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MBA 638 – Operations Management

Professor Bellos – Spring 20

Midterm

MPORTANT:

· This exam is open book/notes/cases/slides/BB/video-lectures.

· No collaboration/communication with other students, use of Internet (other than to access Bb) or unauthorized materials (i.e., materials not provided through the course’s Bb or not mentioned on the Syllabus) is allowed.

· Submit your paper (as a Word document or PDF) through Blackboard by Sunday 02/06/2022 at 11:59 pm EST. Late exams will not be accepted.

· You may use a calculator or Excel. You may write and/or type. If you write, please write legibly and scan (no phone pictures).

· Show your work. Blank answers on the Word document (or PDF) will receive no credit.

· Remember to put units on your answers.

Part	Topic	Value	Grade
I	Quantitative Analysis	61
II	Short Answer	22
III	Objective Questions	17
Total	100

The Honor Pledge:

On my honor, I have neither given nor received unauthorized aid on this exam.

Student Signature

Student Name

GOOD LUCK!

Part I –Quantitative Analysis (61 points)

1. (33 points) Managing the Pets-R-awesOMe Call Center

Tim Woods, a recent GMU MBA grad and serial entrepreneur, has opened Pets-R-awesOMe, an online pet store. He and his team are experimenting with call center staffing.

(13 points) The current call center format has two lines: one for customers who want to place an order and one for customers who want to report a problem. Each line is served by one specialized customer service representative. The call center is staffed so that at any given time, there is one representative taking customer orders and one representative addressing customer problems. On average, there is a new customer who wants to place an order every two minutes, exponentially distributed. Similarly, there is a new customer needing to address a problem every two minutes, exponentially distributed. The call center staffers are fast and can on average complete one call of either type in 75 seconds, with a standard deviation of 30 seconds. Each call center staff member is paid $15 per hour. The estimated cost of a customer waiting for a representative is $20 per hour due to lost goodwill and negative word of mouth.

Fill in the next table. (Don’t forget the units.) Show your work on the table or below.

What type of queuing system is this?

Arrival rate

Average inter-arrival time

Service rate

Average service time

# of servers

Utilization

CA2

CS2

Avg. # of customers in the line

Avg. # of customers in the system

Avg. time a customer waits in the line

Avg. time a customer waits in the system

Probability the system is empty

b) (10 points) One option Pets-R-awesOMe is considering for its call center is to cross-train the two staff so they can both take orders or solve problems. In this case, all customers (i.e., those wishing to place an order and those wanting to report a problem) will join a single line and will be served by the first operator available. Each staffer after cross-training will be paid $16 per hour and will serve customers at the same rate as in part a). Arrivals are not expected to change for each type of call. The opportunity cost of waiting for a representative will still be $20 per hour.

Should Pets-R-awesOMe implement this system or keep the existing call center format (part a)? Justify your answer by calculating the total cost of each option. Please show your work.

c) (10 points) Another option is to invest in technology to augment the staffers’ capabilities in the dedicated server configuration of part a). In this case, they estimate that it will take on average 60 seconds to complete each customer call with a standard deviation of 24 seconds. Arrivals are not expected to change for each type of call. The technology is on a subscription basis, and the cost will be $10 per hour, in addition to the $15 per hour for each staff member. The opportunity cost of waiting will still be $20 per hour of waiting for a representative.

Should Pets-R-awesOMe implement this system or keep the existing call center format (part a)? Justify your answer by calculating the total cost. Please show your work.

2. (11 points) Managing the Process at Pets-R-awesOMe

Tim and his team are trying to develop some additional metrics. They have started to offer personalized dog training tools, called Poka-Yoke Collars, and it has turned into one of their most popular items, so demand is very large. The orders are sent from retail stores on Fridays and then filled on Fridays. The orders are received at a continuous rate of 40 per hour from 7 am until 11 am, the order cut-off time. There is an unlimited order buffer for any orders waiting. The personalization step starts at 7 am and operates continuously until all orders are personalized. The personalization takes 4 minutes per Poka-Yoke Collar, and all orders are for a single item.

a) (3 points) Draw an inventory build-up diagram (triangle) for the process. Please label it appropriately.

b) (8 points) Fill in the following information based on your diagram. If you need to show work, show it below.

Maximum number of orders waiting for personalization

Time the number of orders waiting for personalization reaches its maximum

Time the number of orders waiting for personalization goes to zero

Average number of orders waiting for personalization

Average time an order spends waiting for personalization

3. Managing the damaged order process at Pets-R-awesOMe (17 points)

Pets-R-awesOMe has developed a specific process to track and resolve orders reported damaged through shipping by customers. For some reason, there are large numbers of these. The following process is used to handle such issues. There is one employee per step.

To ensure that no claims “fall through the cracks” the process adheres to “lean principles.” Specifically, a claim for an order is handed from one station over to the next one only when the next station is ready to process the order (i.e., the stations are not allowed to have an inbox where orders accumulate).

a) (3 points) What is the throughput time (i.e., flow time)?

b) (4 points) What is the average inventory of orders in the system?

c) (5 points) What is the minimum time it would take to process a rush order (i.e., a high-priority complaint)?

d) (5 points) If you were to enable an inbox (only one) where claims about orders can “wait,” would you place this inbox right before: i) Station B or ii) Station C? Why?

Part II – Short-Answer Questions (22 points)

1. (2 points) What is an example of a Poka-yoke that Tim Woods and Pets-R-awesOMe could implement? What would your example Poka-yoke accomplish?

2. (6 points) Pets-R-awesOMe is considering making the ordering of Poka-Yoke Collars (i.e., the personalized dog training tools) available online only. The company is thinking of redesigning their website so that customers can use online tools to customize the dog training tools to their liking. Please identify two types of customer-induced variability (examples and classification) that the company may face in this case. What strategies can Pets-R-awesOMe use to manage the types of variability you identified? Please connect your answers to the “Breaking the trade-off between efficiency and service,” framework by F. Frei.

3. (2 points) In this course, we have talked about the queuing benefits of pooling. Why do you think in practice grocery stores or retailers like Pets-R-awesOMe may be hesitant to pool the waiting lines and instead they use configurations with multiple lines each served by a dedicated server?

4. (2 points) How do you think the pandemic has affected call center operations? In particular, the different types of variability that the call centers face and their ability to manage those types of variability?

5. (10 points) In your view, how do you think businesses can use the principles/concepts/methods we have discussed in this course so far to adjust to the post-COVID-19 reality or even get prepared for the next “black swan” event?

Ideally, your answer will meaningfully and succinctly draw on as many as possible principles/concepts/methods we have discussed and attempt to make connections between them.

Part III – Objective Questions (17 points)

1. (3 points) Consider the two processes shown below. Which one will have on average the higher total production outcome (i.e., total number of products produced in a given time period)?

a) Process A

b) Process B

c) They will have the same hourly production outcome.

a) Process A
b) Process B

c) On average, they will have the same total production outcome.

2. (2 points) What happens in Process B above if the buffer size increases from 2 to 10?

a) Throughput time increases

b) Throughput time decreases

c) Starving increases

d) Throughput rate decreases

e) Nothing changes

3. (2 points) Assume that there are on average 45 students in my undergraduate class for every quiz. It took me 90 minutes to grade all responses to Quiz 1. After several Grande Americanos from Starbucks, it took me 50 minutes to grade all responses to Quiz 2. What was the percent change in my productivity?

a) 88.89%

b) 80%

c) -80%

d) 44.4%

e) -44.4%

4. (2 points) On a typical weekday in January, 4200 customers visit the local Wal-Mart. It is estimated that, on average, there are 350 customers in the store. Assuming that the store is open 14 hours a day, how much time does the average customer spend in the store?

a) 7 min

b) 20 min

c) 14 hours

d) 1.167 hours

e) None of these

5. (2 points) The cycle time is always at least as long as the throughput time.

a) True

b) False

6. (2 points) All else the same, which curve in the figure below has less variability?

System B

System A

a) System A

b) System B

7. (2 points) In a pull system, if demand is below capacity, what is the utilization of the slowest machine?

a) Less than 100%

b) 100%

c) More than 100%

d) None of the above

8. (2 point) In a G/G/3 system, the arrival rate λ should never exceed the service rate μ.

a) True
b) False

That’s the end!

M/M/1 0.80000000000000016 0.80500000000000016 0.81000000000000016 0.81500000000000017 0.82000000000000017 0.82500000000000018 0.83000000000000018 0.83500000000000019 0.84000000000000019 0.8450000000000002 0.8500000000000002 0.8550000000000002 0.86000000000000021 0.86500000000000021 0.87000000000000022 0.87500000 000000022 0.88000000000000023 0.88500000000000023 0.89000000000000024 0.89500000000000024 0.90000000000000024 0.90500000000000025 0.91000000000000025 0.91500000000000026 0.92000000000000026 0.92500000000000027 0.93000000000000027 0.93500000000000028 0.94000000000000028 0.94500000000000028 0.95000000000000029 0.95500000000000029 0.9600000000000003 0.9650000000000003 0.97000000000000031 0.97500000000000031 0.98000000000000032 3.2000000000000037 3.3232051282051325 3.4531578947368464 3.5904054054054102 3.7355555555555608 3.8892857142857196 4.0523529411764763 4.2256060606060677 4.4100000000000072 4.6066129032258143 4.8166666666666753 5.0415517241379408 5.2828571428571536 5.5424074074074188 5.8223076923077048 6.1250000 000000142 6.4533333333333491 6.810652173913061 7.2009090909091107 7.6288095238095455 8.1000000000000245 8.6213157894737122 9.2011111111111425 9.8497058823529766 10.580000000000041 11.408333333333381 12.355714285714342

.449615384615448 14.726666666666743 16.236818181818276 18.050000000000114 20.267222222222365 23.040000000000184 26.606428571428818 31.363333333333674 38.025000000000496 48.020000000000785 M/D/1 0.80000000000000016 0.80500000000000016 0.81000000000000016 0.81500000000000017 0.82000000000000017 0.82500000000000018 0.83000000000000018 0.83500000000000019 0.84000000000000019 0.8450000000000002 0.8500000000000002 0.8550000000000002 0.86000000000000021 0.86500000000000021 0.87000000000000022 0.87500000000000022 0.88000000000000023 0.88500000000000023 0.89000000000000024 0.89500000000000024 0.90000000000000024 0.90500000000000025 0.91000000000000025 0.91500000000000026 0.92000000000000026 0.92500000000000027 0.93000000000000027 0.93500000000000028 0.94000000000000028 0.94500000000000028 0.95000000000000029 0.95500000000000029 0.9600000000000003 0.9650000000000003 0.97000000000000031 0.97500000000000031 0.98000000000000032 1.6000000000000019 1.6616025641025662 1.7265789473684232 1.7952027027027051 1.8677777777777804 1.9446428571428598 2.0261764705882381 2.1128030303030338 2.2050000000000036 2.3033064516129071 2.4083333333333377 2.5207758620689704 2.6414285714285768 2.7712037037037094 2.9111538461538524 3.0625000000000071 3.2266666666666746 3.4053260869565305 3.6004545454545553 3.8144047619047727 4.0500000000000123 4.3106578947368561 4.6005555555555713 4.9248529411764883 5.2900000000000205 5.7041666666666906 6.1778571428571709 6.7248076923077242 7.3633333333333715 8.1184090909091378 9.0250000000000572 10.133611111111183 11.520000000000092 13.303214285714409 15.681666666666837 19.012500000000248 24.010000000000392 Utilization

Table oof Conteents

For the exclusive use of P. Vongsumedh, 2022.

This document is authorized for use only by Pubordee Vongsumedh in Online MBA 638-001-Spring 2022 taught by IOANNIS BELLOS, George Mason University from Jan 2022 to Feb 2022.

8007 | Core Reading: PROCESS ANALYSIS 2

1 Introduction …………………………………………………………………………………. 3

2 Essential Reading ……………………………………………………………………….. 5

2.1 Process Flow Diagrams ………………………………………………….. 5

2.2 Batch Processes ………………………………………………………………. 8

2.3 Assessing Capacity …………………………………………………………. 9

Bottleneck Analysis …………………………………………………………. 9

Cycle Times ………………………………………………………………………. 11

2.4 Assessing Efficiency ……………………………………………………… 16

2.5 Assessing Effectiveness ………………………………………………… 17

Quality ………………………………………………………………………………. 17

Speed ………………………………………………………………………………… 18

Flexibility ………………………………………………………………………… 20

Safety………………………………………………………………………………. 20

3 Key Terms ………………………………………………………………………………….. 22

4 For Further Reading ………………………………………………………………… 23

5 Endnotes…………………………………………………………………………………….. 23

6 Index ……………………………………………………………………………………………. 24

This reading contains links to online interactive illustrations, denoted by a . In

order to access these exercises, you will need to have a broadband internet

connection. Verify that your browser meets the minimum technical requirements by

visiting http://hbsp.harvard.edu/list/tech-specs.

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Table of Contents

For the exclusive use of P. Vongsumedh, 2022.
This document is authorized for use only by Pubordee Vongsumedh in Online MBA 638-001-Spring 2022 taught by IOANNIS BELLOS, George Mason University from Jan 2022 to Feb 2022.

8007 | Core Reading: PROCESS ANALYSIS 3

1 INTRODUCTION

t its most basic level, operations management is concerned
with the work that every organization must do to meet its
objectives—how that work is organized, managed,

analyzed, and ultimately carried out. More precisely, operations
management is about designing, managing, and improving the
activities involved in creating products and services and delivering
them to customers. We call these activities and the people,
resources, and procedures that dictate how work is organized the
operating system.

The basic building block of an operating system is the operating process. Most
operating systems consist of multiple processes. A process is a set of tasks to be performed
in a defined sequence and uses inputs (such as labor, capital, knowledge, raw materials,
purchased components, and energy) to create outputs that are of great value to customers
and therefore to the organization itself.a While it may be easier for many of us to visualize
an automobile assembly process or a steel-making process, every organization—
manufacturing or service, public or private, for-profit or not-for-profit—organizes its
work through its operating processes.

A hospital takes as input a sick patient, applies labor (doctors, nurses, and other
personnel), knowledge, capital (in the form of facilities and technology), energy, and
supplies and, as its output (we hope), produces a healthy patient. An airline takes as input
a passenger who is at Point A but wants to be at Point B, and it applies similar categories
of resources (pilots, baggage handlers and equipment, the airplane, and so on) to output a
passenger at Point B. A successful local government agency takes the needs of its citizens,
applies resources, and provides services that improve community life. Unfortunately,
while almost all manufacturing organizations routinely visualize and analyze their
operating processes, many service organizations do not have an explicit process
“viewpoint,” which often hampers their ability to achieve their objectives as well as they
would like.1

An organization’s operating processes are generally meant to fulfill two overarching
goals:

1 Deliver the “customer promise.” Every business seeks to provide specific
things for its customers better than any other business can. This is often
referred to as “strategic positioning.” We will refer to it as the “customer
promise.” Ultimately, it is the job of a firm’s operating processes to deliver
that promise, whether it is lower cost, faster service, higher quality,
customization, or some combination of those attributes. Customer
satisfaction is highly correlated with how well the firm’s operating processes

a In the for-profit sector, this translates to creating profits. We purposely avoid this language because the
concepts in this reading also apply to nonprofit and governmental organizations, for which creating
value may not translate into profits.

8007 | Core Reading: PROCESS ANALYSIS 4

function. This is particularly true for service organizations, in which the
customer interacts directly with those operating processes.

A firm that designs its operating processes to try to be all things to all people
will find it difficult to compete with a focused competitor. So a firm has to
understand which attributes are most important to customers and which are
less so, and it must design its operating processes to reflect that
understanding.

2 Create value for stakeholders. For the for-profit firm, creating value for
stakeholders often means running operating processes efficiently enough to
ensure profits to the firm’s shareholders while simultaneously providing an
environment that motivates and retains workers. For a charitable
organization, it may mean serving its clientele effectively enough so that it
will be able to continue to attract funding from benefactors. For a
governmental administration, it means serving the public to provide social
welfare at a reasonable cost that—perhaps—ensures its reelection. The
circumstances vary widely, but the principle—that an operating system must
create value for its stakeholders—always applies.

The goal of this reading is to introduce you to a broad set of operating processes and
to give you concepts and tools you can use to describe them and to analyze them to assess
their performance. Organizations without the means to improve rarely can keep up with
competition, and improvement requires a deep understanding of underlying operating
processes and an ability to assess their performance. Put more simply, the goal of this
reading is to allow you to observe an operating process, have a sense of what data needs to
be collected, and then, after some analysis of those data, be able to answer the key
question, “How is this process doing?” and “How could it do better?” This is the essence
of process analysis.

How can you tell whether a process is “doing” well? That depends on what the
customer promise is and how the firm seeks to create and capture value for its
stakeholders. For example, if the customer promise is fast and effective service, we would
be concerned with speed, the quality of service, and customer satisfaction. On the other
hand, if we are running a high-volume, capital-intensive factory with a customer promise
of low cost, we might be most concerned with capital utilization, high volumes, and
various measures of efficiency and productivity.

In the material that follows, we first introduce a tool that allows us to represent
pictorially, and better understand, the sequence of tasks and the flow of product and
information in a process. We then use a variety of numerical analyses to measure process
performance, focusing on three critical dimensions: capacity, efficiency, and
effectiveness.b

b We use the term “effectiveness” to refer to measures of how well a process does what it is supposed to
do (e.g., how well it delivers on the customer promise). Efficiency refers to how well the process utilizes
its resources or inputs in producing the output. The efficiency with which a resource is used is often
called that resource’s productivity.

8007 | Core Reading: PROCESS ANALYSIS 5

2 ESSENTIAL READING

The first step in analyzing any process is describing it. This starts with mapping the
sequence of tasks and the flow of product and information in what is often called a
process flow diagram or a process map. There are various conventions for mapping
processes; we will use a set of conventions and terminology that are common in practice
and that we use in teaching process analysis at the Harvard Business School.

The process flow diagram for a very simple sequential process might be drawn as
shown in Figure 1.

