and in the queue none of the above ANSWER: e 14. minimize the sum of the costs of waiting time and providing service. after joining the queue. none of the above ANSWER: a 14. but never returns. jumps from one queue to another. minimize the percent of idle time.68 A balk is an arrival in a queue who (a) (b) (c) (d) refuses to join the queue because it is too long. jumps from one queue to another. ANSWER: c 14. (d) it is impossible to deal with the mathematics (except through monte carlo simulation) if the calling population is finite. goes through the queue. (e) none of the above ANSWER: a 14. but never returns.66 In queuing problems. minimize queue length. refuses to join the queue because it is too long. (b) it is usually easier to deal with the mathematics if the calling population is considered finite. becomes impatient and leaves. (c) it is impossible to deal with the mathematics (except through monte carlo simulation) if the calling population is infinite.67 An arrival in a queue that reneges is one who (a) (b) (c) (d) (e) after joining the queue. becomes impatient and leaves. trying to get through as quickly as possible. 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. trying to get through as quickly as possible. goes through the queue.Waiting Lines and Queuing Theory Models l CHAPTER 14 (b) (c) (d) (e) minimize customer dissatisfaction as measured in balking and reneging. ANSWER: a 194 . by assigned priority. all of the above ANSWER: e 14. none of the above ANSWER: e 14. (a) (b) (c) (d) (e) there is always a mathematical model to solve it. then the average time an arrival will spend in the waiting line or being serviced (W) is (a) (b) (c) (d) (e) increased by 50 percent. none of the above ANSWER: a 14.71 If everything else remains constant.69 Queue discipline may be (a) (b) (c) (d) (e) FIFO (first-in. there are tables available for any combination of complexities. first-served). LIFS (last-in.72 If a queuing situation becomes extremely complex. ANSWER: d 195 . computer simulation is an alternative. reduced by 50 percent. you should make simplifying assumptions and use the mathematical procedure which most closely approximates the system to be studied. first-served). the average waiting time will be doubled. exactly doubled. the same. including the mean arrival rate and service rate.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. the only alternative is to study the real situation. except that the service time becomes constant instead of exponential. FIFS (first-in. the average queue length will increase. first-out).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. the average queue length will double and the average waiting time will double. (a) (b) (c) (d) (d) the average queue length will be halved. what is the average number of customers in the system? (a) (b) (c) (d) (e) 0. first served basis. while service times follow an exponential distribution. what is the average number of customers waiting in line behind the person being served? (a) (b) (c) (d) (e) 0. what proportion of the time is the server busy? (a) (b) (c) (d) (e) 0. The arrival rate follows a Poisson distribution.25 3.00 none of the above ANSWER: c 14.75 2.00 none of the above ANSWER: d 14.74 Customers enter the waiting line at a cafeteria on a first come.25 0. The arrival rate follows a Poisson distribution. first served basis.00 ANSWER: c 196 . first served basis.25 3. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute.75 2.50 0. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute. while service times follow an exponential distribution.50 0. The arrival rate follows a Poisson distribution. while service times follow an exponential distribution.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute.25 3.75 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come.73 Customers enter the waiting line at a cafeteria on a first come.50 0.75 2. on average. What is the average length of the line? (a) (b) (c) (d) (e) 3. first served basis. The arrival rate of customers follows a Poisson distribution.643 none of the above ANSWER: b 14.75 2. If the average number of arrivals is six per minute and the average service rate of a single server is eight per minute.25 3. while service times follow an exponential distribution. while the service time follows an exponential distribution. while the service time follows an exponential distribution. How long does the average person spend waiting for a clerk to become available? (a) (b) (c) (d) (e) 3.143 0.00 ANSWER: b 14.143 0.78 A post office has a single line for customers to use while waiting for the next available postal clerk. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks. The arrival rate follows a Poisson distribution.429 1.50 0. The arrival rate of customers follows a Poisson distribution.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. how much time will elapse from the time a customer enters the line until he/she leaves the cafeteria? (a) (b) (c) (d) (e) 0. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks.429 1. There are two postal clerks who work at the same rate.929 1. There are two postal clerks who work at the same rate.76 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come.929 1.643 none of the above ANSWER: d 197 .25 0.77 A post office has a single line for customers to use while waiting for the next available postal clerk. The average arrival rate is three per minute and the average service rate is two per minute for each of the two clerks.1 4. cars arrive randomly at a rate of 9 cars every 20 minutes. The arrival rate of customers follows a Poisson distribution. There are two postal clerks who work at the same rate.81 At an automatic car wash. with a finite population.750 0. The car wash takes exactly 2 minutes (this is constant). ANSWER: b 14. while the service time follows an exponential distribution. What proportion of the time are both clerks idle? (a) (b) (c) (d) (e) 0.143 none of the above ANSWER: d 14.79 A post office has a single line for customers to use while waiting for the next available postal clerk. the service rate will be less than the arrival rate.9 minutes 0. what would the length of the line be? (a) (b) (c) (d) (e) 8.82 At an automatic car wash.