Brief History of Queueing Theory and Broad Overview 1 · Brief History of Queueing Theory and Broad...

The Sales Counter at Arm-and-a-Leg Tickets Teacher Resources

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TEACHER RESOURCES

Brief History of Queueing Theory and Broad Overview1 All of us have experienced the annoyance of having to wait in line. Unfortunately, this phenomenon continues to be common in congested, urbanized and “high-tech” societies. We wait in line in our cars in traffic jams or at toll booths; we wait on hold for an operator to pick up our telephone calls; we wait in line at supermarkets to check out; we wait in line at fast-food restaurants; and we wait in line at banks and post offices. As customers, we do not generally like these waits, and the managers of the establishments at which we wait also do not like us to wait, since it may cost them business. Why then is there waiting? The answer is relatively simple: There is more demand for service than there is facility for service available. Why is this so? There may be many reasons; for example, there may be a shortage of available servers; it may be infeasible economically for a business to provide the level of service necessary to prevent waiting; or there may be a limit to the amount of service that can be provided. Generally, this limitation can be removed with the expenditure of capital. To know how much service should be made available, one would need to know answers to such questions as, “How long will a customer wait?” and “How many people will form in the line?” Queueing theory attempts to answer these questions through detailed mathematical analysis, and in many cases it succeeds. The word “queue” is in more common usage in Great Britain and other countries than in the United States, but it is rapidly gaining acceptance in this country. However, it must be admitted that it is just as unpleasant to spend time in a queue as in a waiting line. A queueing system can be simply described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served. The term customer is used in a general sense and does not imply necessarily a human customer. For example, a customer could be a ball bearing waiting to be polished, an airplane waiting in line to take off, a computer program waiting to be run, or a telephone call waiting to be answered. Queueing theory, as such, was developed to provide mathematical models to predict behavior of systems that attempt to provide service for randomly arising demands and can trace its origins back to a pioneer investigator, Danish mathematician named A. K. Erlang, who, in 1909, published The Theory of Probabilities and Telephone Conversations based on work he did for the Danish Telephone Company in Copenhagen, Denmark. Work continued in the area of telephone applications, and although the early work in queueing theory picked up momentum rather slowly, the trend began to change in the 1950s when the pace quickened and the application areas broadened well beyond telephone systems.

1 Excerpts taken from D. Gross & C. M. Harris. (1998). Fundamentals of Queueing Theory (3rd ed.). New York: John Wiley & Sons.


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There are many valuable applications of the theory, including traffic flow (vehicles, aircraft, people, communications), scheduling (patients in hospitals, jobs on machines, programs on a computer), and facility design (banks, post offices, amusement parks, fast-food restaurants). Today, we encounter a myriad of queues every day of our lives, and queueing theory, when it can, helps us to navigate around these. Psychology of Queueing The mathematics of queueing theory enables a decision-maker to model the behavior of a queueing system. Mathematical equations are used to calculate the time spent waiting and the number of customers waiting. In some situations, if the lines are too long, customers may go elsewhere to be served. In this case the queueing manager is interested in the number of customers lost due to long waits. Generally, if the waiting time seems excessive and customers are dissatisfied, a manager will explore cost-effective strategies for increasing the service capacity by adding more servers or increasing the speed of service. However, raw numbers fail to tell the whole story. The experience of waiting in line is influenced by the waiting area environment and our expectations as to the length of the wait. Imagine having to wait standing up in a dentist’s office for twenty minutes, while a patient is screaming in an adjacent examination room. Now imagine an alternative wait in comfortable chairs with access to the “latest” magazines for a variety of customer tastes. For your ten-year old child there is a video game machine, and the area is sound proof. Many companies (Disney is one example) have become expert in understanding the psychology of waiting. Waiting in a line that is moving seems less boring than standing still in the same spot. TV monitors with engaging pictures help keep visitors’ minds off the clock. In addition, if they can see and hear some of the excitement of those who have completed their wait, anticipation increases and waiting seems worthwhile. Lastly, expectations are a major factor in determining customer satisfaction. If customers approach a line and are told the wait will be fifteen minutes, at least they have the information to make an informed judgement as to join the line or not. If it turns out to be less than the quoted fifteen minutes then they are pleasantly surprised. Another dimension to the psychology of waiting relates to fairness. It can be very upsetting to see someone arrive after you in line and end up being served before you. This can happen if there are two separate lines. You might get stuck behind a customer who has a complicated request that takes a long time to service. As a result, people who have joined the other line even after you might ending up waiting less time. Many organizations have addressed this potential inequity by creating one line which all arriving customers enter. Thus, anyone who arrives after you must be further back in line and cannot begin service before you do.


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Aspects of a Queueing System In general, a queueing system involves customers who enter the system, wait in line (a queue), are served, and leave the system. While many familiar queueing situations involve only people as customers and servers, there are also many applications in which one or both of these entities is inanimate (e.g., an ATM could be the “server,” parts on an assembly line could be the “customers”). Nevertheless, the terms customer and server are still used. The key features of queueing systems can be classified as characteristics of arrivals, service discipline, and characteristics of service.

Characteristics of the stream of arrivals. Two important issues relevant to a queue involve the timing and types of arrivals. Usually, the timing of arrivals is described by specifying the average rate of arrivals per unit of time (a), or the average interarrival time (1/a). For example, if the average rate of arrivals, a = 10 per hour, then the interarrival time, on average, is 1/a = 1/10 hr = 6 min. In many simple applications, the pattern of times between successive arrivals and the service times both have exponential probability distributions of the form f(t) = ke-kt, where k = a for arrivals and k = h for service. This leads to the formulas x = a/h and L = x / (1 – x). In many cases, the arrival patterns and/or service patterns may not be exponential, such as in the case of a doctor’s scheduled appointments, and the formulas given here do not apply. For these cases, other methods of analysis must be used.

There are at least two issues related to the types of arrivals. First, the arrivals may occur one at a time or in batches (such as a carload, for example). Second, the arrivals might well be treated as essentially all the same, or they may be separated into groups according to some characteristic. For example, at a hospital emergency room, a triage nurse examines in-coming patients and prioritizes their order of treatment.

Service discipline. The service discipline is the rule, or set of rules, specifying which of the waiting customers is next to receive service. The most common service discipline is first-come-first-served. Other service disciplines include last-come-first-served, service-in-random-order, and shortest-processing-time. An example of the last case occurs when computer operators prioritize jobs waiting to be processed according to their expected processing time and run the shortest jobs first.

Other service discipline issues concern whether and how long customers will wait in line and, if there are lines, whether there is one line or multiple lines. There may be a single line even when there is more than one server (e.g., banks, post offices, and airport check-ins).