Figure 1 A Sequential Process

For a product to be produced, or a service to be performed, Task A must be done before
Task B can begin, Task B must be done before Task C can begin, and Task C must be
done before Task D can begin. A single worker is assigned to each of these four tasks.
Tasks are represented by rectangles; the arrows indicate the direction of product flow in a
manufacturing context, or, in a service context,c the movement of a customer or
documentation. In each case this is usually called the process flow. The structure of the
process flow diagram and the way to think about process flow are based on what is called
the “product’s eye view.” Namely, the sequential flow in Figure 1 is what is “seen” by a
product or customer moving through the process. However, all tasks are generally
performed simultaneously, on four separate products or customers. If we were to check in
on the process in the middle of the day, we would see all four tasks under way. We call
this steady state, because the process is not affected by start-up or shut-down activity.

Processes often include tasks that are performed in parallel, and sometimes material
(or customers) must wait along the way. Consider a small, busy café at which customers
order either tea or coffee beverages, as pictured in Figure 2. Orders for tea are prepared at
a dedicated tea station, and orders for coffee are prepared at a barista station.

c For simplicity, we will call a process a service process if the customer is directly involved in the
operating process. Thus, for example, for our purposes, the backroom process in which a loan
application is analyzed and, then, either approved or denied is not a service process.

2.1 Process Flow Diagrams

8007 | Core Reading: PROCESS ANALYSIS 6

Figure 2 Process Flow Diagram for a Café

This process is different in several ways from the simple four-step process that we
encountered earlier. First, it includes tasks that can be done at the same time. We call this
a parallel process. Second, it includes upside-down triangles to indicate two points at
which customers may need to wait between tasks. In a manufacturing process, points at
which components or partially manufactured products wait for the next process step are
called Work in Process (WIP) inventory buffers. Third, this process includes a decision
node, denoted by a diamond with a question mark. At this point in the process, the flow
(of either customers or materials) can move in different directions. A decision node is
commonplace in processes that involve a technology choice—that is, a choice of different
methods of making a product (e.g., an automated technology versus a manual approach)
or delivering a service.

Fourth, this process includes the flow of information, indicated with a dotted line and
an arrow. The customer’s order informs the decision as to which employee will serve that
customer. In most processes, information flow is much more complicated.

Parallel processes need not involve decision nodes. Consider a small workshop that
makes wooden chairs using a simple parallel process, as shown in Figure 3.

Figure 3 Process Flow Diagram for Making Chairs

The making of the seat, the front legs, and the single-piece unit that comprises the

back legs and chair back are done in parallel (at the same time) and then they are
assembled. Again, consider the “product’s eye view.” Initially, the product consists of
three components, so it has three sets of “eyes.” Each component sees the work that is
being done on it. This explains the fundamental difference between a sequential process
and a parallel one.

8007 | Core Reading: PROCESS ANALYSIS 7

For another example, consider the parallel process in a commercial bakery that makes
filled dumplings, as shown in Figure 4. This process involves parallel operation and batch
flow (product moves in batches rather than one by one).

Figure 4 Process Flow Diagram for Making Dumplings

Process flow diagrams are useful in many different ways. One use (which does not

involve the kind of detailed numerical analysis discussed in the next section) is as a
communications tool that allows a team of people to share a common view of the work
they are performing. When a team has such a shared view of what activities (tasks) are
being performed and in what sequence, as well as how products and information flow,
they can more easily discuss the pros and the cons of changes and improvements to the
process.

For example, a large financial services firm that we’ve visited routinely schedules
“process mapping days” during which a team of people who have been working together
on a certain set of (mostly computerized) tasks necessary for processing complex financial
transactions come together to examine how they do their work. They spend the first part
of the day mapping out the process on flip-chart pages that often cover the wall of a
conference room and may involve over 100 boxes, arrows, and decision nodes. Specific
task times or other numerical performance measures are rarely noted.

After this stage the team begins to question the effectiveness and efficiency of the
process. Comments and questions are then discussed:

• “I know the manual says that we are supposed to do Y after we do X, but in
practice we don’t.”

• “Why do we do A after B? Would it be more efficient to do them in parallel?”
• “Why do we have to do Z? It would take less time if we eliminated that.”
• “How is that decision made? Don’t we need more information?”

These are valuable conversations.
In the material that follows, we will go beyond a purely descriptive use of process

mapping and instead use it as the backbone of numerical analysis that facilitates a
discussion of the key question, “How is this process doing?” This requires understanding
the timing of process flows more precisely.

Thus, the first step in more precisely measuring process performance is measuring
task times.d In practice, firms usually use standard times. The standard time of a task is
defined as the average time that an employee (or customer in many service processes)

d A critical element in process design is the determination of what constitutes a “task.” This topic is taken
up in Core Reading: Designing, Managing, and Improving Operations, HBP No. 8012 (Boston: Harvard
Business Publishing, 2013).

8007 | Core Reading: PROCESS ANALYSIS 8

with average skills will take to complete that task, under ordinary circumstances, and
working at a sustainable pace.e

One important measure of any process is labor content, the total time that is spent by
the firm’s employees on the product or service. Labor cost is, of course, greater than labor
content multiplied by the wage rate, because the firm needs to pay employees for any idle
time that they incur because of imbalances in the process structure.

Firms often choose to process not just one product at a time but a batch or a lot of the
same products at once. In services (e.g., rides at Disney World), it will sometimes make
sense to process a fixed number of customers at once. The reasons for batch processing are
many, but usually it is more efficient to process multiple products or customers at the
same time, if possible. It would be inefficient for Disney World to send one customer at a
time through some of its rides, and at the dumpling bakery diagrammed in Figure 4, it
would be inefficient to mix enough filling for only one dumpling and then to switch to
mixing filling for another dumpling. In this case, as in many manufacturing contexts, it
takes time to change a machine from processing one kind of part or product to another,
so processing multiple units of the same product together will save changeover (or setup)
time. For example, if it takes an hour to change a screwdriver stamping machine from
Phillips to slotted, or back, one would prefer to make a large batch of Phillips screwdrivers
and then a large batch of slotted screwdrivers, rather than one Phillips, change, one
slotted, change, one Phillips, change, and so on.

These considerations require that we expand our notion of task times. A task in a
batch process (or in any process) may require the following:

1 a setup time, the amount of time required to get ready for the task (to load
the riders on the Disney World ride, for example, or to prepare the settings
on the machine that mixes the dumpling filling) and, if necessary, clean up
afterward. Formally, setup time is any time taken to perform a task that is
independent of the number of products or customers being processed;

2 and a run time, the time it take to process each unit. Formally, run time is the
time taken to perform a task that varies with the number of products or
customers being processed.

Intuitively, setup time and run time calculations affect our everyday thinking. For
example, if you are making an omelet for yourself, you likely stir a few eggs with a fork; if
you are hosting a brunch for 20, you probably use an electric mixer. For a large batch size,
the extra time required to get the mixer ready and to clean it afterward is worthwhile.
Without explicitly thinking about the tradeoffs between setup time and run time, we
make these kinds of decisions every day.

e Of course, actual times will vary, at any given time, from standard times. This introduces a good deal of
complexity, and for the sake of developing basic concepts, these complexities are covered separately in
Core Reading: The Impact of Variability on Process Performance, HBP No. 8228 (Boston: Harvard
Business School Publishing, 2013).

2.2 Batch Processes

8007 | Core Reading: PROCESS ANALYSIS 9

An important attribute of any operating system is its ability to produce enough “output”
(in other words, make enough products or serve enough customers) to meet demand. For
many businesses, profits increase as volume grows, so being able to produce high volumes
is key to profitability. And for businesses that hope to grow, the ability to produce more
output as demand increases is critical. We define capacity as the number of units of
product that a process can produce (denoted in units of product per time period, e.g.,
units per hour) or, for service processes, the number of customers who can be served
(again, customers per time period). Capacity (often called “design capacity” or “rated
capacity”) is an ideal, assuming nothing goes wrong to slow the process or shut it down.
Actual output is typically lower than capacity.

The key to assessing the capacity of a process is discovering its bottleneck.

Bottleneck Analysis
Let us start with the simple four-step process in Figure 5.

Figure 5 Four-Step Process with Task Times

Task times are written below the task boxes. At Task A, for example, it takes 5 minutes to
process one unit. Let’s assume, to keep this as simple as possible, that the process is a
manual process, setups are not required, no WIP builds up, and that the four workers
associated with the four tasks are specialized—that is, the worker at Task A stays at Task
A and never moves to another task. What does product flow look like? We’ll get the
process into steady state before we begin the analysis. The chart in Figure 6f depicts what
happens over time.

f This kind of chart is known as a Gantt chart.

2.3 Assessing Capacity

8007 | Core Reading: PROCESS ANALYSIS 10

Figure 6 Gantt Chart of the Four-Step Process

P1, P2, and so on, represent the products being worked on. Note that because of the
sequential nature of this process, a task cannot begin on a particular unit until the
previous task is completed. Quite quickly, the process is in steady state. At minute 14, P1
is complete and all four tasks are active. A few important observations:

• The worker at Task A is always busy.
• The worker at Task B has a minute of idle time between successive products.
• The worker at Task C has two minutes of idle time between successive

products.
• The worker at Task D has three minutes of idle time between successive

products.
Upon reflection, these observations should not be surprising; workers at Tasks B, C,

and D receive a product only once every five minutes (time for Task A) and therefore
have to wait. Task A is the bottleneck for this process.

Let’s digress briefly to bottlenecks, which is perhaps the most critical concept in
process analysis. In simple terms, the bottleneck of any process is the task that causes all
other tasks to have idle time (there may be two or more bottlenecks in any process).g In
other words, a bottleneck constrains product flow. It may help to conceptualize this
concept in terms of a flow we think about in everyday life—liquid flow. Imagine the water
pipeline pictured in Figure 7.

Figure 7 Pipeline

Assuming viscosity and a pump that pumps water into the pipeline at least as fast as it can
accept it, what will determine the rate of outflow? The outflow will not be determined by
the rate of inflow, but rather by the rate of flow through the bottleneck. Indeed, the
section of the pipeline after the bottleneck will be only partly filled.

g We will expand on this definition later.

8007 | Core Reading: PROCESS ANALYSIS 11

Why do we call it a bottleneck? Think of a typical soda bottle, illustrated in Figure 8.
Why does the soda bottle have a neck? We might expect that
the bottle would be more efficient to produce if it were a
simple cylinder, with no indentation. But then the soda would
flow out at a rate greater than your ability to drink it (or to
pour it into a cup without splashing). The bottleneck is
designed into the bottle explicitly to limit the rate of outflow.
In this case, bottlenecks are good; in operating processes, they
tend to be bad because limiting the rate of flow limits output.
Limiting output decreases potential revenue (assuming that
there is adequate demand) and, thus, decreases potential
profits. Bottlenecks also slow processes, incur idle time, and
increase lead time.h

As the Figure 6 chart confirms, Task A limits the other
tasks, and because Task A can produce only 12 units per hour, the entire four-step
process can produce only 12 units per hour.

What happens when the bottleneck isn’t the first task, starving all the others?
Consider a different version of the four-step process, as shown in Figure 9.

Figure 9 Four-Step Process with a Bottleneck

For this process, Task C is the bottleneck; it limits the output of the process. Tasks A

and B could produce greater output, but that would result in work-in-process inventory
building up in front of Task C indefinitely, with no increase in process output. This is
neither desirable nor physically sustainable. When the physical space for holding WIP in
front of Task C is full, we say that Workstations A and B are blocked tasks. Task D has to
wait for Task C’s full five-minute cycle to do its work. We say that Task D is a starved
task.

In practice, the workers at Tasks A and B would slow down, or be idle, to match their
output to the fastest rate at which Task C can run. This is referred to as bottleneck pacing
because the pace of the system is set by the pace of the bottleneck.

Blocking and starving are two sides of the same coin: They both lead to idle time for
the workers at those workstations. The worker at Task A will be idle for three of every five
minutes; at Task B, the worker will be idle for one of every five minutes; and at Task D, he
will be idle for two of every five minutes. Theoretically, Task C will never be idle.

Cycle Times
Another way to think about how this process operates is through a simple thought
experiment. Imagine standing at the end of the process with a stopwatch, clocking
successive finished products as they emerge from the final task. The system’s cycle time is
the average time between the completion of successive units of product or, in the case of a
service process, the average time between the departures of successive customers.

h This is particularly problematic when speed is part of our customer promise. We will discuss this in
more detail later in this reading.

Figure 8 Soda Bottle

8007 | Core Reading: PROCESS ANALYSIS 12

You can use Interactive Illustration 1 to see how task times at individual
workstations affect the system’s cycle time. Try varying the task times at each workstation
to see how blocking and starving occurs.

Interactive Illustration 1 Blocked and Starved Tasks

You will notice that the system cycle time (for all three tasks together) is equal to the task
time of the bottleneck task. All other tasks are either blocked or starved and have some
idle time.i System cycle time can never be less than the cycle time of the bottleneck.

So far, we’ve been assuming that each task is performed by only one worker. To
understand the effect on cycle times, and thus on capacity, of multiple workers doing the
same task, consider a one-step, manual process, as pictured in Interactive Illustration 2.
The task time for each worker is always 12 minutes. Therefore, if there is one worker at
the task, the cycle time for the task is also 12 minutes. You will notice, however, that as
the number of workers at the task increases, the cycle time drops. The task time remains
12 minutes, but, with four workers the task cycle time will drop to 3 minutes, in which
case a new unit would come off the line every 3 minutes. One way of “breaking” a
bottleneck is to add another worker to that task.

Interactive Illustration 2 Cycle Time with Multiple Workers

i Unless there are multiple bottlenecks (tasks with the same task time as the system cycle time).

Scan this QR code, click the image, or use this link to access the interactive illustration: bit.ly/hbsp2pHNWah

Scan this QR code, click the image, or use this link to access the interactive illustration: bit.ly/hbsp2GdGUAs

https://s3.amazonaws.com/he-assets-prod/interactives/013_blocked_and_starved/Launch.html

https://s3.amazonaws.com/he-assets-prod/interactives/022_cycle_time_multiple_workers/Launch.html

8007 | Core Reading: PROCESS ANALYSIS 13

Formally, we have defined capacity as the maximum output, in terms of units
produced or customers served, in a specified time period (e.g., units manufactured per
hour or customers served per day). This is a critical measure of process performance
because it constrains revenue. You can’t sell more than you can make—a firm’s maximum
revenue is limited by its capacity.j

Capacity is the inverse of cycle time.

For example, returning to the simple process in Figure 9, the cycle time of Task A is
two minutes per unit. Therefore, its capacity is ½ unit per minute, or 30 units per hour.
Task A can produce at most 30 units per hour. Similarly, Tasks B, C, and D have
capacities of 15, 12, and 20 units per hour, respectively. The line can run no faster than
the bottleneck, so the line capacity (or system capacity) is 12 units per hour. The system
capacity is only as great as the task with the lowest capacity. You’re only as strong as your
weakest link.

If the line is running at the pace of the bottleneck, 12 units per hour, then capacity
utilization at Task A is 40%. The workstation is actually producing 40% of what it is
capable of producing. Similarly, capacity utilizations at B, C, and D are 80%, 100%, and
60%, respectively. Workstations that are higher than the bottleneck will have idle time.
Table 1 summarizes these calculations.

Table 1 Capacity Calculations for Process in Figure 9

Cycle Time

(minutes/unit)
Capacity

(units/hour)
Capacity Utilization
at Bottleneck Pacing

Task A 2 30 40%

Task B 4 15 80%

Task C 5 12 100%

Task D 3 20 60%

Clearly, making a process more efficient means decreasing idle times as much as
possible. This requires redefining tasks, wherever possible, to balance the process—that is,
to make cycle times as nearly equal as possible and capacity utilization at each task as high
as possible. However, because many tasks are not easily divisible it is rarely possible to
balance a process fully.

Returning to Figure 9, let’s say because demand is high and profits for this product
are good, we decide to add a second worker to increase our capacity. If that worker is
specialized to one task, it’s clear that the best plan would be to assign this fifth worker to
Task C. Often, two workers assigned to a single task will work on successive products.k
With two (equally competent) workers performing Task C, on average, two products will
be produced every five minutes; the cycle time for Task C becomes two and a half

j Capacity determines maximum revenue. In practice, for a myriad of reasons, actual output will typically
be less than capacity. The ratio of actual output to capacity is called capacity utilization.
k The calculations here are the same as two workers working together on the same product (or
customer), as shown in Interactive Illustration 2. In many cases, however, this is not feasible (or
clumsy), particularly in service processes, so we will generally view multiple workers working on a
product by themselves.

= =
1 1

capacity , cycle time
cycle time capacity

8007 | Core Reading: PROCESS ANALYSIS 14

minutes. The bottleneck would shift to Task B, with a task time of four minutes, and the
system capacity would rise to 15 units per hour, a 25% increase. Whether the profits on 3
additional units would be greater than the hourly labor cost of an additional worker
would be an important consideration.

We might instead assign a higher-skilled worker capable of doing both Task B and
Task C alongside the specialists already there, as shown in Figure 10. Then for Tasks B
and C together, three workers would process 3 units every nine minutes, so the task cycle
time is three minutes.l Then Tasks B, C, and D become equal bottlenecks, the system cycle
time drops to three minutes, and the capacity of the process jumps to 20 units per hour.
Figure 10 Four-Step Process with Tasks B and C Combined

For the examples discussed above, capacity is determined by the maximum cycle time

for any of the tasks in the process. In reality, any resource used in a process can be a
bottleneck and, thus, constrain output. If raw materials are sufficiently scarce, all tasks
will have idle time; raw materials constrain output. In a restaurant, we often speak of
seating capacity. On a busy night, the number of seats will constrain output, and thus be
the bottleneck.

In general, we can improve performance at a bottleneck by adding resources,
increasing capacity, or lowering task time. For example, a very busy pizzeria with a
bottleneck at the oven could add resources by buying another oven, increase capacity by
fitting more pizzas into the existing oven, or lower task time by baking the pizzas more
quickly at higher heat.