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. how long would each car spend at the car wash? (a) (b) (c) (d) (e) 0.45 minutes 9 minutes 18 minutes none of the above ANSWER: e 198 . cars arrive randomly at a rate of 9 cars every 20 minutes. The car wash takes exactly 2 minutes (this is constant).05 9 1 none of the above ANSWER: b 14.643 0. there is a dependent relationship between the length of the queue and the arrival rate.250 0.80 A finite population model differs from the other models because. On average. (a) (b) (c) (d) the queue line is never empty. On average. the average number in the system is the same as the average number in the queue. Waiting Lines and Queuing Theory Models l CHAPTER 14 14.722 minutes 0.889 0.111 0.722 minutes 0. which provides a queuing problem solution.85 According to Table 14-1.111 0.889 0.667 none of the above ANSWER: a 14.722 0.83 Table 14-1 M/M/3: Mean Arrival Rate: Mean Service Rate: Number of Servers: 4 occurrences per minute 2 occurrences per minute 3 Solution: Mean Number of Units in the System: Mean Number of Units in the Queue: Mean Time in the System: Mean Time in the Queue: Service Facility Utilization Factor: Probability of No Units in System: 2.222 minutes 0.667 0.222 0.889 minutes 0. what proportion of the time is the system totally empty? (a) (b) (c) (d) (e) 0.889 0.333 0.111 minutes ANSWER: c 14. on average.889 0.333 minutes 0. which provides a queuing problem solution.84 According to Table 14-1. what is the utilization rate of the service facility? (a) (b) (c) (d) (e) 0.667 ANSWER: e 199 . which provides a queuing problem solution. how long does each customer spend waiting in line? (a) (b) (c) (d) (e) 0.111 According to Table 14-1.222 minutes 0. 88 According to Table 14-2.455 3.455 3. Counting each person being served and the people in line.900 none of the above ANSWER: a Table 14-3 M/D/1 Mean Arrival Rate: 3 occurrences per minute 200 .833 ANSWER: b 14.87 According to Table 14-2.86 Table 14-2 M/M/2 Mean Arrival Rate: Mean Service Rate: Number of Servers: 5 occurrences per minute 3 occurrences per minute 2 Solution: Mean Number of Units in the System: Mean Number of Units in the Queue: Mean Time in the System: Mean Time in the Queue: Service Facility Utilization Factor: Probability of No Units in System: 5.833 0.909 none of the above ANSWER: d 14. which provides a queuing problem solution.091 0. on average.091 According to Table 14-2. which provides a queuing problem solution.091 minutes 0. which provides a queuing problem solution. how many people would be in this system? (a) (b) (c) (d) (e) 5.455 3.833 0.788 1.243 10.758 minutes 0. on average.091 0.788 9.758 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. what proportion of the time is at least one server busy? (a) (b) (c) (d) (e) 0.788 1. there are two servers in this system.758 0. how many units are in the line? (a) (b) (c) (d) (e) 5. 750 0.90 According to Table 14-3.875 minutes 1. on average. on average.125 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. which presents a queuing problem solution for a queuing problem with a constant service rate.875 1.375 none of the above ANSWER: a 14.125 (e) none of the above ANSWER: a 201 . how much time is spent waiting in line? (a) (b) (c) (d) (e) 1.625 minutes 0.375 minutes 0. how many customers are in the system? (a) (b) (c) (d) (e) 1.375 minutes none of the above ANSWER: d 14.89 Constant Service Rate: 4 occurrences per minute Solution: Mean Number of Units in the System: Mean Number of Units in the Queue: Mean Time in the System: Mean Time in the Queue: Service Facility Utilization Factor: Probability of No Units in System: 1.125 0. which presents a queuing problem solution for a queuing problem with a constant service rate.91 According to Table 14-3. on average. how many customers arrive per time period? (a) 3 (b) 4 (c) 1.625 minutes 0.875 (d) 1.625 0.250 According to Table 14-3.875 1. which presents a queuing problem solution for a queuing problem with a constant service rate.125 minutes 0. Service rates follow the normal distribution.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. expected waiting cost.96 Which of the following is usually the most difficult cost to determine? (a) (b) (c) (d) (e) 14. ANSWER: c 14.93 The most appropriate cost to be considered in making a waiting line decision is the (a) (b) (c) (d) (e) expected service cost. total expected cost. average time the service system is open. ANSWER: d 14. expected balking cost. none of the above ANSWER: e 14. population. which presents a queuing problem with a constant service rate.92 According to Table 14-3. average percent of time the customers wait in line. on average.625 minutes 0.375 minutes 4 minutes 0.95 The utilization factor is defined as the (a) (b) (c) (d) (e) percent of time the system is idle. or very large. multi-phase system. Arrivals are treated on a first-in. percent of time that a single customer is in the system. first-out basis and do not balk or renege. The average service rate is faster than the average arrival rate. 202 .25 minutes none of the above ANSWER: d 14.97 service cost facility cost calling cost waiting cost none of the above ANSWER: d Lines at banks where customers wait to go to a teller window are usually representative of a (a) single-channel. expected reneging cost. Arrivals are Poisson distributed.94 Which of the following is not an assumption in common queuing mathematical models? (a) (b) (c) (d) (e) Arrivals come from an infinite. how many minutes does a customer spend in the service facility? (a) (b) (c) (d) (e) 0. single-phase system. multi-channel. multi-channel. multi-phase system. single-phase system. multi-channel.