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6.2: Variation in Capacity and Demand

Difficulty Level: At Grade Created by: CK-12
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Lesson Objectives

  • Define variation and describe how demand and the service rate vary
  • Explain how variation makes business decisions difficult
  • Explain why the results of business decisions are often counterintuitive when demand and capacity vary
  • Use probability distributions to describe situations in which multiple different outcomes are possible


An average of many numerical values is calculated by summing all the values and then dividing that total by the number of values. If the number of customers that arrive at a business is recorded each hour, and then those values are averaged, this defines the average number of customers per hour that arrive at that business. If the average is similarly computed for the number of customers that a sales associate can process each hour (assuming they are always working), then this is the average service rate, or capacity, of that sales associate. Although the number of customers that arrive or that are served varies from hour to hour, the average can be considered a representative quantity for all such observations.
Customers create the demand for goods and services from a company. The level of demand can be stated in many ways, depending on which is most appropriate for the situation. Sometimes demand is stated as the number of units that customers request. Demand is usually stated as a rate, that is, the number of units requested during some time period, such as units per hour. Alternately, demand can be stated as the number of customers per hour, or the number of customers per year that make requests of a goods or service provider. Measuring demand by the number of customers rather than the number of units of goods that need to be delivered is appropriate when each customer requests one item from a company. If customers can request multiple items, then measuring demand by the number of units over some time period is most appropriate.
demand rate
Demand rate is sometimes defined as the average number of customers that arrive at a business each hour. (See the definition of average to see how this average would be calculated.) However, when each customer might place an order for more than one item (for example, more than one book from Amazon.com), the demand rate might be measured in units (books) demanded per hour rather than by the number of customers per hour.
An event is a situation that will occur in the future and that will have an outcome (see definition of outcome below). This lesson uses a coin flip as an example of an event. The possible outcomes for this event are heads and tails. When we use simulation, we usually are considering uncertain events where we cannot perfectly predict what the outcome will be, just as we cannot perfectly predict the outcome of a coin flip in advance of the event.
A machine or person performing an operation is idle when there are no orders, or no demand, from customers. For example, a cashier at a store is idle when no customers are in line. A bank teller is idle when he or she has served all the customers in the bank and is waiting for more customers to arrive.
An outcome is the observed or possible result of an event. For example, when we flip a coin and the result of the flip is tails, then we say that tails is the outcome of the coin flip. If we are going to flip a coin five minutes from now, we don’t know if the result will be heads or tails, so we say that there are two possible outcomes: heads or tails. When multiple outcomes are possible, then we say that it is an uncertain situation because we cannot perfectly predict the outcome. In this situation, we use probability distributions to describe the likelihood of each of the possible outcomes. For example, when flipping a coin the probability of heads is 50% and the probability of tails is 50%. This means that if we flipped the coin a large number of times, we would expect that heads would result half the time and tails would result half the time.
When multiple outcomes are possible for an uncertain event, the probability of each possible outcome describes how likely that outcome is to occur relative to the other possible outcomes. The higher the probability of a particular outcome, the more likely it is to occur. For example, an outcome with a probability of 0.45 is more likely to occur than an outcome with a probability of 0.21. The probability also reflects the percentage of times that we would expect that outcome to occur if we were to observe which outcome occurred over many events. For example, a probability of 0.21 for a particular outcome implies that outcome would occur in 21% of the events that we observed, if the number of events were large. For a small number of observed events, we would still expect 21% of them to be the outcome whose probability is 0.21, but there would be variation around that percentage.
probability distribution
A probability distribution for an uncertain event is a list of all the possible outcomes and the probability of each. The probabilities for all the possible outcomes must sum to 100% (or 1). This is the same as saying that when the outcome of an uncertain event is observed, one of the possible outcomes must occur.
In service processes, a queue is usually thought of as a line of people waiting to be served. In manufacturing, queues can be orders waiting to be put into production or work waiting to be performed at a manufacturing machine.
service rate
Service rate is expressed in units of whatever is being processed in an operation per some time period. For example, customers per hour is one way to express the rate at which customers are served. Service rate can also be parts per hour or units manufactured per year in manufacturing processes. This rate expresses how fast the demands or orders of customers are being satisfied. The time period over which the rate is expressed can be minutes, hours, days, weeks, months, quarters, years, or any other unit of time.
Uncertainty means that when an event is going to occur in the future we cannot perfectly predict what the outcome of that event will be. The most precise description that can be offered about the future event in this case is a probability distribution, which states the likelihood of each of the possible outcomes.
x axis
The x axis is the horizontal axis on a graph. In this chapter the x axis shows the times at which we want to display some data for the operation we are analyzing. For example, this can be the number of people waiting in line (queue) for service.
Something varies, or exhibits variation, when it is not always the same value. For example, customers arriving at businesses vary because the number of customers that arrive on a day-to-day basis fluctuates. In addition, the type of variation that is considered in this chapter is the kind that cannot be predicted precisely. That is, variation arises in this situation because many different outcomes are possible, and we cannot predict exactly the outcome of the uncertain future event. When this is true, the cause of variation is uncertainty.
y axis
The y axis is the vertical axis of a graph. When data are graphed by a line, it is the y axis that scales the height of each point on the line, which reflects the numerical value of the data of interest that are being graphed.