In the foregoing discussion, we’ve been looking at simple sequential processes. Let’s
now revisit the chair-making workshop to investigate bottlenecks, capacity, and cycle
times for parallel processes. The process is diagrammed in Figure 11.

l This will be true independently of how we schedule the work.

8007 | Core Reading: PROCESS ANALYSIS 15

Figure 11 Task Times for Making Chairs

For simplicity, let’s first assume that specialized workers work on each of the tasks,

never move from one task to another, and work eight-hour days.
Consider the assembly task. It takes only one hour, but since it can’t commence until

all three components of the chair are available, it will have to wait, which means there is
idle time. In particular, the difficult task of fashioning the back legs and chair-back unit
takes two hours to complete. Thus, a finished back-legs-and-chair-back unit appears in
component inventory once every two hours, so the assembler will have to wait an hour
after assembling each chair for a full set of components to be ready for another assembly
operation. (Note that the other workers can finish their components for the chair in less
than two hours.)

The logic here is exactly the same as it was for a line-flow process. Because the system
cycle time (the time between the completion of successive chairs) is two hours and the
task time of the longest task is the bottleneck, only four chairs per day (eight hours/day
divided by 2 hours/chair) can be produced.

Specialization of the workers makes this process very unbalanced and inefficient.
Let’s now examine the impact of cross-training one worker. For example, if the worker
who fashions the front legs could be cross-trained to help with the back-legs-and-chair-
back unit, then the system cycle time would decrease, and the capacity of the process
would increase. To illustrate this, assume that the front-leg maker (whom we will call
Worker A) is trained to work on the first half of the two-hour bottleneck process (with
Worker B, who did it alone before).

To keep things simple, assume that Worker A is just as proficient as Worker B on
that part of the process. With two workers working on the first half of the task, it’s
reasonable to assume that it could be completed in a half hour. Then Worker B would
complete the second hour. In total, the task would be completed in one and a half hours.

Now, the fashion-seat step is balanced with the fashion-back-legs-and-chair-back-
unit step, at one and a half hours. Worker A spends one and a half hours per chair (a half
hour working with Worker B and an hour on the fashion-front-legs task). The capacity
would increase from 4 chairs per day to 8/1.5 = 5⅓ chairs per day. Cross training one
worker has increased capacity by 33%.m

As we have alluded to above, process performance is a matter of efficiency (how well a
process turns its inputs into outputs) and effectiveness (how well a process delivers its

m Note that capacity is a rate, and need not be an integer. Cross training, here, allows the shop to make 16
chairs every three days versus only 12 chairs every three days.

8007 | Core Reading: PROCESS ANALYSIS 16

customer promise). Our focus in this reading is on assessing process performance.
Efficiency and effectiveness are the two broad categories of “performance.” We now delve
into how we can measure these.

Capacity and bottleneck analysis are important because capacity limits output and output
(as well as price) determines revenue, but neither capacity nor bottleneck analysis tells us
much about other key measures of process performance. In particular, most operations
managers are quite concerned with the efficiency of their processes—the cost of the
inputs, and, often as important, how well these processes make use of those inputs. Let’s
first consider labor, a crucial input. A process that delivers service to 100 customers per
day and requires 20 employees is preferable to one that delivers the same service to the
same number of customers utilizing 25 employees (with the same skills and being paid the
same wage rate). Well-balanced processes with minimal idle time will make better use of
labor than unbalanced processes.

With this in mind, we define labor utilization as useful time (namely, time actually
working on a product or delivering a service) spent by workers as a percentage of the total
time for which they are available (and being paid).n

Reconsider the following four-step process, shown again in Figure 12.

Figure 12 Four-Step Process with a Bottleneck

We had earlier concluded that the cycle time of this process is 5 minutes, and hourly

capacity is thus 12 units per hour. During that hour, Worker A worked on 12 units for 2
minutes each, or 24 minutes, but Worker A is available (and paid to work) for 60 minutes.
Similarly, Worker B worked for 12 · 4 = 48 minutes; Worker C worked for 60 minutes;
and Worker D worked for 36 minutes.

The labor utilization for the entire process is:

+ + +
= =

24 48 60 36 minutes worked 168
70%

(60 minutes)(4 people) 240

Using earlier terminology and thinking of calculating utilization per cycle rather than
per hour, we can write labor utilization of the process as:

n This is often called labor productivity, and is sometimes measured by the value of the output divided by
the cost of the input.

2.4 Assessing Efficiency

+ + +
= =

labor content per unit 2 4 5 3
70%

(process cycle time)(# of workers) (5)(4)

8007 | Core Reading: PROCESS ANALYSIS 17

We need to be careful as to how we use the labor utilization calculation to make
decisions. Some managers, thinking that efficiency is reflected in “workers being busy,”
exhort their workers to increase their utilization by making more product. However, at
non-bottleneck steps, greater labor utilization translates into more inventory, not more
saleable product. Rather than increasing labor utilization itself, emphasis should be placed
on breaking bottlenecks. Reducing a bottleneck’s time will automatically raise labor
utilization, but in a useful way, namely one that results in more saleable product and
increased revenues.

Labor, however, is only one of many important inputs. Consider capital (i.e., funds
invested in machines/equipment). If the four operations above were fully automated
using four separate machines, then the same calculations above would give us machine
utilization, an indication of how efficiently capital was being employed. If machine
downtime entered the picture, as it usually does with real machines, then machine
utilization would drop. Machine utilization is a particularly important performance
measure in capital-intensive processes (such as steel production, oil refineries, or
semiconductor manufacturing).

Or, let’s consider material utilization. If, in the processing of a valuable raw material
(such as lithium for batteries), only 90% of the material processed ended up in the
finished product, we would say that the material utilization for that material was 90%. For
example, in many agricultural processing operations, not all harvested product ends up in
finished goods.

In today’s world, operations are particularly concerned with energy utilization. It is
not uncommon to find operations that consume energy to make finished product and
then throw that energy away—for example, in the form of hot wastewater.

As we have highlighted earlier, delivering the customer promise is also an essential goal of
any operating system and its processes. We refer to this ability as effectiveness. There are
many aspects of effectiveness that operating managers will want to observe and improve.
Here, we will highlight two aspects of effectiveness that virtually all firms pay attention
to—quality and speed. For a more complete discussion, refer to Core Reading: Managing
Quality with Process Control (HBP No. 8020) and Core Reading: Managing Quality (HBP
No. 8025).

Quality
Quality is the ability of a product or service to meet or exceed customers’ expectations.2
These expectations, of course, will be influenced by the customer promise. Consider the
two hotel chains Four Seasons and EconoLodge. Four Seasons provides a high-end luxury
customer experience. In contrast, EconoLodge provides limited amenities but at a much
lower price than Four Seasons. These businesses have very different customer promises,
and one would expect to find differences in how they achieve quality.

Types of Quality
We define quality in two different ways:

2.5 Assessing Effectiveness

8007 | Core Reading: PROCESS ANALYSIS 18

• Performance quality
• Conformance quality

When we say that a Four Seasons hotel is a “quality” resort, we are referring to
performance quality. When we say that the BMW 7-series is “the ultimate driving
machine,” we are also referring to performance quality. When we call a bicycle a “quality
bicycle,” we are usually referring to the use of lightweight alloys and high-performance
components. A firm that competes on performance quality produces goods and services
that deliver a high level of some set of performance dimensions. For cars, it could be
superior handling, cornering, and braking. For hotels, it might be a superior level of
luxury, comfort, and spa services. In higher education, it might be based on the faculty’s
high level of research output and the quality of its teaching. In software, it might be
additional features, a high level of functionality, or processing speed.

In contrast, a product or service with high conformance quality delivers on its
specifications, whether that means a high level of performance or not. In software,
conformance quality is more about the absence of bugs and simply meeting whatever
level of functionality or speed is specified in planning documentation. An EconoLodge
with high conformance quality has rooms and cleaning services that conform to
specifications and only basic amenities, which also conform to specifications. When we
talk about quality in everyday life, we typically mean performance quality, but
conformance quality is critical in operations.

Performance quality is primarily the realm of designers and product developers.
Operations is responsible for developing processes that meet design specifications. A
product that does not meet its design specifications is typically called a defect; depending
on the process, it will be either reworked or scrapped (which will be indicated on a
process flow diagram). The yield of the process is the number of good products expressed
as a percentage of the starting total. The concept of defect rates may also be applied to
services with quantifiable design specifications. However, from the customer’s
perspective, a suboptimal service experience may be more detrimental to the company’s
reputation than a product defect, which can be caught before the product reaches the
customer. The job of Quality Assurance in a manufacturing process is to minimize the
number of defects that occur and ensure that defective products are never shipped to the
customer. In a service context, it’s only about the former.

The cost-quality tradeoff is often debated. At its heart, it depends on whether
performance quality or conformance quality is being discussed. Often this isn’t clarified,
and confusion reigns. Typically, creating high performance quality is costly, because of
the expensive materials and components used, the skill (and consequent cost) of the
service staff, the amenities provided, and so on. In contrast, assuring high conformance
quality usually cuts costs, because it cuts rework costs; it saves on materials that would
otherwise need to be scrapped, and it reduces returns and warranty costs.

Speed
A second aspect of effectiveness that is important to many firms is how quickly (and
reliably) they can produce and deliver a product to customers, or, in a service context,
how quickly a customer can be served. In analyzing processes, whether speed is an
important consideration will depend on a number of factors. An elegant, expensive
restaurant may see a long, leisurely meal as a benefit, whereas fast-food restaurants,
almost by definition, are very concerned with how quickly customers get their food. For
physical products, the order-to-delivery lead time (the time between the placement of an

8007 | Core Reading: PROCESS ANALYSIS 19

order by a customer and the delivery of that order) often involves myriad players in a long
supply chain. A toothpaste manufacturer will be concerned with how long the product is
in the supply chain, but the time it takes to produce the product is less important. The set
of interactions that speed up response time in a supply chain is taken up in Core Reading:
Supply Chain Management (HBP No. 8031) and Core Reading: Strategic Sourcing (HBP
No. 8037). For the purpose of process analysis, we will focus on what takes place in the
manufacturing or service process itself.

We define throughput time (TPT) as the start-to-finish time of a process, namely the
total elapsed time between the time when a customer walks in the door and the time when
the customer leaves, or the time from when the raw materials and components begin to
be gathered and the time the finished product is completed.o We calculate two different
versions of throughput time:

• Minimum (or rush order) throughput time
• Average throughput time

Minimum throughput time is particularly important for manufacturing firms that
accept customized rush orders (for example, “Make five units of this specialized electronic
device so that our product development engineers can test those prototypes; we need it
right away”) or want to speed VIP customers through a service process (such as, “Our
platinum customers can go to the head of the line”). For a sequential process with a single
specialized worker at each task, minimum TPT equals labor content. Consider the process
from earlier, shown again in Figure 13.

Figure 13 Four-Step Process with a Bottleneck

A rush-order product will take two minutes at Task A. Then Worker B will put away

whatever she had been processing and work on the rush product, and so on. The rush-
order TPT is 2+4+5+3 = 14 minutes.

Parallel processes are a bit more complicated. For the artisan woodworking shop, the
full set of components will arrive in the components inventory buffer, ready to be
assembled, once every 2 hours. Thus, the minimum TPT of all woodworking is 2 hours.
From the time that the wood is selected and woodworking is begun to the time that a
finished set of components is available for assembly is 2 hours. The minimum TPT for
this process is 2 hours (woodworking) + 1 hour (assembly) + ½ hour (staining), or 3½
hours in total.

Note that minimum TPT will only be achieved by firms that promise rush orders (or,
for service firms, VIP service), and this will require that a rush order (or VIP customer) be
moved to the head of the line, ahead of earlier products or customers.

This kind of simple analysis is neither possible nor useful for most processes that
have many tasks and WIP inventory buffers (or customer waiting lines) and in which
products (or customers) are processed in a first-in-first-out sequence. Instead, operating
managers want to know, “What is the average throughput time?” This is critical for many
organizations because it influences promised delivery time, or estimated time of service.
These are often critical parts of the customer promise.

o In manufacturing contexts, this is often called the manufacturing lead time.

8007 | Core Reading: PROCESS ANALYSIS 20

To better understand average TPT, imagine waiting in line for a popular movie that
has just been released to theaters. The teller takes, on average, 30 seconds to sell a ticket
(for simplicity, assume that it takes twice as long to sell two tickets to a couple). The
movie starts at 7:12 p.m. You’ve heard from your friends that the line is usually long—on
average 25 people. When do you need to get there to enter the theater by 7:10 p.m.?

The 25 people in line will take 12½ minutes to process (25 people · ½
minute/person), and then you’ll need another 30 seconds to buy your ticket, so from the
time you arrive until the time you enter, it will take 13 minutes. Thus, you should arrive
by 6:57. See the Core Reading: Managing Queues (HBP No. 8047) for further information
about analyzing the performance of queues.

Flexibility
A third measure of effectiveness and, for some processes a critical source of
competitiveness, is flexibility. David Upton defined flexibility as “the ability to change or
react with little penalty in time, effort, cost or performance.” This is a very abstract
definition, and, indeed, there are many forms of flexibility. Consider the following
questions:3

• Dimensions: What needs to be flexible? What needs to be adaptable? Do we
need our process to quickly adapt to different raw material specifications,
different worker skills, or different customer needs?

• Time Horizon: What does “little penalty in time” mean? Minutes? Days?
Weeks? Years?

• Elements: Which element(s) of flexibility are most important to us? Which
of the following are we trying to manage or improve?
• Range—the breadth of products we can manufacture or the range of

customer needs we can satisfy.
• Uniformity—consistency of quality or speed across the entire range.
• Mobility—the speed with which we can change from one

task/product/customer to another.

Safety
While customer promise of a business rarely explicitly involves employee safety, most
effective operating managers would rank it as their number-one process metric. In 2011,
U.S. Secretary of Labor Hilda Solis made a simple statement that underscores the
importance that worker safety must play in any operation’s catalog of performance
metrics:

Every day in America, 12 people go to work and never come home.
Every year in America, 3.3 million people suffer a workplace injury from
which they may never recover. These are preventable tragedies that
disable our workers, devastate our families, and damage our economy.4

In 2010, the federal Office of Safety and Health Administration (OSHA) made more
than 40,000 inspection visits. Despite these efforts, 4,547 workers were killed on the job in
2011.5

Responsible operating managers have in place well-publicized safety procedures, and
they train employees in their use. A prominent sign in many well-run factories announces
the number of days since the last lost-work-time injury, and operating managers track

8007 | Core Reading: PROCESS ANALYSIS 21

and manage a number of performance measures, including workplace injuries (typically
injuries requiring some medical interventions) per 200,000 hours of work (100 people
working for a year), workman’s compensation costs, and various metrics for OSHA
compliance. Unfortunately, in too many workplaces around the world, operations
managers allow unsafe work practices to persist, subordinating safety to efficiency.

8007 | Core Reading: PROCESS ANALYSIS 22

3 KEY TERMS
Blocked Task: A task that is idled because the next task is still in process on another unit, and
thus the blocked task has no space for output.

Bottleneck: The task or resource that limits the capacity of a process. The bottleneck can be
any resources (e.g., raw materials, labor, machinery, energy). If the bottleneck is a task, it will
be the task performed by the workstation(s) with the longest cycle time.

Capacity Utilization: The ratio of actual output to capacity.

Cycle Time: The average time between the completion of successive units of product or, for a
service process, the average time between the departures of successive customers.

Labor Content: The total time that is spent by the firm’s employees on the product or service.

Labor Utilization: Productive time spent by workers as a percentage of total time for which
they are available.

Lead Time: Usually the elapsed time between when an order is placed and when it is
delivered.

Machine Utilization: Percentage of time that a machine is running and productive.

Parallel Process: Tasks that can be performed at the same time. Outputs from parallel
processes are typically integrated into one product at some point later in the process flow.

Process Flow Diagram (also called a Process Map): A diagram that shows the sequence in
which tasks take place as well as the flow of products, customers, and/or information through
the tasks. The scale and scope of process flow diagrams are determined by the management
issue they are intended to elucidate. Some show task flows at a high level and across separate
departments; others show every detail of a small, complicated process.

Sequential Process: A set of tasks that must be performed in sequence, one after another. If a
task cannot be started until the previous task is complete, those tasks form a sequential
process.

Starved Task: A task that is idled because it lacks sufficient inputs.

Steady State: When an operating system starts up, initial conditions affect the status of
process flows. After a while, when the system’s status no longer depends on initial conditions,
it reaches steady state.

Throughput Time (TPT): The time it takes for one unit to complete a process, from
beginning to end.

Work in Process (WIP): The number of units or partially completed units in the process at
any point in time. WIP includes units currently being worked on as well as those in WIP
inventory. WIP inventory is separate from raw materials inventory (RMI) and finished goods
inventory (FGI). In a service, WIP can be customers, either receiving service or waiting to be
served.

8007 | Core Reading: PROCESS ANALYSIS 23

4 FOR FURTHER READING

The Goal: A Process of Ongoing Improvement, by Eliyahu M. Goldratt (North River Press)
is an entertaining novel about production flow at a factory.

5 ENDNOTES

1 For an interesting view of a process viewpoint in health care delivery, see Richard Bohmer, Designing
Care: Aligning the Nature and Management of Health Care (Boston: Harvard Business Press, 2009).
2 See David Garvin, “Competing on the Eight Dimensions of Quality,” Harvard Business Review 65
(November–December 1987): 101–109.
3 This section is taken from David Upton’s typology: David M. Upton, “The Management of
Manufacturing Flexibility,” California Management Review 36, no. 2 (Winter 1994): 72–89.
4 http://www.osha.gov/oshstats/commonstats.html, accessed March 26, 2012.
5 Ibid.