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) (b) (c) (d) (e) single-channel. none of the above ANSWER: d 14. single-channel. single-phase system. none of the above ANSWER: c 14. single-phase system. multi-phase system. multi-phase system. multi-channel. single-phase system. multi-phase system. single-phase system. multi-phase system. single-channel.Waiting Lines and Queuing Theory Models l CHAPTER 14 (b) (c) (d) (e) single-channel. multi-channel. multi-channel. single-phase system. multi-phase system. single-phase system. multi-channel. multi-phase system. none of the above ANSWER: b 203 . none of the above ANSWER: a 14. single-channel. multi-channel.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) (b) (c) (d) (e) single-channel.100 A vendor selling newspapers on a street corner is an example of a (a) (b) (c) (d) (e) single-channel. none of the above ANSWER: c 14. Queue discipline is assumed to be FIFO. minimize the total cost (cost of maintenance plus cost of downtime). average time a customer spends waiting in a queue. (d) we will have to consider the amount of space available for the queue. (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. A customer in the queue is usually patient. ANSWER: e 14.101 The utilization factor for a system tells one the (a) (b) (c) (d) (e) mean number of people served divided by the mean number of arrivals per time period.103 Assume that we are using a waiting line model to analyze the number of service technicians required to maintain machines in a factory. minimize the percent of idle time of the technicians. percent idle time. (e) none of the above ANSWER: c 204 . the size of the calling population is important because (a) we have models only for problems with infinite calling populations. proportion of the time the service facilities are in use. minimize the downtime for individual machines.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.102 Which of the following is not an assumption about the queue in the waiting line models presented in this chapter? (a) (b) (c) (d) (e) Queue length is considered to be unlimited. minimize the number of machines needing repair. (b) we have models only for problems with finite calling populations. Our goal should be to (a) (b) (c) (d) (e) maximize productivity of the technicians. none of the above ANSWER: e 14. Customers arrive to enter the queue in a random fashion.104 In queuing problems. blithering. none of the above ANSWER: e 14. degree to which members of the queue are orderly and quiet. sequence in which members of the queue arrived. exactly doubled. ANSWER: b 205 .106 The customer who arrives at a bank. and leaves to return at another time is (a) (b) (c) (d) (e) balking. none of the above ANSWER: a 14. could be any of the above depending on other parameters of the problem. all of the above ANSWER: d 14. cropping.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.105 The behavior of jumping from one queue to another trying to get through as quickly as possible is called: (a) (b) (c) (d) (e) balking.108 If the arrival rate and service times are kept constant and the system is changed from a twochannel system to a single-channel system. reneging. cropping. the same as before. reneging. blithering. sees that there is a long line. increased.107 The term queue discipline describes the (a) (b) (c) (d) (e) degree to which members of the queue renege. sequence in which members of the queue are serviced. then the average time an arrival will spend in the waiting line is (a) (b) (c) (d) (e) decreased. a number of single-channel. Some of the checkouts are reserved for those customers with fewer than twelve items. first out) LIFO (last in.110 Assume that we wish to study the performance of checkout stations in a large grocery store. multi-channel.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. and still other checkouts are open to all customers. average number of customers in the system will be increased. none of the above (a). the (a) (b) (c) (d) (e) average waiting time will be decreased. single-phase models. or pharmacy. multi-phase queuing model. (b).Waiting Lines and Queuing Theory Models l CHAPTER 14 14. multi-channel. single-phase models. except that the service time becomes constant instead of exponential. single-phase model. single-channel. single-phase model.112 The most likely queue discipline to be followed in a hospital emergency room is (a) (b) (c) (d) (e) FIFO (first in. We should employ (a) (b) (c) (d) (e) a multi-channel. last out) WCF (worst case first) none of the above ANSWER: e 206 . a simulation model. first out) FILO (first in. average queue length will be increased. we should employ a (a) (b) (c) (d) (e) single-channel. simplifying assumptions to make the problem fit one or another of the analytical models. multi-phase model. multi-phase model. two separate multi-channel. including the mean arrival rate and service rate.109 If everything else remains constant. ANSWER: c 14. other checkouts are reserved for those customers from the bakery. deli. none of the above ANSWER: e 14. & (c) ANSWER: a 14. while service times follow an exponential distribution.75 0.33 none of the above ANSWER: d 14. first served basis.33 ANSWER: b 207 . The arrival rate follows a Poisson distribution.57 1. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute.115 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come. while service times follow an exponential distribution. The arrival rate follows a Poisson distribution.19 1.43 1.113 Customers enter the waiting line at a cafeteria on a first come.114 Customers enter the waiting line at a cafeteria on a first come. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute. while service times follow an exponential distribution.76 0. what proportion of the time is the server busy? (a) (b) (c) (d) (e) 0. first served basis.25 0.33 1. what is the average number of customers waiting in line behind the person being served? (a) (b) (c) (d) (e) 0. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute.43 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. first served basis. what is the average number of customers in the system? (a) (b) (c) (d) (e) 0. The arrival rate follows a Poisson distribution.67 none of the above ANSWER: a 14.67 0.57 0. on average. The arrival rate of customers follows a Poisson distribution.717 7.817 1. The average arrival rate is seven per minute and the average service rate is four per minute for each of the two clerks.067 0.067 0.67 minutes 0. how much time will elapse from the time a customer enters the line until he/she leaves the cafeteria? (a) (b) (c) (d) (e) 0. The arrival rate of customers follows a Poisson distribution. while the service time follows an exponential distribution. The arrival rate follows a Poisson distribution. while the service time follows an exponential distribution. There are two postal clerks who each work at the same rate.932 5.75 minutes 0.33minutes ANSWER: d 14.429 4. What is the average length of the line? (a) (b) (c) (d) (e) 3.50 minutes 0. first served basis. If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute.116 Customers enter the waiting line to pay for food as they leave a cafeteria on a first come. There are two postal clerks who each work at the same rate.467 none of the above ANSWER: c 14.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. How long does the average person spend waiting for a clerk to become available? (a) (b) (c) (d) (e) 0.118 A post office has a single line for customers to use while waiting for the next available postal clerk.33 minutes 1.875 none of the above ANSWER: b 208 .117 A post office has a single line for customers to use while waiting for the next available postal clerk. while service times follow 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. The car wash takes exactly 4 minutes (this is constant). cars arrive randomly at a rate of 7 every 30 minutes. cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant).171 7. On average.533 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. while the service time follows an exponential distribution. how long would each driver have to wait before receiving service? (a) (b) (c) (d) (e) 28 minutes 32 minutes 17 minutes 24 minutes none of the above ANSWER: a 14. what would the length of the line be? (a) (b) (c) (d) (e) 8.120 At an automatic car wash. What proportion of the time are both clerks idle? (a) (b) (c) (d) (e) 0.875 0.123 At an automatic car wash.817 none of the above ANSWER: b 14.750 0. how long would each car spend at the car wash? (a) (b) (c) (d) (e) 28 minutes 32 minutes 17 minutes 24 minutes none of the above ANSWER: b 14. On average. The arrival rate of customers follows a Poisson distribution. The average arrival rate is seven per minute and the average service rate is four per minute for each of the two clerks.119 A post office has a single line for customers to use while waiting for the next available postal clerk. The car wash takes exactly 4 minutes (this is constant). cars arrive randomly at a rate of 7 cars every 30 minutes. how many customers would be at the car wash (waiting in line or being serviced)? 209 .467 none of the above ANSWER: c 14. The car wash takes exactly 4 minutes (this is constant).067 0. On average.122 At an automatic car wash.121 At an automatic car wash.467 6. On average. There are two postal clerks who each work at the same rate. cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant).171 7.Waiting Lines and Queuing Theory Models l CHAPTER 14 (a) (b) (c) (d) (e) 8.467 0.533 0.905 14. cars arrive randomly at a rate of 7 every 30 minutes.467 6. cars arrive randomly at a rate of 7 every 30 minutes. The car wash takes exactly 4 minutes (this is constant). none of the above ANSWER: c 14. on average.125 At your automatic car wash.000 0. you have space for 7 cars in the waiting area.643 2. The utilization factor for this system is (a) (b) (c) (d) (e) 0. You should consider (a) (b) (c) (d) (e) adding additional parking spaces.467 none of the above ANSWER: b 14.933 none of the above ANSWER: d 14.124 At an automatic car wash.533 1. modifying the system to speed up service and reduce waiting time.643 0. At the moment.217 According to the information provided in Table 14-4. optional service features. adding additional.191 2.609 minutes 6. how many units are in the line? (a) (b) (c) (d) 0.126 Table 14-4 M/M/2 Mean Arrival Rate: Mean Service Rate: Number of Servers: 9 occurrences per minute 7 occurrences per minute 2 Solution: Mean Number of Units in the System: Mean Number of Units in the Queue: Mean Time in the System: Mean Time in the Queue: Service Facility Utilization Factor: Probability of No Units in System: 2.307 0.037 minutes 0.217 210 .191 0. reducing the price you charge for washing the car. 905 minutes 2.128 Using the information provided in Table 14-4: Counting each person being served and the people in line.037 14.609 none of the above ANSWER: b 14.037 minutes 14. what proportion of the time is at least one server busy? (a) (b) (c) (d) (e) 0. what is the average time spent by a person in this system? (a) (b) (c) (d) (e) 0.191 6.191 minutes 6.905 2. on average.643 0. how many people would be in this system? (a) (b) (c) (d) (e) 0.091 none of the above ANSWER: c 14.783 0.129 According to the information provided in Table 14-4.Waiting Lines and Queuing Theory Models l CHAPTER 14 (e) 0.127 According to the information provided in Table 14-4.905 0.609 minutes none of the above ANSWER: d 211 .905 ANSWER: e 14. 714 1.