Check Your Understanding

Recalling Prior Knowledge

The student must be familiar with the definitions for these terms:

  • demand
  • capacity
  • inventory
  • revenue
  • cost
  • profit

The definition for these terms can be found in the prior lesson, "Business Goals, Demand, Capacity, Inventory, and Customer Service."

Students should also be familiar with the following:

  • Graphs of numerical data, specifically column or bar charts.
  • What the average of a set of numbers means and how to calculate it.


This lesson contains three subsections that all describe variation and its effect on business decisions:

A. The Nature of Variation

B. How to Describe Variation

C. How Variation Makes Decisions Difficult

The first two sections describe variation and how to measure it, while the last section gives an example of how business decisions are difficult because the number of customers who arrive at a business varies from hour to hour and from day to day, and the time it takes workers to accomplish tasks varies from one customer to the next.

Lesson Content

The Nature of Variation

In the prior lesson we used a fast food restaurant as an example of a business that we wanted to study. In order to study such an operation and make decisions about capacity and inventory we need to understand variation, because we must take it into consideration in order to make good business decisions.

When customers arrive at a fast food restaurant and place orders, the items that they order are called the customer demand. It is natural for the number of customers who arrive at a business to vary from hour to hour. Customers do not arrive in a steady stream with the same number of customers arriving each hour. Instead, the number of customers arriving each hour is sometimes small and sometimes large. This is called "variation of demand." How quickly customers are served also varies. This is called "variation of supply" or "variation in the service rate." People work faster or slower depending on how much effort they exert. Moreover, sometimes there are many obstacles to work around while serving a particular customer, while serving another customer may be easy and quick. Besides these typical causes of variation in the service rate, the Table below identifies factors that can cause variation in the service rate in situations other than a fast food restaurant. Depending on these factors, service might be either slow or fast. People, machines, and equipment are used to serve customers, and so some of the causes of variation in service in the table are due to people and some are due to machines.

Factors Causing Variation in Service Rate
Service or Goods being Provided Events that Temporarily Slow the Speed of Service
Downloading songs, books, and pictures over the Internet Heavy Internet traffic, Internet failure, Internet provider failure, heavy traffic on website servers
Ordering a video game or book from a web site Heavy Internet traffic, Internet failure, Internet provider failure, heavy traffic on website servers
Grocery store checkout Competence/speed/skill of the cashier, cash register malfunction, number of items purchased by customer, whether an error occurs that needs to be corrected or approved by a manager, whether a bagger is available or not, whether alcohol is purchased or not and whether identification needs to be checked, an item not being recognized by the point-of-sale system, illegible bar code, speed at which customer puts items on belt
Collecting a toll at a toll booth Driver having trouble getting or finding money, a stalled vehicle, running out of change in the toll booth, changing toll booth operators, broken toll booth gate
Cashing a check with a bank teller Amount of the check (larger checks may require manager’s approval), teller running out of currency
Getting served at a sit-down restaurant Menu item that was ordered (some take longer), number of customers being served by a waiter
Getting to school on a school bus Traffic accident, broken traffic signal, delay in some students getting onto the bus, bus driver takes a wrong turn
Getting served lunch at the cafeteria Running out of food that needs to be replenished on the cafeteria line, make-to-order items that are either easier or more difficult to prepare, whether cafeteria worker makes an error and needs to remake an item

Figure below shows two columns of numbers that represent the number of customers that arrive at a restaurant each hour for a period of 20 hours. In Scenario A, the number of customers is not the same each hour: sometimes many customers arrive and sometimes fewer customers arrive. Therefore, we say that the arrival sequence in Scenario A varies. In contrast, the number of customers who arrive over 20 hours in Scenario B does not vary. In business situations we would expect the number of customers who arrive to vary from hour to hour and the number of customers who can be served to vary from hour to hour, as in Scenario A. We would rarely, if ever, expect to see a pattern without variation like that in Scenario B. Even when the number of customers who arrive is scheduled (as in a doctor’s office), demand can vary because some customers forget about their appointments and other customers show up at times other than when their appointment is.