8007 | Core Reading: PROCESS ANALYSIS 24

6 INDEX
Page numbers followed by f refer to figures. Page numbers followed by i refer to
interactive illustrations. Page numbers followed by t refer to tables.

automated process in batch processing, 3, 6, 17

average throughput time (TPT), 19–20

batch flow, 7, 7f

batch processes, 8

batch size, 8

blocked task (blocking), 11, 11f, 12i, 22

bottleneck pacing, 11

bottlenecks, 9–11, 9f, 10f, 11f, 12, 13t, 14, 14t, 15, 16, 16f, 17, 19f, 22

capacity, 9

capacity assessment, 4, 9–16

capacity utilization, 13, 13t, 22

capital, and machine utilization, 17

changeover times, 8. See also setup times in batch processing

conformance quality, 18

conventions in process flow diagrams, 5

cost, tradeoff between quality and, 18

cycle time, 11–16, 12i, 13t, 14f, 15f, 22

customer promise, 3–4, 16, 17, 20

decision node, 6

defect, 18

design capacity, 9

design

specifications, 18

effectiveness, 7, 15–16, 17

effectiveness assessment, 4, 17–19

efficiency, 7, 13, 15, 16

efficiency assessment, 4, 16–17, 16f

energy utilization, 17

flexibility, 20

inputs, 3

labor content, 8, 16, 19, 22

labor cost, 8, 14

labor utilization, 16–17, 22

8007 | Core Reading: PROCESS ANALYSIS 25

lead time, 18–19, 22

machine utilization, 17, 22

manual process in batch processing, 6, 9, 9f, 12, 12i

mapping of processes. See process flow diagrams

material utilization, 17

minimum throughput time (TPT), 19

operating

processes, 3–4

operating system, 3

operations management, 3

output, 3, 9, 11, 13, 14, 16. See also capacity assessment

parallel process, 6–7, 6f, 7f, 19, 22

performance,14, 15, 17. See also process performance

performance quality, 18

processes, 3–4

process flow, 5, 6f

process flow diagrams (process maps), 5–8, 5f, 6f, 7f, 22

process performance, 4, 7, 13, 15–16

process

yield, 18

quality, 17–18

rated capacity, 9

run time in batch processing, 8

rush orders, 19

safety, 20–21

sequential process, 5, 5f, 6, 14, 19, 22

setup times in batch processing, 8

size of batch, 8

specialization, 9, 13, 15

specifications, 18

speed, 18–20

stakeholder value, 4

standard time, 7–8

starved task (starving), 11, 11f, 12i, 22

steady state, 5, 22

strategic positioning, 3

supply chain, 19

system capacity, 13

8007 | Core Reading: PROCESS ANALYSIS 26

task time, 7–8, 12, 14, 15, 15f

throughput time (TPT), 19–20, 19f, 22

value for stakeholders, 4

wage rate, 8

waiting lines, 19

work in process (WIP), 9, 11, 22

work in process (WIP) buffers, 6, 19

yield, 18

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For the exclusive use of P. Vongsumedh,

022.

This document is authorized for use only by Pubordee Vongsumedh in Online MBA

8-00

-Spring

22 taught by IOANNIS BELLOS, George Mason University from Jan 2022 to Feb 2022.

7 | Core Reading: MANAGING QUEUES 2

Introduction

………………………………………………………………………………. 3

Essential Reading

……………………………………………………………………..

2.1 Managerial Objectives and the Nature of Waiting

Lines …………………………………………………………………………………. 5

2.2 Basic Assumptions, Symbols, and Definitions …………… 6

2.3 Performance Characteristics of Waiting-Line

Systems ……………………………………………………………………………. 8

2.4 An Approximation Formula for Lq ……………………………… 11

2.5 Some Insights into Waiting-Line Systems ………………… 13

2.6 Sample Applications of the Lq Approximation

Formula ………………………………………………………………………….. 16

Supplemental Reading

………………………………………………………….. 20

Perceived Versus Actual Waiting Time

…………………….. 20

Key Terms

……………………………………………………………………………….

Endnotes

…………………………………………………………………………………. 23

Index

………………………………………………………………………………………… 25

This reading contains links to online interactive exercises, denoted by a . In order
to access these exercises, you will need to have a broadband Internet connection.
Verify that your browser meets the minimum technical requirements by visiting

http://hbsp.harvard.edu/list/tech-specs

http://www.hbsp.harvard.edu

Ta bl e of C o nt ents

http://hbsp.harvard.edu/list/tech-specs

http://www.hbsp.harvard.edu/

8047 | Core Reading: MANAGING QUEUES 3

1 INTRODUCTION

f this is a service economy, why am I still on hold?” ask
Frances Frei and Anne Morriss in their book Uncommon
Service: How to Win by Putting Customers at the Core of

Your Business.1 This is just one example of waiting that you may
have experienced. Have you recently waited in a checkout line at
the supermarket? At a “fast”-food restaurant? For a web page to
download? All of these are examples of the phenomenon of
queuing: the act of waiting for a service to begin. Queuing is an
important concern for operating managers in both service and
manufacturing environments. They often need to make decisions
regarding capacity levels, staffing, and technology that affect the
time customers or goods spend waiting for a service.

Waiting lines, or queues, most often form because of variability in a system. In most
service businesses, as well as some manufacturing settings, the operating manager’s job is
complicated by an inability to specify either the time of arrival of a customer request or
the work content of the service to be provided. In a fast-food business, for instance,
customers arrive at unpredictable, random times during the day and expect to be able to
order from a wide variety of items. The challenge for the restaurant manager is to fulfill
customers’ expectations while consuming as few resources as possible. The manager must
determine the appropriate staffing level, the proper technology for preparing the food,
and the target for the customer’s waiting time. For example, too few personnel and a grill
that takes an excessive amount of time to cook burgers may result in a waiting time that’s
too long. Too many personnel and a grill that cooks the burgers more quickly (and is
more expensive) may provide a short waiting time but decrease profitability.

Queues can also form in a manufacturing environment, where, for example, physical
parts may have to wait for a machine to become available. In this reading, we primarily
use service examples to explain queuing phenomena. Those interested in manufacturing
can easily apply the frameworks and models discussed here.

Waiting lines can result in delays. The consequences of these delays depend on the
context. A shorter delay in a hospital emergency room may prevent serious medical
complications for patients with heart attacks. Quicker turnaround times for loan
processing may increase the number of loan applications and the lender’s profitability.
Long waits at a restaurant may cause potential customers to go elsewhere for a meal.a
Shorter delays result, in many cases, in better use of labor, equipment, and facilities, and a
greater capability for providing customers with an excellent level of service.

a We are reminded of the quote by the famous American baseball player and “philosopher” Yogi Berra
about a popular restaurant: “Nobody goes there anymore; it’s too crowded.”

“I

8047 | Core Reading: MANAGING QUEUES 4

We also present a formula for approximating line length in a system, some of the insights
into queuing systems that arise from this formula, and some further applications of the
formula. In the Supplemental Reading, we address ways to manage customers’
perceptions of waiting.

8047 | Core Reading: MANAGING QUEUES 5

2 ESSENTIAL READING

The objective for managers in applying the calculations in this reading is to understand
and estimate the various costs and performance characteristics of waiting-line systems in
specific service processes so that they can make appropriate design choices. Costs might
be related to the number and type of servers; performance characteristics include
considerations such as customer waiting time and line length.

Thus, in designing a service, managers might ask the following queue-related
questions:

• On average, how many customers will be waiting for service? How long, on
average, will a customer wait?

• How many servers are necessary to achieve a specified limit to customer
waiting time?

• What number of servers (in a particular situation) leads to an average
waiting time for the customers and an average idle time for the servers that
minimize the total cost of the service system?

•

Should servers be combined into one or more centralized areas?

• Which process improvements should be implemented given different service
rates, operating costs, and target cost and service objectives?

• What is the impact of a reduction in the variability of either the time between
customer arrivals or the service times?

The answers clearly depend on the relative costs (in a particular situation) of customers
and servers. As the above questions imply, a waiting-line system in a service setting has
three basic components:

Servers

Customers, who may arrive at random or unscheduled times

Service encounters, which typically vary in their completion times

At times, a customer’s waiting time and a server’s idle time incur easily measurable
out-of-pocket costs. For example, every minute a prospective customer waits on the
telephone for a customer service representative costs the organization long-distance
telephone charges. But at other times, there are no clear costs associated with a waiting
customer. Customers queued in restaurants, post offices, banks, airports, and so forth, do
not impose direct costs to the service provider, but if the lines become too long, some
customers may leave before completing the transaction or choose a competitor the next
time they need that service. Thus, an operations manager must plan for waiting lines and
make staffing decisions accordingly—all the while striving to manage the ongoing trade-
off between excessive service capacity and needless waiting.

It is important for managers to understand that a line can form even if the capacity of
the server(s) exceeds customer demand. Waiting time cannot be entirely eliminated

2.1 Ma nag erial O bj ecti ves a nd t he Nat ure
of Waiti ng Li nes

8047 | Core Reading: MANAGING QUEUES 6

because of the variability we mentioned earlier. A large number of customers may arrive
over a short period of time, or there might be a few closely spaced service encounters that
take an excessive amount of time. Conversely, even if few servers are present and waiting
lines prevail, the servers may be idle some of the time if there is a long gap between
arrivals or there are some successive short service encounters.

Interactive Illustration 1 shows a graphical representation of a queuing system.
There is one server, and service time is always exactly 4 minutes per customer. Therefore,
during a 25-minute interval, the server could serve 5 customers and have 5 minutes of idle
time. Likewise, it is possible for all 5 customers to be served without having to wait for
service. However, that situation occurs only if customer arrivals are evenly spaced during
the 25-minute interval.

To run the simulation, click “Start” and then click “Customer Arrival” four times,
once for each customer. When the 25 simulated minutes end, you can see data on the
length of queues that form the waiting time per customer. Try different arrival patterns to
see the effect on performance metrics.

Interactive Illustration 1 Cumulative Arrivals and Departures

The calculations discussed in this reading depend on the following assumptions about
waiting-line situations:

1 First come, first served. Customers form single queues while waiting for
service and are taken for service on a first-come, first-served basis.
Customers move into a server channel (for instance, a teller’s window at a
bank) as it becomes available.

2. 2 Basi c Ass um pti ons , Sy mb ols,
and D efi nitio ns

https://s3.amazonaws.com/he-assets-prod/interactives/010_cumulative_arrivals/Launch.html

8047 | Core Reading: MANAGING QUEUES 7

2 Servers perform identical services. Servers can also be referred to as
channels performing services; thus, the number of servers equals the number
of channels. There may be multiple servers operating in parallel to service
one line.

3 Random arrivals and service. Customer arrivals occur randomly at an
average rate, called the arrival rate. Services are performed at an average rate,
called the service rate, at each of the servers. Note that the units of average
arrival rate and average service rate are customers per unit time. The
reciprocal value of the service rate, service time, is the average time for
serving a customer and is measured in time units. Similarly, the reciprocal
value of the average arrival rate is the average time between arrivals, called
the inter-arrival time, and is also measured in time units.

4 Statistical equilibrium (steady state). On average, the capability to process
customers must be greater than the rate at which they arrive; otherwise,
arriving customers might have to wait in lineb indefinitely. The average
capability of the system to process customers per unit time is given by the
average total service rate, which equals (number of servers) · (service rate per
server). The average number of customers requiring service per unit time is
given by the arrival rate. In transient situations, such as system startup—for
example, when an amusement park first opens in the morning—the number
in the queue and in service depends on the number of customers who waited
for the system to go into operation and on how long the system has been in
operation. Over time, the impact of the initial conditions tends to dampen
out, and the number of customers in the system and in line is independent of
time. The calculations described in this reading are not appropriate for
transient situations.

5 Infinite source of potential customers. We assume an infinite source of
potential customers. We may also use these calculations when the population
of the queuing system is not greater than 1% of the population of the source
of potential customers.

6 Appropriate queue behavior. We assume the arrival rate equals the demand
rate. This assumption implies no instances of:

a. balking (potential recipients refusing to join the queue)

b. reneging (customers in the queue leaving before being served)

c. cycling (recipients of service returning to the queue following service)

7 Uniform service effectiveness. The effectiveness of service at each server
channel is uniform over time and throughout the system.

b The use of in line and on line to describe a person in a queue varies regionally in the United States. The
author is from Philadelphia, so he has chosen to use in line.

8047 | Core Reading: MANAGING QUEUES 8

2. 3 Perfo rma nce C hara ct eristi cs of
Waiti ng -Li ne Sy stems

The calculations in this reading also use the symbols and definitions outlined in
Table 1.

Table 1 Definitions of Queuing Parameters

Facility
A total servicing unit; a facility may be composed of several channels (as defined
below) or as few as one

Channel A servicing point within a facility
A Arrival rate; average number of arrivals into the system per unit of time

S Service rate (per server); average number of services per unit of time per channel

Ts Average service time (1/S)

m Number of servers or number of channels in the facility
mS Average service rate of the facility (m · service rate per server)
u A/mS = (arrival rate)/(m · service rate per server ) = utilization or utilization factor
Ns Average number of customers in service

L Average number of customers in the system; those being served plus the waiting line

Lq
Average number of customers waiting in line; this number may be approximated using
a formula provided later in this reading

W Total average time in the system

Wq Average time spent waiting

Now that we have established the assumptions and definitions provided in the preceding
section, we can describe the performance characteristics of waiting-line systems
(utilization factor, average number of customers in service, average number of customers
waiting in the queue, average number of customers in the system, expected waiting time
in the queue, expected total average time in the system, and expected costs) and explore
the relationships among them.

Utilization factor (u): The utilization factor measures the percentage of time that
servers are busy with customers.

Note that u is a percentage and is a unit-less number. Consider a system with an
arrival rate of 10 people per hour, where each server can service 11 per hour. For m = 1
server, the utilization is u = 10/11 = 91%. This indicates that the server is busy 91% of the
time. As you can see in Figure 1, the utilization factor decreases as the number of servers
increases.

mS
=

8047 | Core Reading: MANAGING QUEUES 9

Figure 1 Utilization Factor

Average number of customers in service (Ns): The average number of customers in
service is A/S, derived by multiplying the utilization factor (u) times the number of
servers (m). (This does not include the number of customers waiting in line.)

Ns will vary as a direction function of the utilization, which is a function of the arrival
and service rates of the system, as shown above.

Average number of customers waiting in the queue (L q): The average number of
customers waiting in the queue excludes the customers that are being served. For most
systems, L q cannot be calculated directly because of the complex interactions caused by
the probability distributions describing the arrival and service processes; its value must be
approximated. An approximation formula (or queuing approximation) for this
performance characteristic is presented in the next section. We will show that the queue
length is a function of the number of servers, the utilization factor, and the arrival and
service variability.

The remaining performance characteristics of waiting-line systems all depend on the
value of L q—either observed empirically or estimated from the approximation formula
described in the next section.

Average number of customers in the system (L): By definition, the average number of
customers in the system is equal to the average number of customers in line plus the
average number of customers in service. That is:

Expected waiting time in the queuec (W q): The expected waiting time can be
determined by dividing the average queue length by the average arrival rate:

c There is often a large difference between the actual waiting time and the perceived waiting time.
Methods for managing customers’ perceptions are discussed in the Supplemental Reading.

s
A A

N m u m
mS S

 
= ⋅ = ⋅ = 

 

q sL L N= +

= q

W
A

8047 | Core Reading: MANAGING QUEUES 10

The formula that relates the waiting time and line length is commonly referred to as
Little’s Law. This relationship, L q = A · W q for the queue and L = A · W for the entire
system, was initially proven by John D. C. Little.2 The formula states that the length of a
line is directly related to the time spent in the line (or that the number of customers or
items in the system is directly related to the total time spent in the system). Given any two
of the parameters, the third can be determined. For example, if we know that the average
length of the line at a doughnut store is 3.5 customers, and we know that the average time
spent in line is two minutes, we can infer that the arrival rate to the store is given by:

3.5
1.75 customers minute 105 customers hour

2
q

L
A

W
= = = =

Figure 2 shows the impact of changes in the queue length on the average waiting
time. Obviously, the longer the line, the longer the wait. The slope of the line is the arrival
rate. The slope is not the service rate because Little’s Law deals with average wait time and
queue lengths at steady state. The departure rate from a steady state system must equal
the arrival rate. Thus, the relationship between line length and queue time is a function of
the average rate at which the line decreases. This average rate must weight busy and
nonbusy times.

Figure 2 Little’s Law

Expected total average time in the system (W): Because the average service time per
person or item is equal to the quantity 1/S, the total average time expected in the system
is:

The above relationships among W q , L q , A, and S are not dependent on any
assumptions regarding the distribution of arrival and service times but instead are true in
general.

1
qW = W S

 
+ 
 

8047 | Core Reading: MANAGING QUEUES 11

can be either an out-of-pocket cost or an opportunity cost. When the cost of waiting is
known, the cost of providing service may be added to the cost of waiting to obtain a total
system cost. For example, consider a queuing system with three channels and with a cost
per channel of $25 per hour. The total channel cost is $75 per hour. If the average waiting
time per customer is 15 minutes (0.25 hours), and the opportunity cost of waiting to the
firm is $15 per hour per customer, each customer arrival would cost (0.25 hours) · $15 =
$3.75 per customer. If 10 customers arrived per hour, the cost of waiting would be $37.50
per hour. Added to the $75 cost of service, the total cost would be $112.50 per hour.

Adding an extra server would cost $25 per hour. The server would be added if the
total system cost dropped to less than $112.50 per hour. Given that the cost of servers is
now 4 · $25 = $100, four servers would be optimal if the total opportunity cost of waiting
is now less than $12.50 per hour because the new total system cost is $100 + the cost of
waiting. This implies that the average waiting time for a customer would need to be less
than 0.0833 hours ($12.50/[$15 waiting cost per hour per customer · 10 customers per
hour]). Note that at 0.0833 hours per customer, the waiting cost per customer would be
(0.0833 hours) · $15 = $1.25 per customer. Multiplying by the 10 customers per hour
yields a total cost of waiting of $12.50 per hour.