Service characteristics. In most simple queueing systems, each customer is served by one and only one server, no matter how many servers are present. The service time is usually assumed to be random and exponentially distributed, and when there are multiple servers, it is usually assumed that all servers are identical. That is, we assume that each is able to service customers at the same average rate, h. It might seem more natural to use s or perhaps c to represent this variable, but s and c are used for other variables in queueing theory. Therefore, throughout the Arm-and-a-Leg student activity, we have used the term


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“help” in place of “serve.” The reciprocal, 1/h, of the average rate of service is the average time required to serve one customer. For example, if a server can serve h = 3 customers per hour, on average, then 1/h = 1/3 hr = 20 min is the average service time for one customer.

In addition to queueing systems which employ a single server or multiple servers in parallel, some queueing systems employ multiple servers in sequence. This set-up is appropriate when customers must be served by more than one server and is often encountered in manufacturing settings. For example, parts on an assembly line can be thought of as “customers” waiting for “service” at various workstations (“servers”) on the line. A similar situation occurs in a cafeteria line where customers must wait for service at several points in the line.

Queueing formulas

Analysis of queues requires defining certain performance measures. Two key measures are:

L = the average number of customers in the system and

W = the average time a customer spends in the system.

If we let: a = the average rate of customer arrivals and

h = the average rate at which customers can be served (or helped, to use the language we have used in the student activity), then

1/a = the average time between successive arrivals and

1/h = the average service time per customer.

The ratio of customer arrival rate to customer service rate, x = a/h, also reflects the average number of arrivals during an average service time.2 This formula can also be shown to represent the fraction of time the server is busy.

In steady state (where x must be less than 1), that is, after the system has been operating for a long time,

L = x/(1 – x) = a/(h – a)

A steady state relationship between L and W is given by a formula known as Little’s Law:

L = aW

or equivalently: 2 Note that x = a(1/h), so that, for example, if the average arrival rate a = 10/hr and the average service time 1/h = ½ hr, then x=5 represents the average number of arrivals during an average service time. In this case, since x > 1, the line would grow indefinitely.


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W = L/a

Likewise, if Wq = the average time spent waiting in the queue before service begins, then Lq, the average length of the queue, is given by:

Lq = aWq

Objectives of the Module

The overall objective of the module is to motivate students to learn the mathematics they are studying in school We hope this will occur when students see mathematics applied to everyday life. The techniques of queueing theory are used to solve problems in business, industry, and government. Many of those problems will be familiar to students.

This module can be integrated into any first-year or second-year algebra course (or the integrated courses where students work with literal equations, graphing, or rational functions). Queueing theory is a branch of probability theory and the module complements the material covered in a high school probability and statistics course. The specific mathematics content objectives for both levels of students are:

• Evaluating algebraic expressions.

• Solving literal equations for one variable in terms of other variables.

• Analyzing and interpreting graphs and tables.

• Identifying the domain and range of functions under given conditions.

• Using a graphing utility.

In addition, second-year algebra students can address:

• Simplifying complex fractions.

• Identifying horizontal and vertical asymptotes of rational functions.

The module also offers students in a probability and statistics course opportunities for

• Exploring the nature of randomness

• Gathering and charting data in a meaningful environment

• Examining a stochastic process

The ticket sales problem asks students to think of ways to decrease the length of time waiting in a line. This is an example of queueing theory. Students will use mathematics to discover ways of decreasing the waiting time. More importantly, doing so will introduce the important mathematical concepts listed above.


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TEACHING NOTES

Initiating Activities Although many aspects of queueing theory may be familiar to students through their own everyday life experiences, there are other aspects of the theory that may not be familiar at all and which are not intuitive. For example, many students have difficulty understanding the nature of randomness. Because the queueing theory used in the Arm-and-a-Leg student activity assumes that both customer arrivals and customer service times are random, understanding the meaning of randomness is crucial to understanding the queueing models. Furthermore, students may better appreciate the key concepts in a queueing model if they first simulate a queueing situation. For these reasons, we strongly suggest using the three activities which follow before using the Arm-and-a-Leg student activity. Two similar versions of the second activity are provided. In using either Activity II.A. or II.B, you may prefer to have students run through the simulation once without measuring anything, then follow up with a discussion about what quantities it would make sense to measure.

Activity I: Coin Flips Materials: coin or two-color counter for each student paper pencil Purpose: To illustrate randomness 1. Ask students to number 1 through 20 on a piece of paper. Students will pretend to

flip a coin (simulate) 20 times by writing a “result,” H or T, next to each number. 2. On another piece of paper, ask students to write the letters A through T (a total of

20; using letters instead of numbers the second time will make it easier to keep track of the lists). This time, students will actually flip a coin 20 times and record H or T for each trial.

3. Place students in groups of 4. Give each group 4 simulated and 4 actual samples. Compare and contrast the two different sets of samples. Describe the patterns in each set.

4. As a class, discuss each group’s results. Emphasize the presence of longer strings of repetitive data (e.g., 4 or more heads in succession) when the coins are actually flipped. Long strings are generally absent when humans attempt to simulate random results.

5. Discuss as a class: What makes an event random? Brainstorm real life examples of random events.


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Activity IIA: The Queue at the Pencil Sharpener Materials: clock with a second hand one copy of the attached table (Table IIA) pencil sharpener pencil Purpose: To simulate a queue at the pencil sharpener. Objective: To illustrate vocabulary used in queueing theory: average number of

customers in a system and average wait time. 1. Select 3 student recorders. The remaining students will use a random number

generator to determine when he or she will enter the line to use the pencil sharpener. Five tables of random integers from 1 to 150 are attached if a random number generator is unavailable.

2. Start the simulation at the beginning of a minute. Stop the simulation 2 minutes later.

3. At the indicated time, in seconds (the random integer), the student will enter the queue at the pencil sharpener. Students should be prepared to enter the line several seconds before their indicated time.

4. The first recorder will collect data to determine the average number of people in the system at any given time. S/he will record the number of people at the pencil sharpener (including those in line and the person sharpening) at 90 seconds. This approximates the average number of people in the system at any given time.

5. The second recorder will collect data to determine the average wait time. S/he will record the wait time for the first person to join the line after 60 seconds have elapsed. The wait time includes the time in line and the time used to sharpen the pencil.

6. The third recorder will collect data to approximate the average service time. S/he will record the time it takes every fifth person to sharpen a pencil and compute the average of the times recorded. If there is no one in line, the service time begins as soon as the student reaches the pencil sharpener. If the student is waiting in line, the service time begins as soon as the previous student has finished.

7. To figure the average arrival rate, count the number of people who got to stand in line and/or use the sharpener. (Another recorder is not required, just ask students to raise their hands if they stood in line after each trial.) Then divide the number of students by 2 minutes. You may also want to ask students to express the average arrival rate in customers per hour.

8. Repeat this simulation 5 times. 9. The three recorders should average the results for all of the trials. 10. Average the arrival rates (from #6) for all five trials.