Data for Number of Customers Arriving at a Restaurant

The term customer service, in general, means how well a business "treats" a customer. There are many possible measures of how well customers are treated, one of which is how long a customer waits to be served. A key factor in how quickly customers are served is the rate of customer arrivals relative to the rate of service, which means how fast customers are served. That is, what matters is the demand rate in comparison with the service rate. If customers arrive faster than they are served, then a line of customers forms, causing customers to wait to be served. For example, when a large number of people show up at a movie theater to get tickets and the attendants cannot take care of them right away, a line forms. Waiting in line adds to the amount of time that it takes a customer to get his or her movie ticket. Even when an attendant slows down for a short period (in other words, when there is variation in the service rate), a line can form that delays customers. Variation in service rate and variation in customer arrivals can cause customers to wait longer than average, so that the level of customer service deteriorates. This is the reason why we need to understand variation, because variation affects customer service.

(Sometimes the term "queue" is used instead of the word "line" to describe waiting customers in the types of analysis that we will do. "Queue" would more likely be used in daily conversation in the United Kingdom, but it is also used in scientific studies like the ones discussed in this chapter.)

It is impossible to look at a sequence of numbers like that in Scenario A (Figure above), which represents varying rates of customer arrivals, and accurately approximate how customer service might be affected by that variation. However, we can use computers to mimic how businesses operate with varying demand and service rates to understand the effects of variation. It is the ultimate goal of this chapter to demonstrate how this is done. In addition, this chapter demonstrates how mimicking a process can help a manager determine the number of people, machines, and equipment necessary to process customers’ demands. As discussed in the prior lesson, people, machines, and equipment give a business its capacity to serve customers. Managers need to know how to set the correct capacity of their operations in order to provide good service to the customers who place demands on their businesses.

How to Describe Variation

We described in the previous section how the number of customers that arrives at any business varies from minute to minute and hour to hour. In order to consider the effect of variation in a decision-making process, we need to describe variation in a way that is useful for the computer simulation that we will construct later. We will show how to describe variation in this subsection using a probability distribution.

Perhaps the simplest scenario that can be described by a probability distribution is the flipping of a penny. If we flip a penny, we are interested in whether the penny comes up heads or tails. Flipping the penny is an example of an event. In addition, flipping a penny is an example of an uncertain event because we cannot perfectly predict whether the flip will result in a heads or tails until we actually flip the penny. We call the two possible results from flipping the penny, heads and tails, the two possible outcomes. Once we flip the coin, whichever possible outcome we observe, heads or tails, is called the outcome of the event. Because we do not know whether heads or tails will come up, we need to describe the outcome of this event beforehand, considering that either outcome is possible.

If having the penny come up heads or tails are equally likely outcomes and, therefore, if we flipped the penny a large number of times, then we would expect that the penny would come up heads 50% of the time and tails 50% of the time. We could draw a column chart to reflect these two percentages for heads and tails, as shown in Figure below. This graph is a probability distribution. It shows all the possible outcomes for the uncertain event on the x axis (heads, tails). The heights of the columns are measured using the scale on the y axis (the vertical axis). The height of those columns shows the percentage of times we would expect heads or tails to occur if we flipped the penny a large number of times. These percentages are also called probabilities. Both heads and tails have probabilities of 50%. Probabilities not only explain the percentage of times that we would expect a heads or tails to come up, but they reflect the likelihood that either heads or tails will come up in the next flip. Because the probabilities for heads and tails are both the same, the result of the next flip is equally likely to be either one of those possible outcomes.

Probability Distribution for a Coin Flip

We can say that the outcome of flipping a coin varies because sometimes heads is observed and sometimes tails is observed, as shown in Figure below. Also note in Figure below that there is no recognizable pattern. For example, just because the last flip was heads doesn’t mean that the next flip will be a tails. Each time we flip the coin, either outcome is possible and, in fact, is equally likely. This uncertainty in what will happen next is what causes the variation in outcomes from flip to flip, and, so, uncertainty and variability are related and actually reflect the same idea.