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.321 minutes 0.179 minutes 0.3% could be any of the above.7% 64.714 minutes 0. on the average.893 0. how many customers are in the system? (a) (b) (c) (d) (e) 0.132 According to Table 14-5.607 0.714 According to Table 14-5.893 minutes none of the above ANSWER: c 14.893 0.375 none of the above ANSWER: c 212 . which presents the solution for a queuing problem with a constant service rate. depends upon other factors. none of the above ANSWER: c 14. what percentage of the total available service time is being used? (a) (b) (c) (d) (e) 90.179 minutes 0. on the average.607 minutes 0.5% 21.607 0. how much time is spent waiting in line? (a) (b) (c) (d) (e) 1.130 According to the information provided in Table 14-4. which presents the solution for a queuing problem with a constant service rate.131 Table 14-5 M/D/1 Mean Arrival Rate: Constant Service Rate: 5 occurrences per minute 7 occurrences per minute Solution: Mean Number of Units in the System: Mean Number of Units in the Queue: Mean Time in the System: Mean Time in the Queue: Service Facility Utilization Factor: 1. 133 According to Table 14-5. which presents a queuing problem solution for a queuing problem with a constant service rate.321 0. on average. which presents the solution for a queuing problem with a constant service rate.134 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) (b) (c) (d) (e) 0.893 (e) none of the above ANSWER: a 14.136 According to Table 14-5. which presents the solution for a queuing problem with a constant service rate.893 minutes 0. what percentage of available service time is actually used? (a) (b) (c) (d) (e) 0.607 (d) 0.714 minutes 1. how many minutes does a customer spend in the system? (a) (b) (c) (d) (e) 0.179 none of the above ANSWER: e 14.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.321 minutes 0.714 none of the above ANSWER: c 213 .643 0.217 0.607 minutes none of the above ANSWER: b 14.643 0.286 0. how many customers arrive per time period? (a) 5 (b) 7 (c) 1.217 0.135 According to Table 14-5. on average. The utilization factor for this system is (a) (b) (c) (d) (e) 0.547 0.140 At a local fast food joint. The average time in the queue for each arrival is (a) (b) (c) (d) (e) 2 minutes 4 minutes 6 minutes 8 minutes 10 minutes ANSWER: d 214 . The average wait time for arrivals is (a) (b) (c) (d) (e) 5.139 At a local fast food joint.137 At a local fast food joint.0 minutes 2.5 minutes none of the above ANSWER: c *14.000 0. cars arrive randomly at a rate of 12 every 30 minutes.4 minutes 6.854 none of the above ANSWER: a *14. The fast food joint takes exactly 2 minutes (this is constant).800 0.138 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. cars arrive randomly at a rate of 12 every 30 minutes.Waiting Lines and Queuing Theory Models l CHAPTER 14 *14.467 0. 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.0 minutes 8. The utilization factor for this system is (a) (b) (c) (d) (e) 0.800 0.133 none of the above ANSWER: c *14. The fast food joint has restructured their serving system so that service takes exactly 2 minutes (this is constant) per arrival.723 1. 1 minutes 9.5 minutes 3. with an average service time of 0.6 minutes 2. Jack Burns. the average customer will wait in line (a) (b) (c) (d) (e) 9.141 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. the 215 . On average.. make the shop into a two-channel system. There are five toll booths. Service times are exponentially distributed with an average of 15 minutes.0125 cars 0.0176 cars 0. the JLUBE owner.Waiting Lines and Queuing Theory Models l CHAPTER 14 *14. Jack Burns.144 Cars approach a set of toll booths at the rate of 75 cars per hour.145 Cars approach a set of toll booths at the rate of 75 cars per hour.e.0 minutes none of the above ANSWER: e *14.0179 cars 0. the total time an average customer spends in the system will be (a) (b) (c) (d) 37 minutes 2.0714 cars ANSWER: d *14. how long is the line in front of a specific toll booth? (a) (b) (c) (d) (e) 0. For the two miles before a car reaches a toll booth. i. Under this new scheme.6 minutes ANSWER: c *14.143 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. with an average service time of 0.0270 cars 0.5 minutes 24. has decided to open a second work bay.e.6 minutes 2. Under this new scheme.6 minutes 24.142 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes.5 minutes. a car commits to a specific toll lane nearly a mile before it reaches the toll booth. make the shop into a two-channel system.5 minutes. For the two miles before a car reaches a toll booth.5 minutes 4. Under heavy traffic conditions. i. Service times are exponentially distributed with an average of 15 minutes. There are five toll booths. Service times are exponentially distributed with an average of 15 minutes.0 minutes 2..1 minutes none of the above ANSWER: a *14. has decided to open a second work bay. The average customer waits in line (a) (b) (c) (d) (e) 3. the highway is five lanes wide. the JLUBE owner. 3 minutes with one window to 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 highway is five lanes wide. Waiting time drops from 32.0001 cars 0. What service rate is necessary to keep the average wait time less than 5 minutes? (a) (b) (c) (d) (e) 45 per hour 47 per hour 49 per hour 50 per hour none of the above ANSWER: d 216 . She is considering enlarging the facility by constructing a second takeout window.3 minutes with two. On average.146 Cars appear to approach a local Burger Basket Restaurant at the rate of 20 per hour. the owner.1 minutes with one window to 21.147 Customers arrive at the local PharmCal gas station at the rate of 40 per hour.7 minutes with two.0179 cars 0. How does the waiting time with two windows change from that with only a single window? (a) (b) (c) (d) (e) Waiting time drops from 27. Waiting time drops from 20. how long is the line in front of a specific toll booth? (a) (b) (c) (d) (e) 0. There is no change. none of the above ANSWER: a *14.0100 cars 0. Average service rate averages 22 per hour. Under light traffic conditions. Gale Johnson.5 minutes with two. a car does not have to commit to a specific toll lane until actually approaching the toll booths.3 minutes with one window