Often, the waiting cost for customers is not known. Although there may be an
opportunity cost to the individuals or items in line, there may be no direct cost to the
firm. In these cases, managers may set targets for the average waiting time or for
probabilities of waiting no more than a specified period of time. They may also consider
some trade-offs. Consider the previous example. If we did not know or could not estimate
the $15/hour waiting time cost, we might ask, “Would you spend $25/hour to reduce the
average waiting time from 0.25 hours to x hours?” (where x is the estimated average
waiting time for customers in the new system). Note that rather than have an explicit
numerical answer to this question, we must rely on managerial judgment. One could also
factor in the lost revenue per customer and the probability of losing customers if the lines
are excessively long.

We have ignored the cost of the customer(s) in service (that is, customers who are
being served and so are not waiting in a queue) in our calculations. This may be included
but will not affect how many servers that managers decide to include in the system
because the time in service is not a function of the number of servers in these examples.
(Recall that N s = A/S, and neither A nor S is a function of the number of servers.) If
managers had to decide the actual service rate at each channel, it would be necessary to
include the time spent in service as part of the total system cost.

The L q approximation formula below can be used for approximating the line length
of a queuing system. It is often referred to as the Sakasegawa approximation.3 Although
many approximation formulas are available for queuing systems, we present this

2. 4 A n Ap p roxi matio n Form ula fo r L q

8047 | Core Reading: MANAGING QUEUES 12

particular formula for its relative ease of use, its use for developing insight regarding
queuing systems, and the relative quality of its approximation to “true” values for systems
with significant waiting lines.

( )

( )
( ) ( )( )2 22 1

1 2

m
IAT ST

CV CVu
L

+ +
≈ ⋅

−

u = the utilization factor, as defined above
m = the number of servers

CV IAT = the coefficient of variation of the inter-arrival times
CV ST = the coefficient of variation of the service time

The coefficient of variation of a probability distribution provides a relative measure of its
variability and is defined as the standard deviation of a distribution divided by its mean. It
is used in recognition of the fact that the magnitude of the standard deviation of a
distribution is more relevant with respect to the size of its mean than as an absolute
measure. Thus, a service time distribution with a standard deviation of 10 minutes is
much more significant if the mean is 10 minutes than if the mean is 50 minutes.d

Whereas the general principles of queue management—such as Little’s Law, the
effects of increasing variability, and the nature of pooling or the combination of queues
(which we’ll discuss later)—are applicable in a wide variety of situations, the L q
approximation formula is more limited and should be used only in situations with first
come, first served arrivals; a single type of customer; no limit on the waiting room; and
stationary, steady state conditions. For systems with finite waiting areas, priority queues,
different customer types, and overflow queues, more-sophisticated models and
simulation tools are available. For example, the Erlang loss formula can be used for
systems where customers are turned away when the system is full.4 Many Excel add-ins
that handle specific situations can be downloaded from the Internet (see, for example,
Queuing ToolPak from the University of Alberta School of Business). The Extend
simulation package is also often used to model complex queuing situations.

d In general, distributions with a CV < 0.75 are considered low variability, 0.75 < CV < 1.33 are considered moderate variability, and CV > 1.33 are considered high variability. The actual values of CV
can be measured from empirical data, as in the previous example. Often, for convenience and with no
other information, we assume that CV = 1.0. If the probability distribution describing the number of
arrivals in a given period is Poisson, a common occurrence, then CVIAT = 1.0; if the probability
distribution describing the service time is exponential, a common assumption, then CVIAT = 1.0.

8047 | Core Reading: MANAGING QUEUES 13

At first glance, the L q formula appears intimidating; however, its form provides some
general insights into queuing systems. Here we discuss these and illustrate them using the
L q formula.

Increasing Utilization Degrades Performance at an Increasing Rate

Looking at the L q approximation formula, we see the first term has the factor (1 − u) in
the denominator. As the system gets busier, the term (1 − u) (the percentage of idle time)
gets smaller; therefore, the line length grows at an increasing rate. Indeed, as u approaches
1 and (1 − u) approaches 0, the line length grows without bounds.

To illustrate the effects of changes in utilization in a waiting-line system, consider a
facility with one channel, an average arrival rate of 15 customers per hour, and an average
service rate of 20 customers per hour. The average service time, T s = 1/service rate = 1/20
of an hour (3 minutes). Thus, arrival rate = 15/hour; m = number of service channels = 1;
and service rate = 20/hour. Assume that the coefficient of variation of both the service
time distribution and the arrival time distribution is 1.0.

• The utilization ratio, u = arrival rate / (m ∙ service rate) = A/(m ∙ S), is thus
[(15 / (1 ∙ 20)] or 0.75.

• The average number of customers in service is given by N s = A/S = 15/20 =
0.75.

• The average number waiting (L q ), excluding those being served, is 2.25 using
the formula above.

• The average number of customers in the system (L) is 3.00 (L q + N s). The
average waiting time (W q ) is 0.15 hours, or 9 minutes using Little’s Law, W q
= L q /A.

• The average system time (W) is 0.2 hours, or 12 minutes (W q + [1/S]).

Excel data tables can be used to measure the sensitivity of the performance measures
to the input parameters. Interactive Illustration 2 can also be used to test the sensitivity
to the arrival rate and utilization. Move the slider for the arrival rate to see how the
average time in the queue changes. W q grows exponentially as the arrival rate increases
and the utilization, u, approaches the value 1. Line lengths grow so drastically because the
impact of variability is so much greater at higher utilization rates. A longer service time or
arrivals that occur closer together intuitively have a larger effect on the average line length
when systems are more crowded. The marginal benefit of each additional server decreases
as servers are added.

2. 5 So m e Insi g hts i nto Waiti ng-Li ne Sys tems

8047 | Core Reading: MANAGING QUEUES 14

Interactive Illustration 2 Average Time in Queue

Increasing Variability Degrades Performance at an Increasing Rate

Looking at the second term of the L q formula, we see that when the coefficients of
variation of both the service rate and the arrival rate are equal to 1, the second term has
no effect (because the second term is equal to 1 in that case). This is a common situation
because the exponential probability distribution accurately describes the empirically
observed inter-arrival and service times of many queuing systems. The exponential
distribution has a CV of 1.

The line length goes up with relative variability in either the inter-arrival time or the
service time. Indeed, as we’ve said, one of the causes of lines in systems is the inherent
variability of the service time and the inter-arrival time. When we observe the second
term, systems with lower coefficients of variation have shorter queues, all other things
being equal, as the second term becomes less than 1; when the coefficients of variation
exceed 1, the queue is longer, all other things being equal. In both cases, the magnitude of
the change is affected by the squaring factor. So a 25% reduction in the CV reduces the
queue by less than 25%, but a 25% increase in the CV increases the queue by more than
25%.

To illustrate the effects of changes in variability, look again at Interactive Illustration
2 and see what happens as you change the coefficients of variation of the inter-arrival
time and the service time. For any given utilization factor, as determined by the mean
arrival rate and depicted on the graph by the large red dot, the average time in queue (and
by Little’s Law, the length of the queue) increases with an increase in the system
variability, as measured by the coefficients of variation. The chart also shows the values
for the standard deviations of the inter-arrival times and service times as points of
reference.

https://s3.amazonaws.com/he-assets-prod/interactives/040_avg_time_queue_single/Launch.html

8047 | Core Reading: MANAGING QUEUES 15

All Other Things Being Equal, Bigger Is Better

For the same utilization and variability values, a bigger system (that is, one with more
servers) is better able to handle the variability. We know this intuitively because a larger
system can “pool” the variability across multiple servers and can better “absorb” periods
with a large number of arrivals in a short time. Thus, in larger systems, a long service time
at one or more of the channels does not have as large an impact because the other
channels can serve the additional customers. This effect is apparent from the formula by
again inspecting the first term. Notice that m, the number of channels, enters in the
exponent of u, which is a value less than 1, so as m gets larger, the first term gets smaller
(proportional to the square root of m), and the line length is smaller for larger systems
with the same utilization. Note the implications of this: If two smaller systems can be
pooled (combined) into one larger system, the average line length decreases while
maintaining the same utilization. Alternatively, the pooled system can be run at a higher
utilization than the individual smaller systems while maintaining the same average line
length as the smaller system.

Interactive Illustration 3 demonstrates the effects of changes in system size. The
graphic depicts a system with a service rate of 20 customers per hour and allows you to
change the number of servers, the mean arrival rate, and the coefficients of variation for
the inter-arrival time and the service time. Note the effect of changing the number of
servers. For any given utilization, the average length of the queue is smaller for a larger
numbers of servers. As the number of servers increases, the queue time graph moves
down and to the right. This will be investigated in more detail in an example below.

Interactive Illustration 3 Average Time in Queue with Multiple Servers

https://s3.amazonaws.com/he-assets-prod/interactives/041_avg_time_queue_multiple/Launch.html

8047 | Core Reading: MANAGING QUEUES 16

2. 6 Sa mpl e A ppli ca tions of t he L q
Ap proxim atio n Fo rm ula

The previous section described insights that can be developed from the L q formula
regarding utilization, variability, and pooling. In this section we provide examples of how
the formula can be used with sample data, and how it can be used to evaluate a
consolidation decision at a bank and a staffing decision for a telephone call center.

Using the Formula with Sample Data

The key to applying the Lq approximation formula is understanding the difference
between the probability distribution that describes the number of customers arriving in a
time period (arrival rate) and the probability distribution of the time between customers
arriving (inter-arrival time). Calculating utilization requires information regarding the
average arrival rate and the average service rate, measured in customers per time period.
The coefficients of variation in the L q formula are determined by looking at the inter-
arrival time and the actual service times. While the arrival rate can be determined from
the inter-arrival times, it is not necessarily true that the inter-arrival distribution can be
derived from the arrival rates. Likewise, if the service time distribution is known, then the
service rate and the coefficient of variation of the service times can be calculated.
Knowing the number of people served in a given time period does not necessarily yield
the service time distribution.

Table 2 Sample Call Center Data

Customer
Number Arrival Time

Time from Previous
Customer (minutes)

Service Time
(minutes)

1 8:03 4

2 8:06 3 3

3 8:09 3 1

4 8:12 3 3

5 8:16 4 3

6 8:20 4 2

7 8:23 3 4

8 8:25 2 4

9 8:28 3 1

10 8:32 4 1

11 8:36 4 4

12 8:41 5 4

13 8:45 4 3

14 8:49 4 1

15 8:53 4 3

16 8:55 2 4

17 8:58 3 3

18 9:00 2 2

8047 | Core Reading: MANAGING QUEUES 17

Customer
Number Arrival Time
Time from Previous
Customer (minutes)
Service Time
(minutes)

19 9:04 4 4

20 9:09 5 3

21 9:13 4 3

22 9:17 4 4

23 9:20 3 3

24 9:24 4 2

25 9:26 2 3

26 9:30 4 3

27 9:35 5 3

28 9:38 3 3

29 9:43 5 2

30 9:46 3 4

31 9:48 2 3

32 9:52 4 4

33 9:54 2 3

34 9:58 4 3

35 10:00 2 1

Mean 3.43 2.89
Standard
Deviation 0.95 0.99

• If there is one server, the utilization is given by 17.5/20.76 = 0.84, or 84%.

• The coefficient of variation of the inter-arrival times is given by CVIAT =
0.95/3.43 = 0.28.

Because the mean service time was 2.89 minutes with a standard deviation of 0.99
minutes, the coefficient of variation of the service time is given by

The average line length can be predicted to be equal to 0.58 customers.

0.99
0.43

2.89
= =STCV

8047 | Core Reading: MANAGING QUEUES 18

Effects of Pooling

In the previous section, we claimed that bigger is better. If so, a system that is a
combination of a number of smaller queues should have smaller line lengths and waiting
times than the individual queues themselves. The following example illustrates this
phenomenon, known as pooling.

Consider a bank that is evaluating a decision to consolidate three regional loan
application centers into a single centralized center. The data for the three regions and the
proposed centralized center are shown in Table 3.

Table 3 Data for Three Regions of a Bank

Region A Region B Region C Entire System

Arrival rate (A)
(customers per
hour)

0.38 0.32 0.36

Service rate (S)
(customers per
hour)

0.20 0.20 0.20

Processors (m) 2.00 2.00 2.00

Utilization
(A/mS) 0.95 0.80 0.90

CVIAT 1.00 1.00 1.00

CVST 1.00 1.00 1.00

Lq (customers)
(estimated) 17.60 2.90 7.70

28.2 (sum of
three regions)

Wq (hours) 46.40 9.00 21.50
26.7
(weighted
average of
three regions)

For each region the average time in the queue was calculated using the L q
approximation. Little’s Law was used to calculate the average waiting time. Note that the
system averages 28.2 customers waiting, with an average time of 26.7 hours. What
happens if we consolidate the regions? Consolidation yields an arrival rate of 1.06 per
hour (the sum of the three individual regions). The utilization becomes u = (A/mS) =
1.06/(6 · 0.2) = 0.8833. The L q formula results in a line length of 5.4. W q = L q /1.06 = 5.08
hours. Note the tremendous effect of combining the regions. In the original system, there
were 28.2 (17.6 + 2.9 + 7.7) people waiting at the three regions. Thus, combining the three
regions results in an 81% reduction in the number of loans awaiting service ([28.2 –
5.4]/28.2 = 0.81)! This occurs for two reasons: (1) There is no longer a region with a high
utilization (Region A) and (2) there is no longer the possibility that one region has a
number of loans awaiting processing while another region is idle.

8047 | Core Reading: MANAGING QUEUES 19

Call Center Staffing

A telephone call center for a mail-order catalog has the demand pattern shown in Table 4
for weekdays between 8:00 a.m. and noon.

Table 4 Customer Demand at Call Center

Beginning Time
Average Number of

Customers

8:00 a.m. 75
9:00 a.m. 110

10:00 a.m. 135
11:00 a.m. 185

We can use the L q formula to answer this question. First, note that rather than use the
average arrival rate for the entire four-hour period, we should determine a staffing level
for each of the four-hour periods; that is, we will solve four separate problems.
Aggregating the data would result in overestimating the staffing level for periods with
below-average arrivals and underestimating the staffing level for periods with above-
average arrivals.

We will use a trial-and-error approach with the L q formula in order to determine the
appropriate staffing levels for the target average waiting times. A will be given by the
values, in customers per hour, in Table 4. Because the average service time is 5 minutes,
the service rate per staff person is 60/5 = 12 customers per hour. CVIAT = 1, as given, and
CVST = 6/5 = 1.2. The results are shown in Table 5.

Table 5 Analysis of Call Center Data

Time Period m u
Lq

(Customers)
Wq

(Hours)
Wq

(Minutes)

8:00 a.m. 9 0.69 0.78 0.010 0.625

9:00 a.m. 12 0.76 1.31 0.012 0.714

10:00 a.m. 14 0.80 1.87 0.014 0.833

11:00 a.m. 19 0.81 1.73 0.009 0.559

The m values were determined by increasing m until the average waiting times were
under 1 minute. By looking at the utilization factors, we can see that bigger is indeed
better because the larger systems can run at higher utilizations for the same target value of
1 minute average waiting time. The manager can perform sensitivity analyses of the
number of servers (and utilization) versus the target average waiting time to see if the
target is viable.

8047 | Core Reading: MANAGING QUEUES 20

3 SUPPLEMENTAL READING
Perceived Versus Actual Waiting Time
One of the classic references on waiting is David Maister’s 1985 “The Psychology of
Waiting Lines,” which made the intuitive but, at the time, novel argument that a
customer’s perception of service quality is influenced just as much by the subjective
experience of waiting in a queue as it is by the objective measures of the waiting
experience (such as the number of minutes spent in line). Maister formulated the First
and Second Laws of Service—that is, companies can influence customer satisfaction in a
waiting line by working on what the customer expects and what the customer perceives,
especially in the early parts of the service encounter. Maister also identified eight
psychological factors that increase a customer’s negative perception of a wait, making it
feel longer:5

• Unoccupied time feels longer than occupied time (distraction).6

• Pre-process waits feel longer than in-process waits (moment).

• Anxiety makes waits seem longer (anxiety).

• Uncertain waits are longer than certain waits (uncertainty).

• Unexplained waits are longer than explained waits (explanation).

• Unfair waits are longer than equitable waits (fairness).

• The more valuable the service, the longer people will wait (value).

• Solo waiting feels longer than waiting in a group (solo wait).

Maister proposed that companies remedy these factors by instituting the elements of
customer service that most customers take for granted today, including updating them
about their status in the queue, giving them a sense of control, providing value-added (or
distracting) activities to occupy waiting time (such as perusing menus at a restaurant),
promoting a sense of fairness (by, for instance, using a ticket system to determine order
priority), and setting expectations. Especially if customers must go through a series of
waits, companies should acknowledge that they have been “entered into the system”
(through a registration procedure, such as one finds at a walk-in clinic), even though they
may have another period of waiting before the service can be performed (the medical
consultation). Companies also need to address even “irrational” sources of customer
anxiety: If I switch lines, will the next one move faster? Are there enough seats on the
plane for all ticketed passengers?

Maister’s arguments were theoretical, but in 1984, the first empirical field study on
waiting (in the retail industry) had verified a link between the conditions in which
consumers wait and their subjective perception of the wait. For example, the study
showed that people overestimated waiting time by 36% on average (a five-minute wait
feels seven minutes long).7 In the early 1990s, a survey of hotel and restaurant customers
conducted by a group of United Kingdom Forte hotel managers independently confirmed
many of Maister’s proposed factors.8 More than 70% of respondents were concerned
about waiting times. However, the survey revealed a nuance: Although the customers
believed that quality and value were worth waiting for, at a certain point a wait would
become unacceptable and would lower their perception of quality. This was especially
true for the hotel customers surveyed. In their operations, the hotel managers identified

8047 | Core Reading: MANAGING QUEUES 21

six key points at which customers have to wait during an overnight stay (check in,
luggage, telephone line, messages, room service, and checkout) and nine points during a
restaurant meal. All of these provide opportunities to establish operational standards and
to manage customers’ perceptions.