A Sequence of Coin Flip Outcomes

We said that customer demand varies from hour to hour, and so we can describe it as being uncertain just as we have described the coin flip: we may know all the possibilities for the number of customers who will arrive at a business over the course of an hour, but we can never predict beforehand how many will actually show up and always be correct. Because flipping a coin and the number of customers arriving in a particular hour are both variable and uncertain, we can draw a probability distribution for customer demand just as we drew a probability distribution for flipping a coin. Figure below shows one such probability distribution. In this case, anywhere from 0 to 10 customers can arrive. We know this because these are the values that are shown on the x axis (horizontal axis). So, there were two possible outcomes in the case of a coin flip, but there are 11 possible outcomes for the number of customers in this case. The height of each bar or column, as we have discussed, represents the percentage of time that each number of customers would be observed if we collected data for many hours. The most likely number of customers is 4, and 30% of the time 4 customers will arrive. The next most likely outcome is to have 5 customers in an hour. If we observed this type of customer demand for 20 hours, we might record the data as shown in Figure below. Note that the number of customers varies from hour to hour. In contrast, the customer arrival data in Figure below do not vary: it is the same each hour. In business, we expect a pattern that looks like the one in Figure below. The pattern in Figure below is rare, unless somehow the customer arrivals were carefully controlled and the customers were extremely reliable.

Probability for the Number of Customers in an Hour

Possible Number of Customer Arrivals for a 20-Hour Period

Customer Arrivals that Do Not Vary

How Variation Makes Decisions Difficult

A common way to make business decisions about capacity level is based on the average number of customers that arrive each minute (the average demand rate) and the average number of customers that a sales associate can serve per minute (average service rate). Managers’ intuitions often lead them to think that if enough capacity is put in place so that the average rate at which customers are served (in customers per minute) equals the average rate at which customers arrive (in customers per minute), then that is a sufficient amount of capacity. In the fast food restaurant example, the capacity of the operation is determined by many factors, among them how fast the kitchen can prepare food and how fast the sales associates can serve customers at the sales counter. For this discussion, we will ignore the kitchen operations and assume that they can keep up with the sales associates who take orders from customers. In order to make the illustration clear, we will also ignore the decision of how much inventory of pre-made food to have on hand. In this simplified view, sales associates are the only factor that determines capacity, thus capacity is determined in the example only by the number of sales associates and how fast they work. Therefore, while we have discussed in the previous lesson how both capacity and inventory affect service level, here we will only consider the effect of capacity.

If customers arrive at a fast food restaurant, let’s call it Fish-Fil-A, during lunch time at a rate of 2.5 customers per minute on average, and sales associates can serve 2.6 customers per minute on average, then the intuitive approach to setting a capacity level as described above would suggest that one sales associate is sufficient. The rate at which one sales associate works is roughly equal with the arrival rate of customers and is even a bit faster. Intuition might also suggest that with customers being served faster than they arrive, there should be a negligible queue and little if any waiting time. In addition, if customers are served at a rate of 2.6 customers per minute, then each customer will wait fewer than 0.4 minutes to get his or her food. (This is calculated by taking the reciprocal of 2.5 customers per minute, which is 0.4 minutes per customer.) Remember, however, that the arrival rate of the customers (2.5 customers per minute) is only an average. The number of customers that arrive each minute can vary minute to minute. Similarly, the number of customers that the sales associate serves averages 2.6 per minute, but it varies minute to minute for the reasons mentioned above. It turns out that this variation has a remarkable effect on how well customers are served.

Assume that we want to measure customer service based on our observations of the average amount of time until customers are served and how the length of the line changes over time (a longer line implies more waiting time and poorer customer service). Obviously, customers will be happier the less time they are required to wait in line to get their food. We will see that the fast food operation can provide very good service to customers on some days, but on other days provides very poor customer service, because the number of customers that arrive at lunch varies day to day, the pattern in which they arrive varies, and the speed of serving the customers varies. In fact, in observing the three-hour lunch period on any day, even a good one, we would see different, and much worse, customer service than the logical expectations that we described above.

Let us first observe customer service for a day that is neither one of the best days for customers nor one of the worst. We will call this a typical day. The results are shown in Figure below, which contains a graph that shows the number of people waiting for food at each point in the 180-minute lunch period. The height of the line reflects the number of people in line as scaled by the vertical axis on the left side of the graph, which is called the y axis. To find how many people are in line at a particular time, locate that time on the horizontal axis at the bottom of the graph (the x axis) and determine the height of the line at that point. The y axis scale shows how many people are implied by the height of the line. Another graph shows the amount of time it takes to serve each person, that is, from the time each customer gets in line until the time he or she receives the food. This is the sum of the time a customer waits in line plus how long it takes for that customer to be served once the customer has made it to the counter. Again, the height of the line, as measured by the y axis, shows how long each customer waits.