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Activity IIB: Waiting to be Helped Materials: clock with a second hand one copy of the attached table (Table IIB) 1 chair pencil Purpose: To simulate waiting to sit in a chair (as at a barber shop or hairdresser’s). Objective: To illustrate vocabulary used in queueing theory: average number of

customers in a system and average wait time. 1. Place the chair in the front of the room. A student sitting in the chair will represent a

customer being helped. 2. Select 2 student recorders. The remaining students will use a random number

generator to determine when he or she will enter the line to sit in the chair. (5 tables of random integers pairs are provided.) The first number the student is given (a random integer from 1 to 150) is the time at which he/she enters the line. The second number (a random integer from 3 to 12) is the amount of time spent sitting in the chair.

3. Start the simulation at the beginning of a minute. Stop the simulation 2 minutes later. 4. At the indicated time (the first random integer), the student will enter the queue to sit

in the chair. As soon as the chair is vacant, the next student should sit in the chair. Students should be prepared to enter the line several seconds before their indicated entrance time.

5. The two recorders will collect data to determine the average number of people in the system. They will record the number of people at the chair (including those in line and the person sitting) at 90 seconds. This approximates the average number of people in the system at any given time.

6. To determine the average help time, compute the average of the times each student who actually sat in the chair spent sitting in the chair. This will be the second of their two random integers.

7. To figure the average arrival rate, count the number of people who got to stand in line and/or sit in the chair. (Another recorder is not required, just ask students to raise their hands if they stood in line after each trial.) Then divide the number of students by 2 minutes. Again, you may want to ask students to convert the average rate of arrivals to customers per hour.

8. Repeat this simulation 5 times. 9. From the two recorders, average the data for each trial. Then find the average for all

five trials. 10. Average the arrival rates (from #7) for all five trials. 11. Average the help times (from #6) for all five trials.


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Table IIA: Random Integers for Use with “The Queue at the Pencil Sharpener.”

Trial #1 22 145 18 50 48 80 46 3 124 40 127 24 56 85 42 36 63 49 56 62 47 9 18 21 25 45 145 23 50 75 11 6 61 140 40 120

Trial #2

122 27 109 68 94 46 16 75 128 44 58 126 4 24 36 125 14 124 3 91 135 127 47 12

140 48 10 25 99 5 141 140 111 91 81 93

Trial #3

36 52 73 56 50 37 125 144 127 3 21 99 67 94 28 35 147 19 91 96 134 18 45 81 11 43 3 52 119 130 24 19 120 146 14 71

Trial #4

91 67 144 39 146 27 76 114 117 68 45 56 30 39 105 66 141 43 35 130 140 4 21 9 74 102 98 34 80 35 87 48 48 98 15 45

Trial #5

22 15 104 10 15 40 144 36 122 29 50 113 132 5 149 45 15 47 17 83 30 102 71 58 35 104 13 19 29 5 29 43 35 8 127 86


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Table IIB: Random Integers for Use with “Waiting to be Helped.”

Top number: Time to enter line; Bottom number: Amount of time sitting in the chair

Trial #1 42 11

62 6

117 11

33 11

6 9

86 12

1 9

49 4

100 12

135 7

99 4

9 7

58 4

68 7

39 9

24 9

135 6

134 6

89 6

79 5

111 8

52 8

98 5

80 9

14 7

148 7

84 8

57 5

135 6

41 7

Trial #2 100 6

103 7

43 12

57 12

55 7

129 9

20 11

92 10

136 7

38 10

30 12

66 12

116 8

14 5

36 6

12 11

56 5

128 6

71 8

123 10

120 10

56 12

2 7

60 7

46 10

100 12

96 10

86 8

105 3

30 8

122 9

47 7

75 8

111 11

94 3

94 7

Trial #3 102 9

56 4

60 4

80 9

80 8

83 10

123 6

80 7

101 3

128 7

14 11

52 6

123 8

104 6

120 10

100 11

110 3

3 3

131 12

37 6

123 7

56 4

1 10

56 10

28 5

18 6

55 3

122 12

133 4

46 10

102 5

51 10

141 8

44 7

58 11

110 4


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Table IIB: (Continued)

Top number: Time to enter line; Bottom number: Amount of time sitting in the chair

Trial #4 143 12

1 12

61 4

133 3

77 4

93 7

4 3

20 3

12 11

134 6

141 6

62 3

54 11

49 4

129 10

49 9

33 4

70 5

66 11

134 12

32 5

37 4

27 7

103 9

88 11

72 3

148 6

39 5

87 7

38 3

76 9

99 8

105 11

85 10

31 9

14 8

Trial #5

79 11

120 11

36 11

84 8

138 9

111 9

5 7

87 8

109 8

30 10

92 9

26 3

100 5

125 10

1 11

54 5

140 10

11 9

102 4

120 8

73 10

116 4

128 12

149 8

45 7

59 6

136 8

97 12

86 4

93 9

133 11

57 7

45 8

27 6

114 5

108 4


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Activity III: Writing About a Queueing Experience Purpose: To have students reflect on their experiences in Queueing Theory Objective: To illustrate the ways Queueing Theory can be applied in real situations. 1. Write a paragraph about a specific experience when you have had to wait your turn to

be helped. Describe the circumstances. Be sure to include

a. Location b. Time of day c. Type of help being sought d. How long did you wait in line? e. How long did it take to be helped? f. Do you think the customer arrival rate is the same all of the time, or do

you think it changes from time to time during the day? g. Does each person require the same length of time to be helped.

2. Be prepared to share your paragraph with the class. Suggestions for Using the Arm-and-a-Leg Problem In paragraph 3 of the Arm-and-a-Leg student activity, the individual situations noted in activities I.5, IIA.8-10, and IIB. 9-11 of the initiating activities could be cited here to reinforce the randomness of customer arrival times (assumption #1) and the variability of speed of service (assumption #2). Question #1 should be carefully guided. You may want to prompt/ask the students what actual data (i.e., what numbers Dr. Cue collected) were used to arrive at the average rates of 18 and 20. Students should realize that she did not necessarily even see 18 customers arrive during any one hour period of time, but that 18 is the average of the data collected after many observations. The same is true for the customer service rate. After introducing the variables in question #2, be sure everyone defines them as: a = 18 customers arriving/hour h = 20 customers helped/hour so that the variable names are consistent in the formulas for the rest of this exercise. Throughout this exercise, the correct labeling of units will reinforce the validity of the formulas. For example:

if a = the average number of customers arriving/hour, and h = the average number of customers that can be helped/hour,

then the units for a/h are: customershour

customershour

unitless"= "