A 1999 literature review of 18 empirical studies of wait management supported
Maister’s basic conceptual framework but also identified a hierarchy of factors
influencing consumers’ behavior.9 The review’s authors ranked their importance as
temporal factors (real time/duration waited) first, individual factors (disposition of the
customer) second, and situational factors (controlled by the company) third. The studies
emphasized the importance of personal expectations of waits, although the authors of the
18 articles used different measures of the expectation concept, ranging from “probable
duration” to “reasonable duration” to “maximum tolerable duration.” Although there are
many individual factors that companies can’t control, they should try to identify
customers whose preferences they can accommodate (such as by providing various
checkout options at supermarkets) and those who worry more about waits. The review’s
only real revision to Maister’s framework was to reclassify “anxiety” as a dependent
variable, not a causal factor.

Different companies take different approaches to sharing information with
customers. Some give actual expected waiting times: Disney does so for its amusement
park rides, which helps families plan their day. Other companies “hide” visual
information by, for instance, wrapping a line around the corner. Overestimating is also
common; for example, a restaurant may tell customers they’ll have a ten-minute wait but
then seat them after eight minutes so that they feel gratified with the “fast service.”
Depending on the configuration of their operations, managers need to decide the best
queue discipline. Will they process requests by arrival, according to priority (such as
hospital triage), or on an appointment basis? Scheduling appointments involves a fine
balance between leaving service providers unproductive (if appointments are scheduled
too far apart) and not meeting consumers’ high expectations (if appointments are
scheduled so close together that the provider runs late).10

It’s important not only to provide waiting customers with status information but also
to show them work in progress as they wait for their service to be performed. In fact,
according to recent research, even the “appearance of effort” improves customer
satisfaction—customers who have to wait but receive visual cues (as the customers ahead
of them are served) may in fact be happier than customers who experience no wait in the
absence of visual cues.11 Examples of this labor illusion, as the authors call it, are showing
the names of airlines searched on the Kayak travel website or steaming the milk for each
individual coffee order at Starbucks.

8047 | Core Reading: MANAGING QUEUES 22

clocks and watches per capita than similarly developed societies, and
those they had were less likely to be accurate.

The British Broadcasting Corporation (BBC) also sheds light on some cultural
attitudes toward queuing: 15

• In India, although first come, first served lines are common in airports, those
who jockey for position are often served first in railway and bus stations.

• Although Russians usually form orderly lines, exceptions may occur at
doctor’s offices when people ask for “a minute with a doctor, just to get his
signature.” One minute often turns into a half hour.

Gillam, Simmons, and Weiss conclude that companies trying to differentiate their
brands in international markets must consider the challenges and opportunities presented
by local queuing preferences. In particular, they should

• understand what is important to customers’ satisfaction with queues

• determine the steps that optimize the experience for customers in a cost-
effective manner

• research what industry peers are doing and how customers respond to their
queues

• continually adapt the queue management system on the basis of past
experiences and customers’ evolving needs

8047 | Core Reading: MANAGING QUEUES 23

4 KEY TERMS
Idle Time: The amount of time a server is inactive while waiting for customers to arrive.

Line Length: The number of customers or items waiting in line for a service to begin.

Little’s Law: A formula that measures the relationship among the line length, the arrival
rate, and the waiting time.

Queuing Approximation: A formula that estimates the average number of customers
waiting in line. The formula is a function of the utilization, the number of servers, and the
variability in arrivals and service as measured by the coefficient of variation of the
customer times and the coefficient of variation of the service times.

Utilization Factor: The percentage of time that a server is busy with customers or items
in service.

Waiting Time: The amount of time a customer or an item spends in line before a service
begins.

8047 | Core Reading: MANAGING QUEUES 24

5 ENDNOTES

1 Frances Frei and Anne Morriss, Uncommon Service: How to Win by Putting Customers at the Core of
Your Business (Boston: Harvard Business Review Press, 2012), pp. 1–12.
2 John D. C. Little, “A Proof for the Queuing Formula: L = λ W,” Operations Research 9, no. 3 (May–June
1961): 383–387.
3 Hirotaka Sakasegawa. “An Approximation Formula for Lq Annals of the Institute of Statistical
Mathematics,” Volume 29, (1977): part A, pp. 67–75.
4 Donald Gross et al., Fundamentals of Queueing Theory, (New York: Wiley-Interscience, 2008).
5 David Maister, “Psychology of Waiting Lines,” in The Service Encounter: Managing Employee/Customer
Interaction in Service Businesses, ed. John A, Czepiel, Michael R. Solomon, and Carol F. Surprenant, pp.
113–115 (Lexington, MA: D.C. Heath/Lexington Books, 1985).
6 Parenthetical terms are borrowed from Agnès Durrande-Moreau, “Waiting for Service: Ten Years of
Empirical Research,” Journal of Service Management 10, no. 2 (1999): 171–194.
7 Jacob Hornik, “Subjective vs. Objective Time Measures: A Note on the Perception of Time in
Consumer Behavior,” Journal of Consumer Research 11, no. 1 (June 1984): 615–618, as cited in
Durrande-Moreau, “Waiting for Service: Ten Years of Empirical Research.”
8 Peter Jones and Michael Dent, “Improving Service: Managing Response Time in Hospitality
Operations,” International Journal of Operations & Production Management 14.5 (1994): 52.
9 Agnes Durrande-Moreau, “Waiting for Service: Ten Years of Empirical Research,” Journal of Service
Management (1999): 171–194.
10 David H. Maister, “Note on the Management of Queues,” HBS No. 680-053 (Boston: Harvard Business
School Publishing, 1979), http://hbsp.harvard.edu, accessed May 2013.
11 Ryan W. Buell and Michael I. Norton, “Think Customers Hate Waiting? Not So Fast…,” Harvard
Business Review, May 2011: 2, http://hbsp.harvard.edu, accessed May 2013.
12 Graham Gillam, Kyle Simmons, and Elliott Weiss, “Line, Line Everywhere a Line, the Impact of
Culture on Waiting Line Management,” (working paper, Darden School of Business, University of
Virginia, 2013).
13 Richard C. Larson, “Perspectives on Queues: Social Justics and the Psychology of Queueing,”
Operations Research 35, no. 6 (Nov/Dec 1987): 895–905, ABI/INFORM via ProQuest, accessed May
2013.
14 Malcolm Gladwell, “You Are How You Wait—Queues Have Subtle Rules of Fairness and Justice,” The
Seattle Times, December 28, 1992, http://community.seattletimes.nwsource.com/archive/?date=
19921228&slug=1532358
15 Benjamin Walker, “Priority Queues: Paying to Get to the Front of the Line,” BBC News Magazine,
October 10, 2012, http://www.bbc.co.uk/news/magazine-19712847, accessed May 2013.

8047 | Core Reading: MANAGING QUEUES 25

6 INDEX
Page numbers followed by f refer to figures. Page numbers followed by i refer to
interactive illustrations. Page numbers followed by t refer to tables.

anxiety, 20, 21
appointment times, 21
approximation formula, 9, 11–12, 13, 16–17,

18, 23
arrival rate, 7, 8, 8t, 9, 10, 10f, 13, 14, 14i, 15,

16, 17t, 18, 23
arrival time variability, 3, 5, 6, 6i, 8, 11, 12, 13,

14, 14i,

15, 15i, 17, 23

average line length, 12, 13, 14i, 15, 17
average number of customers in service, 8, 8t,

9, 13
average number of customers waiting in the

queue, 5, 8, 8t, 9. See also queuing
approximation

average waiting time, 5, 10, 11, 12, 13, 14i, 18,
19

British Broadcasting Corporation (BBC), 22

call-center waiting times, 12, 17, 17t, 19, 19t
channels, 6–7, 8t. See also number of servers;

server idle time
control, sense of, 20
costs, 5, 11. See also expected costs;

opportunity cost; waiting time cost
cultural factors, 21–22
customer arrival times, 3, 5, 6, 6i, 8, 11, 12, 13,

14, 14i, 15, 15i, 16t–17t, 20, 23
customer behavior, 21
customer expectations, 20, 21
customer perception of waiting time, 20–22
customers. See average number of customers in

service; average number of customers
waiting in the queue; customer arrival times

customer waiting time. See waiting time

Disney, 21
duration. See waiting time

Erlang loss formula, 12
estimating wait times, 20–21
expectations of customers, 20, 21
expected costs, 8, 10–11
expected total average time in the system, 8, 10
expected waiting time, 8, 9, 21

fairness, 20, 21
fast-food restaurant queues, 3
First and Second Laws of Service, 20

hotel waiting times, 20–21

idle time, 5, 6, 13, 18, 23
inter-arrival time, 7, 12, 14, 15, 16, 17, 19
international markets, 22

Kayak travel website, 21

labor illusion, 21
line length, 5, 6, 6i, 9–10, 10f, 11, 13, 14, 15, 15i,

18, 23. See also average line length
Little’s Law, 10, 10f, 12, 13, 14, 18, 23
Lq formula, 9, 11–12, 13, 14, 16, 18, 19

management principles, 12, 21
managers, 3, 5, 11, 19, 20, 21
manufacturing queues, 3, 10

number of call-center staff, 19
number of servers (channels), 5–6, 6–7, 8, 8t, 9,

9f, 10–11, 12, 13, 15, 15i, 17, 19, 23

operating managers, 3, 5, 11, 19, 20, 21
opportunity cost, 11
overestimating wait times, 20, 21

perception of waiting time, 20–22
pooling, 12, 15, 18
psychological factors, 20

queue management principles, 12, 21
queues, 3
queuing approximation, 9, 11–12, 13, 16, 18, 23
Queuing ToolPak, 12

restaurant waiting times, 3, 20–21

Sakasegawa approximation, 11. See also

approximation formula
scheduling times, 21
server idle time, 5, 6, 13, 18, 23
server number. See number of servers
service encounters, 5, 6, 20
service rate, 7, 8t, 9, 10, 11, 13, 14, 14i, 15, 16,

17, 19
service time, 6, 7, 8t, 10, 12, 13, 14, 14i, 15, 16,

16t–17t, 17, 19, 23
service time variability, 3, 5, 8, 11, 12, 13, 14,

15, 15i, 17, 23
For the exclusive use of P. Vongsumedh, 2022.
This document is authorized for use only by Pubordee Vongsumedh in Online MBA 638-001-Spring 2022 taught by IOANNIS BELLOS, George Mason University from Jan 2022 to Feb 2022.

8047 | Core Reading: MANAGING QUEUES 26

simulation tools, 12
solo waiting, 20
Starbucks, 21
status information for customers, 20, 21
system variability, 3. See also arrival time

variability; service time variability

uncertainty, 20
updates on queue status, 20
utilization factor, 8, 8t, 9, 9f, 12, 14, 19, 23

value-added activities, 20
variability in arrival time, 3, 5, 6, 6i, 8, 11, 12,

13, 14, 14i, 15, 15i, 17, 23
variability in service time, 3, 5, 8, 11, 12, 13, 14,

15, 15i, 17, 23
visual clues, 21

waiting lines, 3
wait management principles, 12, 21
waiting time, 3, 5–6, 6i, 8, 9–10, 10f, 11, 12, 13,

14i, 18, 20–21, 23. See also average waiting
time; expected waiting time

waiting time cost, 10–11

Increasing Utilization Degrades Performance at an Increasing Rate
Increasing Variability Degrades Performance at an Increasing Rate
All Other Things Being Equal, Bigger Is Better
Using the Formula with Sample Data
Effects of Pooling
Call Center Staffing
Perceived Versus Actual Waiting Time

Table of Contents

Introduction

Essential Reading

2.1 Managerial Objectives and the Nature of Waiting Lines 5

2.2 Basic Assumptions, Symbols, and Definitions 6

2.3 Performance Characteristics of Waiting-Line Systems 8

2.4 An Approximation Formula for Lq 11

2.5 Some Insights into Waiting-Line Systems 13

2.6 Sample Applications of the Lq Approximation Formula 16

Supplemental Reading

Perceived Versus Actual Waiting Time

Key Terms

Endnotes

Index

2
5

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INTRODUCTION

“I

f this is a service economy, why am I still on hold?” ask Frances Frei and Anne Morriss in their book Uncommon Service: How to Win by Putting Customers at the Core of Your Business.[endnoteRef:1] This is just one example of waiting that you may have experienced. Have you recently waited in a checkout line at the supermarket? At a “fast”-food restaurant? For a web page to download? All of these are examples of the phenomenon of queuing: the act of waiting for a service to begin. Queuing is an important concern for operating managers in both service and manufacturing environments. They often need to make decisions regarding capacity levels, staffing, and technology that affect the time customers or goods spend waiting for a service. [1: Frances Frei and Anne Morriss, Uncommon Service: How to Win by Putting Customers at the Core of Your Business (Boston: Harvard Business Review Press, 2012), pp. 1–12.]

Waiting lines
, or
queues
, most often form because of variability in a system. In most service businesses, as well as some manufacturing settings, the operating manager’s job is complicated by an inability to specify either the time of arrival of a customer request or the work content of the service to be provided. In a fast-food business, for instance, customers arrive at unpredictable, random times during the day and expect to be able to order from a wide variety of items. The challenge for the restaurant manager is to fulfill customers’ expectations while consuming as few resources as possible. The manager must determine the appropriate staffing level, the proper technology for preparing the food, and the target for the customer’s
waiting time
. For example, too few personnel and a grill that takes an excessive amount of time to cook burgers may result in a waiting time that’s too long. Too many personnel and a grill that cooks the burgers more quickly (and is more expensive) may provide a short waiting time but decrease profitability.

Queues can also form in a manufacturing environment, where, for example, physical parts may have to wait for a machine to become available. In this reading, we primarily use service examples to explain queuing phenomena. Those interested in manufacturing can easily apply the frameworks and models discussed here.

Waiting lines can result in delays. The consequences of these delays depend on the context. A shorter delay in a hospital emergency room may prevent serious medical complications for patients with heart attacks. Quicker turnaround times for loan processing may increase the number of loan applications and the lender’s profitability. Long waits at a restaurant may cause potential customers to go elsewhere for a meal.[footnoteRef:1] Shorter delays result, in many cases, in better use of labor, equipment, and facilities, and a greater capability for providing customers with an excellent level of service. [1: We are reminded of the quote by the famous American baseball player and “philosopher” Yogi Berra about a popular restaurant: “Nobody goes there anymore; it’s too crowded.”]

In this reading, we present frameworks and tools that managers can use to understand the characteristics of waiting-line systems and improve operations where queues may form. We first discuss queuing-related questions that managers ask in designing a service operation and the nature of waiting-line systems. We then lay out some basic assumptions about those systems and describe various characteristics of them. We also present a formula for approximating
line length
in a system, some of the insights into queuing systems that arise from this formula, and some further applications of the formula. In the Supplemental Reading, we address ways to manage customers’ perceptions of waiting.

Essential Reading

2.1 Managerial Objectives and the Nature
of Waiting Lines

The objective for managers in applying the calculations in this reading is to understand and estimate the various costs and performance characteristics of waiting-line systems in specific service processes so that they can make appropriate design choices. Costs might be related to the number and type of servers; performance characteristics include considerations such as customer waiting time and line length.

Thus, in designing a service, managers might ask the following queue-related questions:

On average, how many customers will be waiting for service? How long, on average, will a customer wait?

How many servers are necessary to achieve a specified limit to customer waiting time?

What number of servers (in a particular situation) leads to an average waiting time for the customers and an average
idle time
for the servers that minimize the total cost of the service system?

Should servers be combined into one or more centralized areas?

Which process improvements should be implemented given different service rates, operating costs, and target cost and service objectives?

What is the impact of a reduction in the variability of either the time between customer arrivals or the service times?

The answers clearly depend on the relative costs (in a particular situation) of customers and servers. As the above questions imply, a waiting-line system in a service setting has three basic components:

Servers
Customers, who may arrive at random or unscheduled times
Service encounters, which typically vary in their completion times

At times, a customer’s waiting time and a server’s idle time incur easily measurable out-of-pocket costs. For example, every minute a prospective customer waits on the telephone for a customer service representative costs the organization long-distance telephone charges. But at other times, there are no clear costs associated with a waiting customer. Customers queued in restaurants, post offices, banks, airports, and so forth, do not impose direct costs to the service provider, but if the lines become too long, some customers may leave before completing the transaction or choose a competitor the next time they need that service. Thus, an operations manager must plan for waiting lines and make staffing decisions accordingly—all the while striving to manage the ongoing trade-off between excessive service capacity and needless waiting.

It is important for managers to understand that a line can form even if the capacity of the server(s) exceeds customer demand. Waiting time cannot be entirely eliminated because of the variability we mentioned earlier. A large number of customers may arrive over a short period of time, or there might be a few closely spaced service encounters that take an excessive amount of time. Conversely, even if few servers are present and waiting lines prevail, the servers may be idle some of the time if there is a long gap between arrivals or there are some successive short service encounters.

Interactive Illustration 1 shows a graphical representation of a queuing system. There is one server, and service time is always exactly 4 minutes per customer. Therefore, during a 25-minute interval, the server could serve 5 customers and have 5 minutes of idle time. Likewise, it is possible for all 5 customers to be served without having to wait for service. However, that situation occurs only if customer arrivals are evenly spaced during the 25-minute interval.

To run the simulation, click “Start” and then click “Customer Arrival” four times, once for each customer. When the 25 simulated minutes end, you can see data on the length of queues that form the waiting time per customer. Try different arrival patterns to see the effect on performance metrics.

Interactive Illustration 1 Cumulative Arrivals and Departures

2.2 Basic Assumptions, Symbols, and Definitions

The calculations discussed in this reading depend on the following assumptions about waiting-line situations:

1. First come, first served. Customers form single queues while waiting for service and are taken for service on a first-come, first-served basis. Customers move into a server channel (for instance, a teller’s window at a bank) as it becomes available.

Servers perform identical services. Servers can also be referred to as channels performing services; thus, the number of servers equals the number of channels. There may be multiple servers operating in parallel to service one line.