to 6. has become concerned about the waiting time under the current configuration.5000 cars none of the above ANSWER: b *14. (a) Find the proportion of the time that the employees are busy. Find the proportion of the time that the employee is busy. (c) Find the expected time a person spends just waiting in line to have his question answered.50 (c) L = 1 (d) Lq = 0. Find the average time a person seeking information spends at the desk.50 (e) W = 0. ANSWER: (a) Po = 0.667 (b) Lq = 1. Assuming the secretary works 8 hours a day: (a) (b) (c) (d) What is the secretary's utilization rate? What is the average waiting time before the secretary types a letter? What is the average number of letters waiting to be typed? What is the probability that the secretary has more than 5 letters to type? 217 . (b) Find the average number of people waiting in line to get some information. ANSWER: (a) = 0. Find the average number of people waiting in line to get some information. It takes an average of four minutes to answer a question.Waiting Lines and Queuing Theory Models l CHAPTER 14 PROBLEMS 14. 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.150 Due to a recent increase in business. it is believed that people will arrive at the desk at the rate of 20 per hour. It is assumed that arrivals are Poisson and answer times are exponentially distributed. a secretary in a certain law firm is now having to type 20 letters a day on average.0667 hours (f) Wq = 0.148 A new shopping mall is considering setting up an information desk manned by one employee. It is assumed that arrivals are Poisson and answer times are exponentially distributed.0533 hours 14. Based upon information obtained from similar information desks.0333 hours 14. It takes her approximately 20 minutes to type each letter. Find the expected time a person spends just waiting in line to have his question answered. Find the average number of people receiving and waiting to receive some information.149 A new shopping mall is considering setting up an information desk manned by two employees. (a) (b) (c) (d) (e) (f) Find the probability that the employee is idle. It takes an average of two minutes to answer a question.50 (b) = 0.0667 (c) Wq = 0. 2 hours Calls arrive at the hotel switchboard at a rate of two per minute. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every six minutes.75 L=3 Lq = 2. There is only one switchboard operator at the current time.5 = 0.5 L=1 Lq = 0. average number of dogs waiting to be shot. proportion of the time that Sam is busy. according to a Poisson distribution.5 Wq = 0.153 Po = 0. The Poisson and exponential distribution appear to be relevant in this situation.1 hours Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. average time a dog waits before getting shot.15 hours W = 0. ANSWER: (a) (b) (c) (d) (e) (f) 14.335 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school.25 Wq = 0. average time a dog waits before getting shot. proportion of the time that Sam is busy. according to a Poisson distribution. Sam can "shoot" a dog every three minutes. Find the: (a) (b) (c) (d) (e) (f) probability that Sam is idle. Also assume that Sam's shooting times are exponentially distributed. Also assume that Sam's shooting times are exponentially distributed.Waiting Lines and Queuing Theory Models l CHAPTER 14 ANSWER: (a) (b) (c) (d) 14.05 hours W = 0.152 Po = 0. average amount (mean) of time a dog spends between waiting in line and getting shot. ANSWER: (a) (b) (c) (d) (e) (f) 14.833 Wq = . average number of dogs waiting to be shot. average amount (mean) of time a dog spends between waiting in line and getting shot. Sam can "shoot" a dog every three minutes. average number of dogs being shot or waiting to be shot.151 = 0. Find the: (a) (b) (c) (d) (e) (f) probability that Sam is idle. The average time to handle each of these is 15 seconds. average number of dogs being shot or waiting to be shot.167 P(n>5) = 0. 218 . It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every four minutes.25 = 0.20833 days Lq = 4. 20 14. (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? ANSWER: (a) Lq = 1.154 At the start of football season.155 At the start of football season.500 (b) Wq = 0.2 (b) W = 1 time unit (10 minutes) (c) = 0.8 14. while the service time follows an exponential distribution.500 14.1 hours (6 minutes) (c) Po = 0. each of whom operate at the same rate of speed.04 minutes) (c) = 0. The arrival rate of cars follows a Poisson distribution.01667 (b) W = 0. and the average time to transact business is one minute.Waiting Lines and Queuing Theory Models l CHAPTER 14 (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? ANSWER: (a) = 0. the ticket office gets very busy the day before the first game. the ticket office gets very busy the day before the first game. The average arrival rate is 20 per hour and the average service time 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? ANSWER: (a) Lq = 0. (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? ANSWER: (a) Lq = 3.104167 time units (1.333 (b) W = 0. and the average time to transact business is two minutes. Customers arrive at the rate of four every ten minutes.156 A post office has a single drive-in window for customers to use. Customers arrive at the rate of four every ten minutes.25 time units (minutes) (c) Lq = 0.333 219 . There are two servers in the ticket office. 