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Thus, a/h, because it is a ratio of quantities having the same units, is itself unitless. For reasons such as this, it is important to ask students always to label the units. In question #3, if you want the average interval between customer arrivals, then using the units as a guide, hours/arrival is the reciprocal of arrivals per hour. In questions #3-#6, the time units must be consistent. In question #7, you may want to leave the answer as a common fraction to emphasize the simplification of complex fractions that will be encountered later. In discussing the material preceding question #9, explain that customer satisfaction is one of the main reasons for analyzing a system using queueing theory. Losing customers may occur at many different points in the system. If a customer, upon seeing the length of the line, decides not to join, it is called “balking.” If the customer joins the line, but leaves before receiving service, it is called “reneging.” In either case, customers are not being served and business is lost. Note that in question #9, the units of L will be customers. This is inconsistent with the formula L = x/(1 – x), which is unitless. This is something of a paradox, but in this sense, “customers” is not a real unit due to the unitless nature of L. In question #13 and #14, the effect of a small change in a on the values of L and W should be stressed. You may want to use the black line master provided to make an overhead of the table in question #14. Students could then fill in the values of x, L, and W at the overhead. The table can also be used to prompt answers for questions #16 and #17. Class discussions using a continual reference to what would be happening at the ticket sales office are appropriate. Questions 21 through 26 refer to the function L = x/(1 – x) in the real number system including x < 0 and not just for the limits defined by queueing theory. In question #22, you may want to introduce the concept of a vertical asymptote. Likewise, in question #26, you could introduce the concept of a horizontal asymptote. In question #23, as in question #21, the function L = x/(1 – x) is not defined at x = 1, and, therefore, the domain is restricted to all real numbers except x = 1. In question #30, you may want to relate x to a/h. If necessary, you could ask what specific values of a and h might have produced the values of x in the table accompanying question #32.


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In question #31, students are asked for the domain of x in the problem situation. In question #33, you may want to remind students what x and L represent. In lieu of questions #36 and 37, you may prefer to ask students to write a paragraph summarizing what Dr. Cue’s recommendations to Mr. Boss should be. It is important to emphasize the analysis of the Arm-and-a-Leg problem as well as the computation throughout the activity. Extensions Four problem situations, which extend the ideas in the Arm-and-a-Leg problem, follow. In extension 1, students manipulate some of the formulas underlying a single-server model. In extensions 2-4, multi-server models are introduced. In the second and third extensions, students will use the table immediately following extension 2 to determine appropriate values of L. In extension 4, students manipulate the two-server formulas. The formulas for systems which use more than two servers are quite difficult and beyond the scope of this module. In extensions 2 and 3, linear interpolation, the insertion of an approximate value between two values in a table, must be used. In extension 4, notice that for a multi-server model a may be greater than h, but x must still be less than 1. This leads quite naturally to a different defining equation for x. In a multi-server system with n servers,

.nha

x =


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Extension 1: Manipulate Equations

an expression for L in terms of a and h. 2. Use the previous result and the formula L = aW to find W in terms of a and h and

simplify the resulting expression. 3. W represents the average total time in the system, including the average time waiting

in line, Wq, and the average time being served, 1/h.

a. Express W in terms of Wq and h. b. Rewrite the equation for W and substitute to find Wq in terms of a and h.

4. a. Using a = 18 and h = 20, find Wq for the Arm-and-a-Leg problem.

b. What does this value of Wq mean?

havenow should Youfraction. resulting heSimplify t . with replace , and 1

If 1.ha

xha

xx

xL =

−=


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Extension 2: Add Servers – Evaluate Impact Using Tables One of the ways to improve service at Arm-and-a-Leg Tickets that Dr. Cue might have suggested to Mr. I. M. Boss is the addition of one or more servers at his busiest locations. This creates a multi-server queueing system. In this case, the mathematical formulas are much more complex, but tables of values for L can be used to analyze such systems. The key quantity that must be computed is f, the fraction of time each server is busy, on average. In a single server system, if customers arrive at the average rate of a per hour and are served at the average rate of h per hour, then a/h represents the proportion of time the server is busy, on average. 1. Assume that a = 18 and h = 20. What is the average proportion of time the server is

busy? Another way to look at the ratio a/h is that it represents the fraction of the total available service time that is actually being used. 2. Assume that there are n servers in the system. In terms of a, h, and n, represent f, the

fraction of each server’s available time that is actually being used. Suppose that Mr. I. M. Boss hires a second server at each of his busiest outlets. 3. Using the same values of a and h, on average, what fraction of the time will each of

the two servers be busy? The accompanying table (see next page) contains values of L for various values of n and f. 4. If n = 2 and f = 0.4, what is the value of L given in the table? 5. If n = 2 and f = 0.5, what is the value of L given in the table? 6. What is an appropriate value of L for n = 2 and the value of f you computed in

question 3.


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7. What does this value of L mean in the Arm-and-a-Leg problem? 8. Recall that W = L/a. Compute W for the Arm-and-a-Leg problem using two servers. 9. What does this value of W mean? 10. How do the values of L and W in the two-server example compare to the

corresponding values from the original (single-server) problem?

L

n: 1 2 3 4 5 6 f

0.10 0.111 0.202 0.300 0.400 0.500 0.600 0.20 0.250 0.417 0.606 0.802 1.001 1.200 0.30 0.429 0.659 0.930 1.216 1.509 1.805 0.40 0.667 0.952 1.294 1.661 2.040 2.427 0.50 1.000 1.333 1.737 2.174 2.630 3.099 0.55 1.222 1.577 2.008 2.477 2.969 3.475 0.60 1.500 1.875 2.332 2.831 3.354 3.895 0.65 1.857 2.251 2.732 3.258 3.812 4.385 0.70 2.333 2.745 3.249 3.800 4.382 4.984 0.75 3.000 3.429 3.953 4.528 5.135 5.765 0.80 4.000 4.444 4.989 5.586 6.216 6.871 0.85 5.667 6.126 6.689 7.306 7.959 8.636 0.90 9.000 9.474 10.054 10.690 11.362 12.061 0.95 19.000 19.487 20.083 20.737 21.428 22.146 0.98 49.000 49.495 50.100 50.764 51.466 52.194 0.99 99.000 99.498 100.106 100.773 101.478 102.210

L = the average number of customers in the system f = fraction of time the server(s) are busy n = number of servers


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Extension 3: Determine Number of Servers to Add Recall from Extension 2 that in a multi-server queueing system having n servers with an arrival rate of a customers per hour and a customer service rate of h

customers per hour for each server, the fraction of time that each server is busy is given by: At the Eastworst Airlines ticket counter at Metropolis International Airport, the number of customers arriving averages 76 per hour. The average time necessary for ticket purchase and baggage checking at the counter is 3 minutes for each customer. 1. On average, how many customers can each ticket agent handle per hour? 2. What is the fewest number of ticket agents needed to keep the line at the Eastworst

ticket counter from growing indefinitely? 3. If the Eastworst queueing system uses the fewest number of ticket agents from

question 2, what fraction of the time will each ticket agent be busy? 4. Using the table on p. 16, on average, how many people will be in the Eastworst

queueing system? 5. The formula W = L/a also applies to a multi-server system. Using the Eastworst

values of L and a, what is the average time an Eastworst customer will spend at the ticket counter, waiting in line and being served?