Random arrivals and service.
Customer arrivals occur randomly at an average rate, called the arrival rate. Services are performed at an average rate, called the service rate, at each of the servers. Note that the units of average arrival rate and average service rate are customers per unit time. The reciprocal value of the service rate, service time, is the average time for serving a customer and is measured in time units. Similarly, the reciprocal value of the average arrival rate is the average time between arrivals, called the inter-arrival time, and is also measured in time units.

Statistical equilibrium (steady state). On average, the capability to process customers must be greater than the rate at which they arrive; otherwise, arriving customers might have to wait in line[footnoteRef:2] indefinitely. The average capability of the system to process customers per unit time is given by the average total service rate, which equals (number of servers) · (service rate per server). The average number of customers requiring service per unit time is given by the arrival rate. In transient situations, such as system startup—for example, when an amusement park first opens in the morning—the number in the queue and in service depends on the number of customers who waited for the system to go into operation and on how long the system has been in operation. Over time, the impact of the initial conditions tends to dampen out, and the number of customers in the system and in line is independent of time. The calculations described in this reading are not appropriate for transient situations. [2: The use of in line and on line to describe a person in a queue varies regionally in the United States. The author is from Philadelphia, so he has chosen to use in line. ]

Infinite source of potential customers. We assume an infinite source of potential customers. We may also use these calculations when the population of the queuing system is not greater than 1% of the population of the source of potential customers.

Appropriate queue behavior.
We assume the arrival rate equals the demand rate. This assumption implies no instances of:

a. balking (potential recipients refusing to join the queue)
b. reneging (customers in the queue leaving before being served)
c. cycling (recipients of service returning to the queue following service)

Uniform service effectiveness. The effectiveness of service at each server channel is uniform over time and throughout the system.

The calculations in this reading also use the symbols and definitions outlined in Table 1.

Table 1 Definitions of Queuing Parameters

Facility

A total servicing unit; a facility may be composed of several channels (as defined below) or as few as one

Channel

A servicing point within a facility

Arrival rate; average number of arrivals into the system per unit of time

Service rate (per server); average number of services per unit of time per channel

Average service time (1/S)

Number of servers or number of channels in the facility

Average service rate of the facility (m · service rate per server)

A/mS = (arrival rate)/(m · service rate per server ) = utilization or utilization factor

Average number of customers in service

Average number of customers in the system; those being served plus the waiting line

Average number of customers waiting in line; this number may be approximated using a formula provided later in this reading

Total average time in the system

Average time spent waiting

2.3 Performance Characteristics of Waiting-Line Systems

Now that we have established the assumptions and definitions provided in the preceding section, we can describe the performance characteristics of waiting-line systems (utilization factor, average number of customers in service, average number of customers waiting in the queue, average number of customers in the system, expected waiting time in the queue, expected total average time in the system, and expected costs) and explore the relationships among them.

Utilization factor (u): The
utilization factor
measures the percentage of time that servers are busy with customers.

Note that u is a percentage and is a unit-less number. Consider a system with an arrival rate of 10 people per hour, where each server can service 11 per hour. For m = 1 server, the utilization is u = 10/11 = 91%. This indicates that the server is busy 91% of the time. As you can see in Figure 1, the utilization factor decreases as the number of servers increases.

Figure 1 Utilization Factor

Average number of customers in service (Ns): The average number of customers in service is A/S, derived by multiplying the utilization factor (u) times the number of servers (m). (This does not include the number of customers waiting in line.)

Ns will vary as a direction function of the utilization, which is a function of the arrival and service rates of the system, as shown above.

Average number of customers waiting in the queue (Lq): The average number of customers waiting in the queue excludes the customers that are being served. For most systems, Lq cannot be calculated directly because of the complex interactions caused by the probability distributions describing the arrival and service processes; its value must be approximated. An approximation formula (or
queuing approximation
) for this performance characteristic is presented in the next section. We will show that the queue length is a function of the number of servers, the utilization factor, and the arrival and service variability.
The remaining performance characteristics of waiting-line systems all depend on the value of Lq—either observed empirically or estimated from the approximation formula described in the next section.

Average number of customers in the system (L): By definition, the average number of customers in the system is equal to the average number of customers in line plus the average number of customers in service. That is:

Expected waiting time in the queue[footnoteRef:3] (Wq): The expected waiting time can be determined by dividing the average queue length by the average arrival rate: [3: There is often a large difference between the actual waiting time and the perceived waiting time. Methods for managing customers’ perceptions are discussed in the Supplemental Reading.]

The formula that relates the waiting time and line length is commonly referred to as
Little’s Law.
This relationship, Lq = A · Wq for the queue and L = A · W for the entire system, was initially proven by John D. C. Little.[endnoteRef:2] The formula states that the length of a line is directly related to the time spent in the line (or that the number of customers or items in the system is directly related to the total time spent in the system). Given any two of the parameters, the third can be determined. For example, if we know that the average length of the line at a doughnut store is 3.5 customers, and we know that the average time spent in line is two minutes, we can infer that the arrival rate to the store is given by: [2: John D. C. Little, “A Proof for the Queuing Formula: L = λ W,” Operations Research 9, no. 3 (May–June 1961): 383–387.]

Figure 2 shows the impact of changes in the queue length on the average waiting time. Obviously, the longer the line, the longer the wait. The slope of the line is the arrival rate. The slope is not the service rate because Little’s Law deals with average wait time and queue lengths at steady state. The departure rate from a steady state system must equal the arrival rate. Thus, the relationship between line length and queue time is a function of the average rate at which the line decreases. This average rate must weight busy and nonbusy times.
Figure 2 Little’s Law

Expected total average time in the system (W):
Because the average service time per person or item is equal to the quantity 1/S, the total average time expected in the system is:

The above relationships among Wq, Lq, A, and S are not dependent on any assumptions regarding the distribution of arrival and service times but instead are true in general.

Expected costs:
The expected costs of a queuing system to the organization are usually determined by some function of the number of servers and the expected customers’ waiting time or the number of customers in the system. The cost of waiting can be either an out-of-pocket cost or an opportunity cost. When the cost of waiting is known, the cost of providing service may be added to the cost of waiting to obtain a total system cost. For example, consider a queuing system with three channels and with a cost per channel of $25 per hour. The total channel cost is $75 per hour. If the average waiting time per customer is 15 minutes (0.25 hours), and the opportunity cost of waiting to the firm is $15 per hour per customer, each customer arrival would cost (0.25 hours) · $15 = $3.75 per customer. If 10 customers arrived per hour, the cost of waiting would be $37.50 per hour. Added to the $75 cost of service, the total cost would be $112.50 per hour.
Adding an extra server would cost $25 per hour. The server would be added if the total system cost dropped to less than $112.50 per hour. Given that the cost of servers is now 4 · $25 = $100, four servers would be optimal if the total opportunity cost of waiting is now less than $12.50 per hour because the new total system cost is $100 + the cost of waiting. This implies that the average waiting time for a customer would need to be less than 0.0833 hours ($12.50/[$15 waiting cost per hour per customer · 10 customers per hour]). Note that at 0.0833 hours per customer, the waiting cost per customer would be (0.0833 hours) · $15 = $1.25 per customer. Multiplying by the 10 customers per hour yields a total cost of waiting of $12.50 per hour.
Often, the waiting cost for customers is not known. Although there may be an opportunity cost to the individuals or items in line, there may be no direct cost to the firm. In these cases, managers may set targets for the average waiting time or for probabilities of waiting no more than a specified period of time. They may also consider some trade-offs. Consider the previous example. If we did not know or could not estimate the $15/hour waiting time cost, we might ask, “Would you spend $25/hour to reduce the average waiting time from 0.25 hours to x hours?” (where x is the estimated average waiting time for customers in the new system). Note that rather than have an explicit numerical answer to this question, we must rely on managerial judgment. One could also factor in the lost revenue per customer and the probability of losing customers if the lines are excessively long.
We have ignored the cost of the customer(s) in service (that is, customers who are being served and so are not waiting in a queue) in our calculations. This may be included but will not affect how many servers that managers decide to include in the system because the time in service is not a function of the number of servers in these examples. (Recall that Ns = A/S, and neither A nor S is a function of the number of servers.) If managers had to decide the actual service rate at each channel, it would be necessary to include the time spent in service as part of the total system cost.

2.4 An Approximation Formula for Lq

In this section, we present an approximation formula for calculating the value of Lq. Recall that once we have this value for the line length, other important performance characteristics, such as Wq, W, and L, can be determined. The relationships among Wq, Lq, W, L, A, and S, described in the preceding section, are not dependent on the nature or shape of the probability distributions describing the arrival and service processes. The values of the measures Wq, Lq, W, and L depend highly on the assumptions regarding these distributions, as we will see.
The Lq approximation formula below can be used for approximating the line length of a queuing system. It is often referred to as the Sakasegawa approximation.[endnoteRef:3] Although many approximation formulas are available for queuing systems, we present this particular formula for its relative ease of use, its use for developing insight regarding queuing systems, and the relative quality of its approximation to “true” values for systems with significant waiting lines. [3: Hirotaka Sakasegawa. “An Approximation Formula for Lq Annals of the Institute of Statistical Mathematics,” Volume 29, (1977): part A, pp. 67–75.]

u = the utilization factor, as defined above

m = the number of servers

CVIAT = the coefficient of variation of the inter-arrival times

CVST = the coefficient of variation of the service time
The coefficient of variation of a probability distribution provides a relative measure of its variability and is defined as the standard deviation of a distribution divided by its mean. It is used in recognition of the fact that the magnitude of the standard deviation of a distribution is more relevant with respect to the size of its mean than as an absolute measure. Thus, a service time distribution with a standard deviation of 10 minutes is much more significant if the mean is 10 minutes than if the mean is 50 minutes.[footnoteRef:4] [4: In general, distributions with a CV < 0.75 are considered low variability, 0.75 < CV < 1.33 are considered moderate variability, and CV > 1.33 are considered high variability. The actual values of CV can be measured from empirical data, as in the previous example. Often, for convenience and with no other information, we assume that CV = 1.0. If the probability distribution describing the number of arrivals in a given period is Poisson, a common occurrence, then CVIAT = 1.0; if the probability distribution describing the service time is exponential, a common assumption, then CVIAT = 1.0.]

Whereas the general principles of queue management—such as Little’s Law, the effects of increasing variability, and the nature of pooling or the combination of queues (which we’ll discuss later)—are applicable in a wide variety of situations, the Lq approximation formula is more limited and should be used only in situations with first come, first served arrivals; a single type of customer; no limit on the waiting room; and stationary, steady state conditions. For systems with finite waiting areas, priority queues, different customer types, and overflow queues, more-sophisticated models and simulation tools are available. For example, the Erlang loss formula can be used for systems where customers are turned away when the system is full.[endnoteRef:4] Many Excel add-ins that handle specific situations can be downloaded from the Internet (see, for example, Queuing ToolPak from the University of Alberta School of Business). The Extend simulation package is also often used to model complex queuing situations. [4: Donald Gross et al., Fundamentals of Queueing Theory, (New York: Wiley-Interscience, 2008).]

In addition, the Lq approximation is useful for calculating the average line length and waiting time. In many instances, we are interested in the probability that line lengths or wait times will exceed a certain threshold. For example, a call center’s goal may be to have an average hold time of 30 seconds, with no more than a 5% chance of waiting 45 seconds or longer. In this situation, more-detailed results on the shape of the waiting time distribution would be needed to determine staffing requirements.
At first glance, the Lq formula appears intimidating; however, its form provides some general insights into queuing systems. Here we discuss these and illustrate them using the Lq formula.
2.5 Some Insights into Waiting-Line Systems

Increasing Utilization Degrades Performance at an Increasing Rate
Looking at the Lq approximation formula, we see the first term has the factor (1 − u) in the denominator. As the system gets busier, the term (1 − u) (the percentage of idle time) gets smaller; therefore, the line length grows at an increasing rate. Indeed, as u approaches 1 and (1 − u) approaches 0, the line length grows without bounds.
To illustrate the effects of changes in utilization in a waiting-line system, consider a facility with one channel, an average arrival rate of 15 customers per hour, and an average service rate of 20 customers per hour. The average service time, Ts = 1/service rate = 1/20 of an hour (3 minutes). Thus, arrival rate = 15/hour; m = number of service channels = 1; and service rate = 20/hour. Assume that the coefficient of variation of both the service time distribution and the arrival time distribution is 1.0.
The utilization ratio, u = arrival rate / (m ∙ service rate) = A/(m ∙ S), is thus [(15 / (1 ∙ 20)] or 0.75.
The average number of customers in service is given by Ns = A/S = 15/20 = 0.75.
The average number waiting (Lq), excluding those being served, is 2.25 using the formula above.
The average number of customers in the system (L) is 3.00 (Lq + Ns). The average waiting time (Wq) is 0.15 hours, or 9 minutes using Little’s Law, Wq = Lq/A.
The average system time (W) is 0.2 hours, or 12 minutes (Wq + [1/S]).
Excel data tables can be used to measure the sensitivity of the performance measures to the input parameters. Interactive Illustration 2 can also be used to test the sensitivity to the arrival rate and utilization. Move the slider for the arrival rate to see how the average time in the queue changes. Wq grows exponentially as the arrival rate increases and the utilization, u, approaches the value 1. Line lengths grow so drastically because the impact of variability is so much greater at higher utilization rates. A longer service time or arrivals that occur closer together intuitively have a larger effect on the average line length when systems are more crowded. The marginal benefit of each additional server decreases as servers are added.

Interactive Illustration 2 Average Time in Queue
Increasing Variability Degrades Performance at an Increasing Rate
Looking at the second term of the Lq formula, we see that when the coefficients of variation of both the service rate and the arrival rate are equal to 1, the second term has no effect (because the second term is equal to 1 in that case). This is a common situation because the exponential probability distribution accurately describes the empirically observed inter-arrival and service times of many queuing systems. The exponential distribution has a CV of 1.
The line length goes up with relative variability in either the inter-arrival time or the service time. Indeed, as we’ve said, one of the causes of lines in systems is the inherent variability of the service time and the inter-arrival time. When we observe the second term, systems with lower coefficients of variation have shorter queues, all other things being equal, as the second term becomes less than 1; when the coefficients of variation exceed 1, the queue is longer, all other things being equal. In both cases, the magnitude of the change is affected by the squaring factor. So a 25% reduction in the CV reduces the queue by less than 25%, but a 25% increase in the CV increases the queue by more than 25%.
To illustrate the effects of changes in variability, look again at Interactive Illustration 2 and see what happens as you change the coefficients of variation of the inter-arrival time and the service time. For any given utilization factor, as determined by the mean arrival rate and depicted on the graph by the large red dot, the average time in queue (and by Little’s Law, the length of the queue) increases with an increase in the system variability, as measured by the coefficients of variation. The chart also shows the values for the standard deviations of the inter-arrival times and service times as points of reference.
All Other Things Being Equal, Bigger Is Better
For the same utilization and variability values, a bigger system (that is, one with more servers) is better able to handle the variability. We know this intuitively because a larger system can “pool” the variability across multiple servers and can better “absorb” periods with a large number of arrivals in a short time. Thus, in larger systems, a long service time at one or more of the channels does not have as large an impact because the other channels can serve the additional customers. This effect is apparent from the formula by again inspecting the first term. Notice that m, the number of channels, enters in the exponent of u, which is a value less than 1, so as m gets larger, the first term gets smaller (proportional to the square root of m), and the line length is smaller for larger systems with the same utilization. Note the implications of this: If two smaller systems can be pooled (combined) into one larger system, the average line length decreases while maintaining the same utilization. Alternatively, the pooled system can be run at a higher utilization than the individual smaller systems while maintaining the same average line length as the smaller system.

Interactive Illustration 3 demonstrates the effects of changes in system size. The graphic depicts a system with a service rate of 20 customers per hour and allows you to change the number of servers, the mean arrival rate, and the coefficients of variation for the inter-arrival time and the service time. Note the effect of changing the number of servers. For any given utilization, the average length of the queue is smaller for a larger numbers of servers. As the number of servers increases, the queue time graph moves down and to the right. This will be investigated in more detail in an example below.

Interactive Illustration 3 Average Time in Queue with Multiple Servers

The previous section described insights that can be developed from the Lq formula regarding utilization, variability, and pooling. In this section we provide examples of how the formula can be used with sample data, and how it can be used to evaluate a consolidation decision at a bank and a staffing decision for a telephone call center.
2.6 Sample Applications of the Lq Approximation Formula

Using the Formula with Sample Data
The key to applying the Lq approximation formula is understanding the difference between the probability distribution that describes the number of customers arriving in a time period (arrival rate) and the probability distribution of the time between customers arriving (inter-arrival time). Calculating utilization requires information regarding the average arrival rate and the average service rate, measured in customers per time period. The coefficients of variation in the Lq formula are determined by looking at the inter-arrival time and the actual service times. While the arrival rate can be determined from the inter-arrival times, it is not necessarily true that the inter-arrival distribution can be derived from the arrival rates. Likewise, if the service time distribution is known, then the service rate and the coefficient of variation of the service times can be calculated. Knowing the number of people served in a given time period does not necessarily yield the service time distribution.

Table 2 Sample Call Center Data

Customer Number

Arrival Time

Time from Previous Customer (minutes)

Service Time (minutes)

8:03

8:06

8:09

8:12

8:16

8:20

8:23

8:25

8:28

8:32

8:36

8:41

8:45

8:49

8:53

8:55

8:58

9:00

Customer Number

Arrival Time

Time from Previous Customer (minutes)

Service Time (minutes)

9:04

9:09

9:13

9:17

9:20

9:24

9:26

9:30

9:35

9:38

9:43

9:46

9:48

9:52

9:54

9:58

10:00

Mean

3.43

2.89

Standard Deviation

0.95

0.99

Consider the call center data presented in Table 2, which shows the arrival and service times for the customers who called between 8 a.m. and 10 a.m. one morning. Thirty-five customers arrived in the two-hour interval. The mean time between arrivals was 3.43 minutes with a standard deviation of 0.95. Because 35 customers arrived in a two-hour interval, the arrival rate is 35/2 = 17.5 customers per hour. The average service time was 2.89 minutes, so the service rate is 60/2.89 = 20.76 per hour.
If there is one server, the utilization is given by 17.5/20.76 = 0.84, or 84%.
The coefficient of variation of the inter-arrival times is given by CVIAT = 0.95/3.43 = 0.28.
Because the mean service time was 2.89 minutes with a standard deviation of 0.99 minutes, the coefficient of variation of the service time is given by

The average line length can be predicted to be equal to 0.58 customers.