167 (e) W = 0. the manager expected that it would take approximately four minutes for the Information Desk employee to help the average person. Find the proportion of the time that the employee is busy. Find the average time a person seeking information spends waiting and at the desk.8545 (b) W = 0. Each of these runs an average of 90 minutes without requiring any attention from the technician. It will also take longer to answer their questions – approximately four minutes per person on average. Each time the technician is required to adjust a computer. Find the proportion of the time that the Information Desk employee is busy. Find the average time a person seeking information spends waiting and at the desk. leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. an average of 15 minutes (following an exponential distribution) is required to fix the problem. how long is a computer out of service? (c) What is the average waiting time in the queue to be serviced? ANSWER: (a) Lq = 0.2 hours = 12 minutes (f) Wq = 0.159 The new Providence shopping mall is considering setting up an information desk manned by one employee. Find the average number of people receiving and waiting to receive some information. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. Find the average number of people waiting in line to get some information. it is expected that people will arrive at the desk at about twice the rate for most malls.167 (b) = 0. how many computers are waiting for service? (b) On average. The expected rate is 25 per hour. ANSWER: No solution – line would increase indefinitely. Find the expected time a person spends just waiting in line to have his question answered.157 A company has six computers that are used to run an automated manufacturing facility. Because of the complex design of the mall. (a) On average.2906 hours 14.5406 hours (c) Wq = 0. ANSWER: (a) Po = 0.167 hours = 10 minutes 220 . (a) (b) (c) (d) (e) (f) Find the probability that the employee is idle.Waiting Lines and Queuing Theory Models l CHAPTER 14 14.833 (c) L = 5 (d) Lq = 4. the 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) (b) (c) (d) (e) (f) Find the probability that the Information Desk employee is idle. By utilizing a new map and special guide signs. The layout for this mall is quite complex. Find the average number of people receiving and waiting to receive some information. Find the average number of people waiting in line to get some information.158 The new Providence shopping mall is considering setting up an information desk manned by one employee. 14. Under the original plan. Find the expected time a person spends just waiting in line to have his question answered. 833 hours (50 minutes) for employees versus 0. and (b) mall staff.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour. he has met that goal: time spent being served: 0.160 The new Providence shopping mall has been considering setting up an information desk manned by one employee.152 hours.833(8 hours per day)(2 workers) = 13.161 The new Providence shopping mall has been considering setting up an information desk manned by one employee. Utilized time = 0. (a) find the probability that both Information Desk employees are idle.09 minutes Yes. If the manager has a goal that. he believes that the required service time can be reduced to an average of two minutes per patron. and mall staff or delivery persons to arrive at the rate of 5 per hour. (b) determine whether the patron or the staff person is likely to have the longer wait. and an average of ten minutes to answer those of a staff or delivery person. (e) Utilization factor is 0. He has now come to realize that employing only a single person at the information desk would lead to a very lengthy line – theoretically. leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. Assuming that the two-desk concept is implemented. (a) (b) (c) (d) 221 . the other to help staff and delivery persons.056. Under the original plan.788 Wq = 0.833 Lq = 3. the time spent having one's question answered is less than half the time spent waiting. Under the original plan. the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person.066 hours. The layout for this mall is quite complex. It is likely to take an average of two minutes to answer the questions of a patron. leading the mall manager to expect a higher than normal arrival rate for persons seeking assistance. 14. The manager believes that he can expect patrons to arrive at the rate of 20 per hour. to employ two staff members at the information desk. 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? (f) What would the arrival rate have to be for the manager to meet the 45 minute idle time goal? ANSWER: = 0. Find the average number of people waiting in line to get some information. the manager has decided that he wants to consider another option: establishing two information desks – one desk to help mall patrons. the manager expected that it would take approximately 4 minutes for the Information Desk employee to help the average person.33 hours Nonutilized time in hours is given by: 16 – 13. Find the expected time a person spends just waiting in line to have his question answered.152 hours or 9. ANSWER: (a) Probability that both employees are idle: 0. time spent waiting: 0. He did not meet this goal. By utilizing a new map and special guide signs. for the average patron.67 hours 160 minutes.33 = 2. It appears that a reasonable expectation is an arrival rate of approximately 25 patrons per hour.066 hours (4 minutes) for patrons. an infinite line! He has decided.833. or delivery persons. The manager has also noticed that the people seeking help at the information desk may come from one of two groups: (a) mall patrons. (a) (b) (c) (d) Find the proportion of the time that the employees are busy. therefore. The layout for this mall is quite complex. (b) The employees will have to wait longer: 0. Therefore. Is this likely to present a serious problem? 