The management at Eastworst Airlines has received numerous complaints about the time it takes to be served at their ticket counter at Metropolis International Airport and is considering adding one or two additional ticket agents to the system.

nh

af =


19

6. For the values of n that are 1 and 2 more than your value of n in question #2, calculate the values of f, L, and W.

7. If you were the manager of the ticket counter at Eastworst Airlines, would you add

one ticket agent or two? Explain your reasoning. Extension 4: Two-Server Formula To address Arm & a Leg’s customer satisfaction, Dr. Cue recommended that Mr. Boss increase the average number of customers serviced per hour. One way to do this is to hire a faster server (one with a larger h). Another way is to add a second comparable server to lighten the single-server workload and speed up service. Assuming that the two servers have identical service rates, h, here are the formulas for a two-server single-line queue system in terms of a and h:

where Lq is the average number of customers in the line, L is the average number of customers in the system, W is the average total time waiting in the system, a is the average customer arrival rate, and h is the average customer service rate. L is related to Lq by the equation:

,

,4

4

,4

22

23

3

aL

W

ahah

L

haha

Lq

=

−=

−=

ha

LL q +=


20

The formulas, written in this form for Lq and L, allow us to find the values of Lq and L, given the arrival and service rates, a and h. The formula relating L and W applies to all types of queueing models and thus allows us to manipulate the formulas for L and W. 1. Write the formula for W in terms of a and h. (Hint: substitute L from above into the

formula L = aW). 2. Now verify that:

(Hint: substitute in L and Lq, and verify in terms of a and h.) 3. Suppose Mr. Boss hires another server who also has a customer service rate of h = 20

customers per hour. Complete the following table given the values of a and h.

a customer/

hr

h customer/

hr

Lq customers in

line

L Total

customers

W wait time (hr)

W wait time (min)

28 20 1.35 2.75 0.10 5.9 30 20 32 20 35 20 38 20 39 20

ha

LL q +=


21

Case Write-ups The Toilets Down Under The New Zealand Building Codes dictate the minimum number of “sanitary fixtures” in a building. This number is based on the number of people in the building and the type of facility. The code was extremely old and very inconsistent when the New Zealand Consultancy Services was contracted to use queueing theory to revise the tables in the building code. For three weeks, data about wait times were collected from thirteen different types of buildings including office buildings, schools, theatres, swimming pools, and shopping plazas. Arrival times for the stalls were measured by cutting a pair of infrared beams, and occupancy times were measured by magnetic switches on doors or infrared beams. Gender ratios, total building occupancy, and average peak rates were considered in the model. A spreadsheet simulation was used to develop new building codes to ensure that ninety percent of the people would wait one minute or less for a “sanitary fixture.” Although the spreadsheet simulation was only intended to produce starting values for a more complex simulation model, the spreadsheet model was extended and modified because of its ease of use, reliability, and accuracy. As expected, the old building codes provided more than enough “sanitary fixtures” for males to achieve the one minute wait goal, but not enough for females to achieve the same goal, especially in theatres, cinemas, and sports arenas. The new building codes corrected this and saved an estimated $82,000,000. These savings were largely due to a reduction in the amount of commercial office space allocated to “sanitary fixtures.” McNickle, Donald (1998). “Queueing for Toilets,” OR Insight, 11(2), 2-5. Department of Motor Vehicles In 1995, the Virginia Department of Motor Vehicles (DMV) determined that the department was no longer capable of meeting its customer service goals. The management determined there were:

• numerous customer complaints at all facilities, and • high levels of stress both for customers in the facility and employees.

A solution for this problem was found using a queueing theory software program. The goals for this program were to provide:

• DMV with a new image, • an effective management tool,


22

• improved customer satisfaction, • employee empowerment, • an average wait time of 15 minutes, • a maximum wait time of 30 minutes, and • an average service time of 7 minutes.

The developer of the program separated the service categories into 6 different kinds using the criteria: (a) the complexity of the job and (b) the length of time needed to process the job. In practice, the new system was designed to have customers form one or two lines upon entering the building. The server(s) for the line(s) assesses the customer’s request for service and assigns a ticket on which each job is letter-coded by type. Also on this ticket is a number representing the order in which customers will be called for service within this job category. Customers are also provided with a clipboard and all the paperwork needed for their job. A seating area is also provided for customers. This arrangement disguises the actual queue length and contributes to a more effective and relaxed climate for customers. Each service station is provided with an electronic display of the coded ticket of the customer being served. A voice pager has been added to alert customers to their turn and the proper window to go to for service. The manager of each DMV office can access the present traffic conditions in their facility and adjust the program parameters so that service can be improved. For example, an increase in the number of requests for vehicle titles would trigger a change in the frequency of service for this job. This is a rather complex model, but the results have made a remarkable improvement in:

• the perception of customers about the efficiency of service, • the actual time of service, and • the number of customers served per unit of time.

For example, in February 1997, in the Arlington, Virginia DMV facility, 17,929 customers were served with an average wait time of 21 minutes. After the implementation of the new system in Arlington, in February 1998, 20,843 customers were served with an average wait time of 10 minutes, 12 seconds. A by-product of the new system is a decrease in employee stress. Smoot, Janet, Jacobs, Bill, and Fadely, Michelle. (1997). Virginia Department of Motor Vehicles Queueing Management System. Richmond: DMV.


23

Coal Unloading Times The Detroit Edison Company owns and operates a coal-fired power plant in Monroe, Michigan. Coal is generally brought by train to the plant from mines in nearby states and is unloaded by a single unloader system. Originally, coal was brought into the power plant entirely by rail. Difficulties developed in meeting the plant’s coal needs as the generating capacity was increased. The more recent plan combined rail and lake-vessel delivery of coal to satisfy the plant’s requirements. Costly delays were occurring.

Several factors contributed to the difficulties – the major contributor being the design of the single-car unloader and its rate of breakdown. To alleviate the problems, Detroit Edison considered the addition of a second multimillion-dollar unloader system. Although a simulation model of the entire rail-coal movement system was available, an analytic model was built to focus on the unloader. The analytic model was a modified version of a standard single- and multiple-server finite-source queueing model. Other solutions to the problem were also studied:

§ Reducing the frequency of unloader breakdowns, § Reducing the repair time, and § Changing the cycle between the mine and the power plant. The factors considered were:

§ The impact of a second unloader system on the coal throughput, § L, the average number of trains in queue, and § W, the average time a train spent in the unloader system.

Assumptions about the model included:

§ Coal availability does not affect the train cycle rate. § All trains are the same size. § A train can wait at the facility while an unloader is being prepared. § When two unloaders are broken, two crews work independently on them. § When one unloader is broken, only one crew repairs it. § A train that is in a facility when one unloader breaks down is rerouted to the other

unloader facility. (The average repair time of the unloader is substantially greater than the time necessary to reroute the train.)