Effects of Pooling
In the previous section, we claimed that bigger is better. If so, a system that is a combination of a number of smaller queues should have smaller line lengths and waiting times than the individual queues themselves. The following example illustrates this phenomenon, known as pooling.
Consider a bank that is evaluating a decision to consolidate three regional loan application centers into a single centralized center. The data for the three regions and the proposed centralized center are shown in Table 3.
Table 3 Data for Three Regions of a Bank

Region A

Region B

Region C

Entire System

Arrival rate (A) (customers per hour)

0.38

0.32

0.36

Service rate (S) (customers per hour)

0.20

Processors (m)

2.00

Utilization (A/mS)

0.95

0.80

0.90

CVIAT

1.00

CVST

1.00

Lq (customers) (estimated)

17.60

2.90

7.70

28.2 (sum of three regions)

Wq (hours)

46.40

9.00

21.50

26.7 (weighted average of three regions)

For each region the average time in the queue was calculated using the Lq approximation. Little’s Law was used to calculate the average waiting time. Note that the system averages 28.2 customers waiting, with an average time of 26.7 hours. What happens if we consolidate the regions? Consolidation yields an arrival rate of 1.06 per hour (the sum of the three individual regions). The utilization becomes u = (A/mS) = 1.06/(6 · 0.2) = 0.8833. The Lq formula results in a line length of 5.4. Wq = Lq/1.06 = 5.08 hours. Note the tremendous effect of combining the regions. In the original system, there were 28.2 (17.6 + 2.9 + 7.7) people waiting at the three regions. Thus, combining the three regions results in an 81% reduction in the number of loans awaiting service ([28.2 – 5.4]/28.2 = 0.81)! This occurs for two reasons: (1) There is no longer a region with a high utilization (Region A) and (2) there is no longer the possibility that one region has a number of loans awaiting processing while another region is idle.

Call Center Staffing
A telephone call center for a mail-order catalog has the demand pattern shown in Table 4 for weekdays between 8:00 a.m. and noon.
Table 4 Customer Demand at Call Center

Beginning Time

Average Number of Customers

8:00 a.m.

9:00 a.m.

110

10:00 a.m.

135

11:00 a.m.

185

The average time per customer call is 5 minutes, with a standard deviation of 6 minutes. The coefficient of variation for the inter-arrival times has historically been 1. The call center manager would like to know the number of staff required in each 1-hour block in order to have an average waiting time of under 1 minute.
We can use the Lq formula to answer this question. First, note that rather than use the average arrival rate for the entire four-hour period, we should determine a staffing level for each of the four-hour periods; that is, we will solve four separate problems. Aggregating the data would result in overestimating the staffing level for periods with below-average arrivals and underestimating the staffing level for periods with above-average arrivals.
We will use a trial-and-error approach with the Lq formula in order to determine the appropriate staffing levels for the target average waiting times. A will be given by the values, in customers per hour, in Table 4. Because the average service time is 5 minutes, the service rate per staff person is 60/5 = 12 customers per hour. CVIAT = 1, as given, and CVST = 6/5 = 1.2. The results are shown in Table 5.

Table 5 Analysis of Call Center Data

Time Period

Lq (Customers)

(Hours)

(Minutes)

8:00 a.m.

0.69

0.78

0.010

0.625

9:00 a.m.

0.76

1.31

0.012

0.714

10:00 a.m.

0.80

1.87

0.014

0.833

11:00 a.m.

0.81

1.73

0.009

0.559

The m values were determined by increasing m until the average waiting times were under 1 minute. By looking at the utilization factors, we can see that bigger is indeed better because the larger systems can run at higher utilizations for the same target value of 1 minute average waiting time. The manager can perform sensitivity analyses of the number of servers (and utilization) versus the target average waiting time to see if the target is viable.

supplemental reading

Perceived Versus Actual Waiting Time
One of the classic references on waiting is David Maister’s 1985 “The Psychology of Waiting Lines,” which made the intuitive but, at the time, novel argument that a customer’s perception of service quality is influenced just as much by the subjective experience of waiting in a queue as it is by the objective measures of the waiting experience (such as the number of minutes spent in line). Maister formulated the First and Second Laws of Service—that is, companies can influence customer satisfaction in a waiting line by working on what the customer expects and what the customer perceives, especially in the early parts of the service encounter. Maister also identified eight psychological factors that increase a customer’s negative perception of a wait, making it feel longer:[endnoteRef:5] [5: David Maister, “Psychology of Waiting Lines,” in The Service Encounter: Managing Employee/Customer Interaction in Service Businesses, ed. John A, Czepiel, Michael R. Solomon, and Carol F. Surprenant, pp. 113–115 (Lexington, MA: D.C. Heath/Lexington Books, 1985).]

Unoccupied time feels longer than occupied time (distraction).[endnoteRef:6] [6: Parenthetical terms are borrowed from Agnès Durrande-Moreau, “Waiting for Service: Ten Years of Empirical Research,” Journal of Service Management 10, no. 2 (1999): 171–194.]

Pre-process waits feel longer than in-process waits (moment).
Anxiety makes waits seem longer (anxiety).
Uncertain waits are longer than certain waits (uncertainty).
Unexplained waits are longer than explained waits (explanation).
Unfair waits are longer than equitable waits (fairness).
The more valuable the service, the longer people will wait (value).
Solo waiting feels longer than waiting in a group (solo wait).
Maister proposed that companies remedy these factors by instituting the elements of customer service that most customers take for granted today, including updating them about their status in the queue, giving them a sense of control, providing value-added (or distracting) activities to occupy waiting time (such as perusing menus at a restaurant), promoting a sense of fairness (by, for instance, using a ticket system to determine order priority), and setting expectations. Especially if customers must go through a series of waits, companies should acknowledge that they have been “entered into the system” (through a registration procedure, such as one finds at a walk-in clinic), even though they may have another period of waiting before the service can be performed (the medical consultation). Companies also need to address even “irrational” sources of customer anxiety: If I switch lines, will the next one move faster? Are there enough seats on the plane for all ticketed passengers?
Maister’s arguments were theoretical, but in 1984, the first empirical field study on waiting (in the retail industry) had verified a link between the conditions in which consumers wait and their subjective perception of the wait. For example, the study showed that people overestimated waiting time by 36% on average (a five-minute wait feels seven minutes long).[endnoteRef:7] In the early 1990s, a survey of hotel and restaurant customers conducted by a group of United Kingdom Forte hotel managers independently confirmed many of Maister’s proposed factors.[endnoteRef:8] More than 70% of respondents were concerned about waiting times. However, the survey revealed a nuance: Although the customers believed that quality and value were worth waiting for, at a certain point a wait would become unacceptable and would lower their perception of quality. This was especially true for the hotel customers surveyed. In their operations, the hotel managers identified six key points at which customers have to wait during an overnight stay (check in, luggage, telephone line, messages, room service, and checkout) and nine points during a restaurant meal. All of these provide opportunities to establish operational standards and to manage customers’ perceptions. [7: Jacob Hornik, “Subjective vs. Objective Time Measures: A Note on the Perception of Time in Consumer Behavior,” Journal of Consumer Research 11, no. 1 (June 1984): 615–618, as cited in Durrande-Moreau, “Waiting for Service: Ten Years of Empirical Research.”] [8: Peter Jones and Michael Dent, “Improving Service: Managing Response Time in Hospitality Operations,” International Journal of Operations & Production Management 14.5 (1994): 52.]

A 1999 literature review of 18 empirical studies of wait management supported Maister’s basic conceptual framework but also identified a hierarchy of factors influencing consumers’ behavior.[endnoteRef:9] The review’s authors ranked their importance as temporal factors (real time/duration waited) first, individual factors (disposition of the customer) second, and situational factors (controlled by the company) third. The studies emphasized the importance of personal expectations of waits, although the authors of the 18 articles used different measures of the expectation concept, ranging from “probable duration” to “reasonable duration” to “maximum tolerable duration.” Although there are many individual factors that companies can’t control, they should try to identify customers whose preferences they can accommodate (such as by providing various checkout options at supermarkets) and those who worry more about waits. The review’s only real revision to Maister’s framework was to reclassify “anxiety” as a dependent variable, not a causal factor. [9: Agnes Durrande-Moreau, “Waiting for Service: Ten Years of Empirical Research,” Journal of Service Management (1999): 171–194.]

Different companies take different approaches to sharing information with customers. Some give actual expected waiting times: Disney does so for its amusement park rides, which helps families plan their day. Other companies “hide” visual information by, for instance, wrapping a line around the corner. Overestimating is also common; for example, a restaurant may tell customers they’ll have a ten-minute wait but then seat them after eight minutes so that they feel gratified with the “fast service.” Depending on the configuration of their operations, managers need to decide the best queue discipline. Will they process requests by arrival, according to priority (such as hospital triage), or on an appointment basis? Scheduling appointments involves a fine balance between leaving service providers unproductive (if appointments are scheduled too far apart) and not meeting consumers’ high expectations (if appointments are scheduled so close together that the provider runs late).[endnoteRef:10] [10: David H. Maister, “Note on the Management of Queues,” HBS No. 680-053 (Boston: Harvard Business School Publishing, 1979), http://hbsp.harvard.edu, accessed May 2013.]

It’s important not only to provide waiting customers with status information but also to show them work in progress as they wait for their service to be performed. In fact, according to recent research, even the “appearance of effort” improves customer satisfaction—customers who have to wait but receive visual cues (as the customers ahead of them are served) may in fact be happier than customers who experience no wait in the absence of visual cues.[endnoteRef:11] Examples of this labor illusion, as the authors call it, are showing the names of airlines searched on the Kayak travel website or steaming the milk for each individual coffee order at Starbucks. [11: Ryan W. Buell and Michael I. Norton, “Think Customers Hate Waiting? Not So Fast…,” Harvard Business Review, May 2011: 2, http://hbsp.harvard.edu, accessed May 2013.]

In their investigation of the impact of culture on queuing behavior, Graham Gillam, Kyle Simmons, and Elliott Weiss note that culture may affect how an individual perceives a queue and thus can affect his or her service experience.[endnoteRef:12] Drawing on other research demonstrating that social justice, or a sense of fairness, often informs a customer’s attitude toward waiting in a particular line,[endnoteRef:13] they observe that this sense of justice varies with culture. In some countries, it is common for people of higher status to be ushered to the front of lines and be served immediately. This is viewed as “fair” in locations less concerned about equality among people. Malcolm Gladwell observes:[endnoteRef:14] [12: Graham Gillam, Kyle Simmons, and Elliott Weiss, “Line, Line Everywhere a Line, the Impact of Culture on Waiting Line Management,” (working paper, Darden School of Business, University of Virginia, 2013). ] [13: Richard C. Larson, “Perspectives on Queues: Social Justics and the Psychology of Queueing,” Operations Research 35, no. 6 (Nov/Dec 1987): 895–905, ABI/INFORM via ProQuest, accessed May 2013.] [14: Malcolm Gladwell, “You Are How You Wait—Queues Have Subtle Rules of Fairness and Justice,” The Seattle Times, December 28, 1992, http://community.seattletimes.nwsource.com/archive/?date= 19921228&slug=1532358 ]

In cultures that aren’t obsessed with punctuality or “wasted” time, chaotic lines for services are considered less of a problem. When Robert Levine, a psychologist at California State University at Fresno, studied the notoriously nonqueuing Brazilians, he found they had far fewer clocks and watches per capita than similarly developed societies, and those they had were less likely to be accurate.
The British Broadcasting Corporation (BBC) also sheds light on some cultural attitudes toward queuing: [endnoteRef:15] [15: Benjamin Walker, “Priority Queues: Paying to Get to the Front of the Line,” BBC News Magazine, October 10, 2012, http://www.bbc.co.uk/news/magazine-19712847, accessed May 2013.
]

In India, although first come, first served lines are common in airports, those who jockey for position are often served first in railway and bus stations.
Although Russians usually form orderly lines, exceptions may occur at doctor’s offices when people ask for “a minute with a doctor, just to get his signature.” One minute often turns into a half hour.
Gillam, Simmons, and Weiss conclude that companies trying to differentiate their brands in international markets must consider the challenges and opportunities presented by local queuing preferences. In particular, they should
understand what is important to customers’ satisfaction with queues
determine the steps that optimize the experience for customers in a cost-effective manner
research what industry peers are doing and how customers respond to their queues
continually adapt the queue management system on the basis of past experiences and customers’ evolving needs

Key Terms

Idle Time: The amount of time a server is inactive while waiting for customers to arrive.

Line Length: The number of customers or items waiting in line for a service to begin.

Little’s Law: A formula that measures the relationship among the line length, the arrival rate, and the waiting time.

Queuing Approximation: A formula that estimates the average number of customers waiting in line. The formula is a function of the utilization, the number of servers, and the variability in arrivals and service as measured by the coefficient of variation of the customer times and the coefficient of variation of the service times.

Utilization Factor: The percentage of time that a server is busy with customers or items in service.

Waiting Time: The amount of time a customer or an item spends in line before a service begins.

5 EndNotes

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8047 | Core Reading: MANAGING QUEUES 10
6 index
Page numbers followed by f refer to figures. Page numbers followed by i refer to interactive illustrations. Page numbers followed by t refer to tables.

anxiety, 20, 21
appointment times, 21
approximation formula, 9, 11–12, 13, 16–17, 18, 23
arrival rate, 7, 8, 8t, 9, 10, 10f, 13, 14, 14i, 15, 16, 17t, 18, 23
arrival time variability, 3, 5, 6, 6i, 8, 11, 12, 13, 14, 14i, 15, 15i, 17, 23
average line length, 12, 13, 14i, 15, 17
average number of customers in service, 8, 8t, 9, 13
average number of customers waiting in the queue, 5, 8, 8t, 9. See also queuing approximation
average waiting time, 5, 10, 11, 12, 13, 14i, 18, 19

British Broadcasting Corporation (BBC), 22

call-center waiting times, 12, 17, 17t, 19, 19t

channels, 6–7, 8t. See also number of servers; server idle time
control, sense of, 20
costs, 5, 11. See also expected costs; opportunity cost; waiting time cost
cultural factors, 21–22
customer arrival times, 3, 5, 6, 6i, 8, 11, 12, 13, 14, 14i, 15, 15i, 16t–17t, 20, 23
customer behavior, 21
customer expectations, 20, 21
customer perception of waiting time, 20–22
customers. See average number of customers in service; average number of customers waiting in the queue; customer arrival times
customer waiting time. See waiting time

Disney, 21
duration. See waiting time

Erlang loss formula, 12
estimating wait times, 20–21
expectations of customers, 20, 21
expected costs, 8, 10–11
expected total average time in the system, 8, 10
expected waiting time, 8, 9, 21

fairness, 20, 21
fast-food restaurant queues, 3
First and Second Laws of Service, 20
hotel waiting times, 20–21

idle time, 5, 6, 13, 18, 23
inter-arrival time, 7, 12, 14, 15, 16, 17, 19
international markets, 22

Kayak travel website, 21

labor illusion, 21
line length, 5, 6, 6i, 9–10, 10f, 11, 13, 14, 15, 15i, 18, 23. See also average line length
Little’s Law, 10, 10f, 12, 13, 14, 18, 23
Lq formula, 9, 11–12, 13, 14, 16, 18, 19

management principles, 12, 21
managers, 3, 5, 11, 19, 20, 21
manufacturing queues, 3, 10

number of call-center staff, 19
number of servers (channels), 5–6, 6–7, 8, 8t, 9, 9f, 10–11, 12, 13, 15, 15i, 17, 19, 23

operating managers, 3, 5, 11, 19, 20, 21
opportunity cost, 11
overestimating wait times, 20, 21

perception of waiting time, 20–22
pooling, 12, 15, 18
psychological factors, 20

queue management principles, 12, 21
queues, 3
queuing approximation, 9, 11–12, 13, 16, 18, 23
Queuing ToolPak, 12

restaurant waiting times, 3, 20–21

Sakasegawa approximation, 11. See also approximation formula
scheduling times, 21
server idle time, 5, 6, 13, 18, 23
server number. See number of servers
service encounters, 5, 6, 20
service rate, 7, 8t, 9, 10, 11, 13, 14, 14i, 15, 16, 17, 19
service time, 6, 7, 8t, 10, 12, 13, 14, 14i, 15, 16, 16t–17t, 17, 19, 23
service time variability, 3, 5, 8, 11, 12, 13, 14, 15, 15i, 17, 23
simulation tools, 12
solo waiting, 20
Starbucks, 21
status information for customers, 20, 21
system variability, 3. See also arrival time variability; service time variability

uncertainty, 20
updates on queue status, 20
utilization factor, 8, 8t, 9, 9f, 12, 14, 19, 23

value-added activities, 20
variability in arrival time, 3, 5, 6, 6i, 8, 11, 12, 13, 14, 14i, 15, 15i, 17, 23
variability in service time, 3, 5, 8, 11, 12, 13, 14, 15, 15i, 17, 23
visual clues, 21

waiting lines, 3
wait management principles, 12, 21
waiting time, 3, 5–6, 6i, 8, 9–10, 10f, 11, 12, 13, 14i, 18, 20–21, 23. See also average waiting time; expected waiting time
waiting time cost, 10–11
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INTRODUCTION

Essential Reading

2.1 Managerial Objectives and the Nature of Waiting Lines

2.2 Basic Assumptions, Symbols, and Definitions

2.3 Performance Characteristics of Waiting-Line Systems

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2.1 Managerial Objectives and the Nature
of Waiting Lines