222 .833 hours = 50 minutes P(more than three students waiting) = P(more than four students in system) = 0. (a) (b) (c) (d) What is the staff person's utilization rate? What is the average time a student has to wait before getting his application processed? What is the probability that there are more than three students waiting in line? The office has a total of five chairs for students. Sam can "shoot" a cat every four minutes. how many cats will be in the waiting room? (c) If a cat has to wait more than 20 minutes.162 Bank Boston now has a branch at Bryant College. Is it likely that any students will have to stand? ANSWER: (a) (b) (c) (d) 14. This year.402 P(someone will have to stand) = P(more than five students in system) = 0. according to a Poisson distribution. freshmen arrived at the office at a rate of 40 per day (8-hour day). The branch is always busiest at the beginning of the college year when freshmen and transfer students arrive on campus and open accounts.163 = 0. It is estimated that the cats will arrive independently and randomly throughout the day at a rate of one cat every five minutes. four for waiting.833 Wq = 0.Waiting Lines and Queuing Theory Models l CHAPTER 14 (f) Arrival rate would have to be approximately 27. Also assume that Sam's shooting times are exponentially distributed.335 Sam the Vet is running a rabies vaccination clinic for cats at the local grade school. On average. it takes the Bank Boston staff person about ten minutes to process each account application. (a) What is the probability that a cat will have to wait? (b) On the average. and one at the service desk.2 persons per hour. 14. it will become obnoxious. Single seller: Wq = 8 minutes. 14. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every six minutes. There is only one lot attendant at the current time. Sam can "shoot" a dog every three minutes.2 (c) The average time waiting is 0. 14.164 Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Customers arrive at the rate of four every ten minutes. The average time to get a ticket and proceed to a parking space is two minutes.33 autos 14. Sam would like to have each waiting dog placed in a holding pen during the waiting period. the ticket office gets very busy the day before the first game. The Poisson and exponential distribution appear to be relevant in this situation. how many cages should he prepare? ANSWER: He needs to prepare two cages–the probability of having more than three dogs in the system is less than 10 percent. (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? ANSWER: (a) = 0. the customer would be better off with the slower ticket sellers. 223 .33 hours (or 20 minutes) so this will likely be a problem.688 minutes.166 At the start of football season.165 Cars arrive at the entrance to a parking lot at the rate of 20 per hour. according to a Poisson distribution. or two ticket sellers.8 (b) 3.667 (b) Wq = 4 minutes (c) Lq = 1. Two sellers: Wq = 1. Would the customer be better off if the stadium employed a single ticket seller who could service a customer in two minutes. each of whom could service a customer in three minutes? ANSWER: Considering waiting times.Waiting Lines and Queuing Theory Models l CHAPTER 14 ANSWER: (a) 0. Also assume that Sam's shooting times are exponentially distributed. If Sam wants to be certain to have enough cages to accommodate all dogs at least 90 percent of the time. 14. Customers arrive at the rate of four every fifteen minutes. The probability of having more than two machines in the system either receiving or awaiting repairs is 0.1 percent of the time.167 At the start of ballet season.05. how many car-lengths should they make the driveway leading to the window? ANSWER: They should make the driveway approximately six car-lengths long. Probability of having more than seven cars in the system is 0.84 (b) W = 0. 14.278 hours (16.Waiting Lines and Queuing Theory Models l CHAPTER 14 14. and the average time to transact business is 6 minutes.039. both of whom operate at the same speed. The arrival rate of cars follows a Poisson distribution. The average arrival rate is 20 per hour and the average service time is two minutes.67 minutes) including waiting and service. If the post office wants to accommodate all of the waiting cars at least 95 percent of the time. 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.169 A company has six computers that are used to run an automated manufacturing facility. Each time the technician is required to adjust a computer. while the service time follows an exponential distribution. Each of these runs an average of 90 minutes without requiring any attention from the technician. ANSWER: She will require two machines. (c) One or more of the servers is busy 71. (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? ANSWER: (a) Lq = 2. an average of 12 minutes (following an exponential distribution) is required to fix the problem. the ticket office gets very busy the day before the first performance. 224 .168 A post office has a single drive-in window for customers to use. There are two servers in the ticket office. 177 What is meant by a multi-phase system? ANSWER: service is provided at more than one station. define what is meant by reneging.Waiting Lines and Queuing Theory Models l CHAPTER 14 SHORT ANSWER/ESSAY 14.171 With regard to queue theory. ANSWER: the average number of customers in line. hence.178 What is represented by ? ANSWER: the proportion of the time service facilities are in use 225 . ANSWER: the situation when customers in a queue choose to leave the queue unserviced 14. 14. one after the other 14. the average waiting times.170 With regard to queue theory. define what is meant by balking. and the percent of system idle time 14.175 What is meant by a multi-channel queuing system? ANSWER: more than one service facility system all fed by the same queue 14.176 What is meant by a single-phase system? ANSWER: arrivals leave the system after receiving service at only one station 14.173 List three key operating characteristics of a queuing system.172 How is FIFO used in describing a queuing theory problem? ANSWER: FIFO describes the queue discipline in that the first customers to arrive are the first customers to be serviced and. ANSWER: the situation when arriving customers choose to not enter the queue 14. the first out.174 What is meant by a single-channel queuing system? ANSWER: one service facility system fed by one queue 14.