Data were collected from records on breakdowns, trip-completion times, the use of a single unloader and two unloaders. The analytic mathematical model provided answers within an hour, whereas the simulation model would have taken at least a day.

The results of the study showed management that they must differentiate between ways of increasing unloader availability, a concept not previously recognized. The analytic model, originally developed only to evaluate the addition of an unloader, provided ongoing information about changing the system configuration to meet system constraints (such as coal throughput and wait time). Chelst, K., Tilles, A. Z., and Pipis, J. S. (1981). “A Coal Unloader: A Finite Queueing System with Breakdowns,” Interfaces, 11( 5), 12-25.


24

Calling L. L. Bean L. L. Bean is a retailer of high quality outdoor goods and apparel. Twenty percent of their goods are sold through mail orders, 15% through in-store transactions, and 65% through phone orders. Because L. L. Bean receives 18% of its annual call volume in the three weeks prior to Christmas, this season makes or breaks the year financially. During this three- week period, the number of telephone agents is increased from 500 to 1,275, and the number of telephone lines is expanded from 150 to 576. In 1988, management was confronted with these problems: On average, 80% of callers received a busy signal when they used L. L. Bean’s 800 number to place an order, and those customers who got through waited an average of 10 minutes for an available agent. In addition, L. L. Bean’s 800 number long distance charges from potential customers waiting in the queue often amounted to $25,000 per day. This expense does not account for the loss in sales from customers who hung up while waiting for an agent. After a consulting team examined the situation, L. L. Bean used queueing theory to develop a model that focused on improving the efficiency of its telemarketing operations. To improve efficiency, the model determined optimal levels for:

(1) the number of phone lines carrying incoming calls to telephone agents; (2) the number of agents scheduled; and (3) the queue capacity; i.e., the number of wait positions for calls.

As the busy season approached, management had to consider “building-up” the number of phone lines and agents; similarly they also had to “build-down” these areas after the Christmas season. The queueing analysis process cost L. L. Bean $40,000, but they conservatively estimated an increased profit of $10,000,000 in 1989. This model improved service rates and call volume throughout the year, and especially during the three week peak period just before Christmas. Quinn, Phil, Andrews, Bruce, and Parsons, Henry (1991). “Allocating Telecommunications Resources at L. L. Bean, Inc.” Interfaces, 21(1), 75-91.


25

Homework Problems: 1.a.) Read the case studies and classify each as a single-server, multiple single-servers,

parallel servers, or servers-in-sequence model. b.) Give a situation where you have encountered each type of server model. 2.) At an outdoor concert, there are two

portable toilets designated for females and one for males. Women wait in one line to use one of the two portable toilets. Assume that the average customer arrival rate at both the male and female portable toilets is 30 per hour, but the average service time for females is three minutes and for males is one minute. Are the portable toilets distributed fairly? Use queueing theory to support your answer.

. 3.) At a local library, one clerk is checking out books. On average, 40 people per hour

arrive at the counter to check out books. It takes an average of one minute to service a person.

a.) On average, how many people are in line and being serviced?

b.) On average, how many people are in line?

c.) On average, how much time will a person spend in line and at the counter checking out books?

d.) On average, how much time will a person spend in line?

4) Your school is hosting the Homecoming Game in two weeks. Your job is to set up a reasonable queueing model to manage ticket taking and admission to the game.

a.) Using your school as a model, estimate each of the following:

The number of people expected at the game (based on your school population),

The average time it takes to take a ticket and admit someone,

Calculate the average number of customers served by one server (ticket taker) per minute and per hour. The number of totally separate entrances (e.g. opposite sides of the arena) where ticket takers will be stationed.


26

b.) Before you can develop an appropriate queueing model, you will need to determine the

average arrival rate per minute.

During how long a time period will the vast majority of fans arrive at the game? Use this time length and the number of people listed in part a) to determine the average arrival rate of fans to the ticket-taking gate.

If you assumed that there was more than one entrance, you will need to determine an arrival rate for each gate.

c.) Discuss the type of queueing model(s) you could use to plan the number of ticket takers

at each entrance. (When separate lines are formed far apart so that people can not jump back and forth to the shorter line, a separate queuing model is used for each location.)

In order to decide how many ticket takers to have, you will want to establish a customer service standard. Specify this standard as the average time, W, customers will spend in line before being admitted.

d.) Every mathematical model used to analyze a real-world situation involves making

simplifying assumptions. Discuss assumptions you made in order to determine the average arrival rate.

e.) Find each of the following:

The minimum number of ticket takers required at each gate so that the total service

rate exceeds the arrival rate. Why is this number important?

The traffic intensity, x.

The average number of customers in the system, L, at each entrance.

The average waiting time for each customer, W.

The fraction of each server’s available time being used per hour, f. f.) Earlier you specified a service standard for W. Convert that standard into a value for L.

Use the tables to determine the minimum number of ticket agents that will be needed to achieve this standard. Create a simple table that specifies the value of f as the number of servers is increased above the absolute minimum.

g.) Assume that all of the fans arrive during the hour before the game but the average arrival

rate is not constant. Break this hour into 4 fifteen-minute intervals. What percentage of the fans do you estimate will arrive during each of these four time-periods. Use these percentages to determine an average arrival rate for each 15-minute interval. Repeat steps e) and f) for each period.


27

h.) Write your conclusions in a two-page form suitable to be used as a proposal to your athletic director for management of ticket sales. Support your conclusions.

L

n: 1 2 3 4 5 6 f

0.10 0.111 0.202 0.300 0.400 0.500 0.600 0.20 0.250 0.417 0.606 0.802 1.001 1.200 0.30 0.429 0.659 0.930 1.216 1.509 1.805 0.40 0.667 0.952 1.294 1.661 2.040 2.427 0.50 1.000 1.333 1.737 2.174 2.630 3.099 0.55 1.222 1.577 2.008 2.477 2.969 3.475 0.60 1.500 1.875 2.332 2.831 3.354 3.895 0.65 1.857 2.251 2.732 3.258 3.812 4.385 0.70 2.333 2.745 3.249 3.800 4.382 4.984 0.75 3.000 3.429 3.953 4.528 5.135 5.765 0.80 4.000 4.444 4.989 5.586 6.216 6.871 0.85 5.667 6.126 6.689 7.306 7.959 8.636 0.90 9.000 9.474 10.054 10.690 11.362 12.061 0.95 19.000 19.487 20.083 20.737 21.428 22.146 0.98 49.000 49.495 50.100 50.764 51.466 52.194 0.99 99.000 99.498 100.106 100.773 101.478 102.210

L = the average number of customers in the system

f = fraction of time the server(s) are busy

n = number of servers


28

Project Ideas 1) Write a two-page report applying the concepts of psychology of queueing to a recent

experience you had waiting in line for a long time. In the report discuss the following issues. a) Describe the situation, the service facility layout, the number of servers and how

the queue was organized. b) Before arriving how long did you think you were going to have to wait and how

long was your actual wait? Why do you think the wait turned out longer than you originally expected?

c) Did you find the wait boring? What did you do to pass the time? What did most customers do while waiting in line?

d) If you had had advanced warning as to how long the wait was going to be, what, if anything, would you have done differently?

e) What could management have done to make the wait more pleasant? f) What cost effective strategy could management have used to reduce the average

waiting time? 2) Give students the assignment to visit a service facility in their community. They need

to make an appointment to interview the manager about any problems that have been have encountered with respect to how long customers have to wait and the customers’ satisfaction. Inquire what solutions have been tried and how effective each solution has been. Some suggestions of places to visit are the post office, bank, grocery store, the school cafeteria, a theater, a fast food franchise, or a restaurant.

3) Separate the class into groups of 4. Each group should select a different business in

your community. Each member of the group should be assigned an interval of 3 hours over which to observe the business and gather data on (a) how often customers arrive and (b) the length of time it takes for each customer to be served. The group should then compile their data into a single table using the variables in queueing theory. A report should be written by the group noting (a) their reservations, and (b) if they thought that the business is using queueing theory. If business is using queueing theory, which queueing model is being used: single-server, multiple-servers, parallel servers, or servers-in-series? Make a diagram to illustrate the queueing system observed.

Solution Key An answer key for the student activity, all of the extensions, and all of the homework problems is contained on the following pages. For ease in use, we have reproduced the student activity and the four extensions, with solutions.


29

The Sales Counter at Arm-and-a-Leg Tickets: Does this Line Ever Move?

Mr. I. M. Boss is vice president in charge of operations for Arm-and-a-Leg Ticket Sales. He is concerned about complaints regarding long waits at the ticket windows on Friday afternoons at many of the malls. To reduce the number of complaints, Mr. Boss has hired Dr. Hye I. Cue, an expert on queueing theory.

Queueing theory deals with the mathematical study of waiting lines. The entities waiting in line can take on a variety of forms. They could be people waiting at a doctor’s office or airplanes waiting on a runway. The waiting line is not even visible in every case. For example, telephone calls waiting for an operator are also “waiting in line.”

Before we begin to analyze Mr. Boss’s problem, which is called a single-server model, we will make the following assumptions:

1. Individual customers arrive at random to purchase tickets. 2. The time to complete a purchase is also random. This might be due to the number of

tickets the customer purchases or the customer asking for information about dates and seat location.

To use queueing theory, Dr. Cue needed to collect data about the customers. She spent several Friday afternoons observing the situation and collecting the data. She found that the average number of customer arriving per hour, a, is 18 and that the average number of customers the single ticket agent can help per hour, h, is 20.

1. What variables did Dr. Hye I. Cue observe during her Friday afternoon visits to the

mall? The number of customers arriving and number of customers served in a given period of time.

2. Dr. Cue observed that a = 18 customers per hour and h = 20 customers per hour. 3. If 18 customers arrive per hour, on average, how much time occurs between

successive arrivals? 1/18 hours or 3.333... minutes.


30

4. If a customers arrive per hour, on average, how much time occurs between arrivals?

1/a hours 5. If 20 customers are helped per hour, on average, how long does it take to help one

customer? 1/20 hours or 3 minutes

6. If h customers are helped per hour, on average, how long does it take to help one customer?

1/h hours

To develop a mathematical model of our queue, we must have an idea of the traffic intensity, x, which is the ratio of the average rate of customer arrivals, a, to the average rate of customers being helped, h. In order that this ratio make sense, the time units of a must be the same as those of h.

7. Write a ratio between the variables that represents the traffic intensity: x = a/h. Using the values of a and h from question 2, in this case, x = 18/20=9/10 or 0.9.

8. The average number of customers in the system, L, (including those in line and the

one at the ticket window) can be represented by the function:

Using the value of x from question 7, calculate L. L = 9 customers

What does this value of L tell you about this queueing system?

L is the average number of customers in the system at a given time

Customer satisfaction actually is more dependent upon the length of time it takes to get a ticket than the length of the line. Therefore, let:

W = the average time a customer waits in a system including the time in line and the time

to be served.

The function L = aW expresses the relationship between L and W. 9. What are the units for each of the variables a, L, and W?

a = customers per hour; L = customers; W = hours

xx

L−

=1


31

10. Given that L = aW, solve for W in terms of a and L.

W = L/a

11. Using your values of a and L, calculate the value of W for the system at Arm-and-a-Leg Ticket Sales.

W = 1/2 hour

What do you know from this value of W? Average time in the system is half an hour or 30 minutes. Suppose Dr. Cue learns that during a slow time of the day an average of only 16 customers per hour arrive to purchase tickets. However, the customer help rate remains the same.

12. What is the traffic intensity, x, during this time of day? x = 4/5 or 0.8 13. Calculate the values of L and W for this time of day.

L = 4 customers W = 1/4 hour or 15 minutes

14. Enter the values of x, L, and W from questions 7, 8, 11, 12, and 13 in the appropriate places in the following table. Then complete the rest of the table for the given values of a and h.

a

(customers/hr) h

(customers/hr) x L

(customers) W

(hours) 14 20 0.70 2.333 0.167 16 20 0.80 4.000 0.250 18 20 0.90 9.000 0.500 18 22 0.82 4.500 0.250 18 24 0.75 3.000 0.167

15. Using the values you entered in the table above, what happened to the values of L and

W when a increased while h remained constant? L and W increase nonlinearly.

16. What happened to the values of L and W when h increased while a remained

constant?

L and W decrease.


32

17. If a = 22 and h = 20, what are the values of x, L, and W? x = 1.1 L = –11 W = –0.5 Do all of these values make sense? No.

Explain. The number of customers and wait time came out to be negative. .

18. Why did this particular value of x occur? The average number of customers arriving is greater than the average number of customers served in a hour.

19. Explain what would happen to the ticket line given the situation in number 17?

The line would increase indefinitely. 20. Compare the value of x in question 17 to all values of x in the table.

What do you observe? x is less than 1 in the table and greater than 1 in number 17.

The graph of L = x/(1 – x) appears at the right. 21. Using the equation for L, what happens when x =1?

L is undefined.

22. Where is this represented on the graph? At x = 1 lies a vertical asymptote.

23. For the function L = x/(1 – x), what is the domain of

x? All real numbers except for x = 1.

24. If L = –1, then –1 = x/(1 – x). What happens when you solve this equation for x? You end up with –1 = 0, which is obviously false. No solution exists.

25. Where is this represented on the graph?

At y= –1, which is the horizontal asymptote.

26. For the function L = x/(1 – x), what is the range of L? All real numbers except y = –1.

x

L


33

27. Referring to questions 17-20, what happens to the value of L when x > 1?

The values of L are negative and decreasing.

28. Discuss why such a value of L makes sense, or does not make sense, for the Arm-and-

a-Leg problem. The number of customers in the system cannot be negative.

29. Thinking about the definition of x, x = a/h, what would a value of x = 0 mean? There are no customers arriving and therefore, no one can be waiting in the system.

30. Trace the portion of the graph above that is appropriate for the Arm-and-a-Leg

problem. 31. Write the domain for the portion of the graph you traced. 0 < x < 1 Calculate the values of L for each of the values of x in the table below. Enter the values of L into the table. 32. x L 0.8 4 0.9 9 0.95 19 0.99 99 33. What happens to L as the values of x approach (get closer to) x = 1? L increases

rapidly. 34. Where is this represented on the graph?

The graph approaches positive infinity at x = 1. 35. In all of the analyses above, changes were made in the values of a and h. In reality,

which, if any, of these variables would a manager like Mr. I. M. Boss be able to control? The service rate, h.

02468

101214161820

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0x

L


34

36. What could Mr. Boss do to improve the customer satisfaction at his sales outlets?

Get a faster server, or train the servers to increase their speed. (This would increase the value of h.) Add more servers, and use a parallel server model. (This would require a different model to calculate the average wait time for customers.)

37. What strategies to improve customer satisfaction have you experienced while waiting in line?

Some possible answers would be to provide entertainment (e.g., install a TV) and/or provide customers with serving numbers and a time to return.

Extension 1 1.

2.

3a.

b.

4a.

b. In our case, customers spend 9/20 hours or 27 minutes waiting in line. Extension 2 1.

)( ahha

Wq −=

hours 45.0209

==qW

x = =1820

0 9.

aha−

ahW

−=

1

W Whq= +1


35

2.

3.

4. 0.952 5. 1.333 6. Using linear interpolation,

7. The average number of customers in the system if 2 servers are used. 8.

9. The average time a customer waits in the 2 server system. 10. L and W are much smaller. Extension 3 1.

2. The smallest n such that

3.

4. L = 20.737 5. W = 0.273 hours or 16.38 minutes

4 therefore;12076

=<== nnnh

af

nha

f =

f = 045. or 45%

( )05 1333 0 952 11425 1143. . . . .+ = ≈

WLa

= = = =114318

0 0635 381.

. . hrs min

L = =60

20 min

3 min / customer customers

fanh

= = = = =76

4 207680

1920

0 95( )

. or 95%


36

6.

n f L W 5 0.76 5.351 0.070 hrs or 4.20 min 6 0.63 4.189 0.055 hrs or 3.31 min

7. Add one more server, because the advantage gained by adding two, compared to

adding only one, probably does not justify the cost of the extra server. Extension 4 1.

2.

3.

a customer/

hr

h customer/h

r

Lq

customers in line

L Total

customers

W Total wait

time in hours

Min Total wait

time in minutes

28 20 1.35 2.75 0.10 5.9 30 20 1.93 3.43 0.11 6.8 32 20 2.84 4.44 0.14 8.3 35 20 5.72 7.47 0.21 12.8 38 20 17.59 19.49 0.51 30.8 39 20 37.54 39.49 1.01 60.8

2244

ahh

W−

=

ha

ahhaha

ahhaah

ahha

ahah

hh

ha

haha

ahah

=−−

=−−

=−

−

−

=−

−−

)4()4(

)4(4

)4(44

444

Show that

22

22

22

32

22

3

22

23

3

22


37

Homework Solutions 1a) The Toilets Down Under-- Multiple parallel servers of different types Department of Motor Vehicles--Servers in a Series The Coal Unloader--Single Server L. L. Bean--Parallel Server 1b) Answers will vary depending on students’ experiences. 2) The toilets are not fairly distributed. Males will be in the system a total of two

minutes with a one-minute wait. Females will be in the system approximately 6.86 minutes with a waiting time in line of 3.86 minutes.

3a) On average, two people will be in the system (both in line and being serviced). 3b) On average, 1.333 people will be in line. 3c) A person will spend a total of three minutes in line and checking out. 3d) A person will spend two minutes in line. 4a) In gathering the data, the teacher may facilitate this portion of the homework by

asking the Director of Athletics for this information. There may be different scenarios given to different students.

To determine this statistic, students can perform a simple experiment that simulates a group of people waiting in line with tickets to enter a room. They could time how long it takes to admit someone. To calculate the average number of customers served by one server per minute and per hour invert the service time and scale the number appropriately.

The number of totally separate entrances is important because the students will need to divide the total arrival rate by the number of entrances to find the arrival rate per entrance.

4b) The total attendance statistic will need to be divided by the time period chosen and

the number of entrances to determine the arrival rate per hour or minute. 4c) Each entrance should be treated as a separate entity that is either a single or multiple

server system with just one line. Even if there are two ticket takers at an entrance and it looks as if there are two lines, as long as people can jockey between the two lines, the system behaves as if it were a single queue in front of multiple servers.


38

4d) Assumptions The average arrival rate of attendees during the time period specified is approximately a constant average rate of random arrivals. Arrivals of people and admitting them will be viewed as individual cases even if some people arrive in groups of two or more. Arriving customers will be divided evenly amongst the gates.

. 4e) The minimum number of servers, n, is the smallest number larger than the ratio of

(a/h). This insures that the total service rate exceeds the total arrival rate.

The average fraction of time servers are busy is just (a/nh) Use this value of f and n to look up in the table to estimate the value of L. If f is midway between the table values you may want the students to extrapolate to find L. W is found using the standard formula W = L/a.

4f) Use the standard set for W, to determine the target value of L. Once L has been

specified, it becomes somewhat tricky to use the table. Remember each value of L in the table corresponds to a pair of f and n values. Students should determine for each value of n in their problem, the corresponding value of f and circle the resulting estimate for L. They should identify the first column in which L drops below the standard they have set.

4g) Students may want to assume that the arrival pattern is similar to the following:

60 – 45 minutes before game time 10% arrive 45 – 30 minutes before game time 25% arrive 30 – 15 minutes before game time 45% arrive 15 – 0 minutes before game time 20% arrive These percentages are then used to calculate the average arrival rate per minute for each fifteen-minute interval. We will be treating each time period as a separate independent entity. However, if in reality long lines exist at the end of one time period and carryover into the next, this assumption can produce a poor approximation. Advanced queueing models were developed in order to relax this independence assumption when queueing models were used to forecast delays in responding to police emergencies in NYC.

4h) The points to be considered in writing a rubric for the paragraph should be:

• The number of servers needed to work at the game • The length of time each of the sellers need to work • Customer satisfaction • A justification for using your queueing model at the game • Decreasing the stress on customers and sellers • Decreasing the congestion of patrons at the ticket booth

Brief History of Queueing Theory and Broad Overview 1 · Brief History of Queueing Theory and Broad...

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