Some statistics are also reported in Figure below, which are computed over the 180-minute lunch hour. The quality of customer service in this example can be described by the following statistics: the average time for a customer to get food is 4.31 minutes, and the average number of people in line is 11.34 people. Some days, customers are served more quickly than this, and some days customers wait much longer, as we will see. We can see a more detailed view of customer service in the graphs. Specifically, we can see that there were 28 customers waiting at one point in time and some customers waited for longer than 10 minutes, although the average was about 4 minutes. Note that at the end of the lunch period, there were still seven customers waiting to be served.

A Day with Average Customer Service

It is also interesting to see the rate at which customers arrive and the rate at which they are served. In this example, customers arrived at an average rate of 2.45 customers per minute over the lunch hour, which is a little slower than average. The average number of customers served per minutes by the sales associate is 2.41 customers per minute, which is below their average service rate. This average, however, was computed by dividing the number of customers served by the number of minutes in the lunch hour. We note that the sales associate was idle for 8.65 minutes because there were no customers present to be served. This is not the sales associate’s fault: an associate can’t serve customers if there are none. If we adjust the sales associate’s service rate by dividing only by the time the associate is able to work, we see that the rate is faster: 2.53 customers per minute.

Why is the sales associate idle? Because there is variation in how customers arrive. Sometimes many customers arrive all at once, while at other times, fewer arrive. If too few customers arrive, the sales associate can serve all the customers in line before other customers arrive. This causes the sales associate to be idle. The time the associate is idle causes a problem because that is time that has been lost forever. Once the crowd arrives, the associate can’t travel back in time to use those minutes to get more work done. So, besides a "clump" of customers arriving at one time and causing a line to form, variation also causes the sales associate to waste time, which reduces the number of customers served in a lunch period.

The next example describes a day that is better for the customers. Graphs and statistics for this day are shown in Figure below. Customers wait 2.85 minutes on average to get their food, and the average number of customers in line is 7.94. At the end of the lunch period only one customer had not been served. The graphs show that the maximum length of the line is 24 customers and the longest that any customer had to wait was a little over 7 minutes.

A Day with Good Customer Service

Why did customers get much better service on this day? Customers arrived at about the average rate we would expect, 2.51 customers per minute. However, while the average number of customers served per minute over the whole lunch period was about the same as the customer arrival rate, 2.50 customers per minute, the sales associate processed customers very quickly, at a rate of more than 3 customers per minute. So this is one case when variation from the average rate was good: the workers were able to work much faster than they usually do. Perhaps this is because the food orders for this day were smaller, easier to prepare, or there were fewer special orders.

The next example describes a day that is bad for the customers. Graphs and statistics for this day are shown in Figure below. Customers wait 13.95 minutes on average to get their food, and the average number of customers in line is 34.30. In fact, we can see from the graph that the number of customers waiting in line grew steadily over the lunch period, and 56 customers are still waiting to be served at the end of the lunch hour. A manager might call this day a disaster, and it will take about 22 more minutes past the end of lunch to take care of the remaining customers. You might imagine that the customers would be very unhappy with this restaurant on this day. Perhaps they would decide never to come back based on this experience, especially if they were late getting back to work because they were waiting in line.

Why was customer service so poor on this day? Like the previous examples, there was variation but, unlike the previous examples, the variation occurred in a way that hurt customer service. In particular, the customer arrival rate was higher than average, 2.68 customers per minute, and the sales associate processed customers more slowly than average, at a rate of 2.39 customers per minute over the entire lunch period (excluding idle time). Similarly, if we were to put a hose into a swimming pool and run water into the pool at a rate greater than the rate at which water drained out of the pool, then water would accumulate in the pool, just as customers accumulated in line in this scenario.

A Day with Poor Customer Service

A manager might not be able to foresee such variability in customer service and the possibility for such poor customer service, because customer arrivals and service rates vary day to day and minute to minute. Over time, a manager should observe his operation, see that there is the possibility of very bad days, and get a feel for how often they occur. However, customers’ impressions of the restaurant might be permanently hurt by the poor customer service that occurred while the manager was getting a feel for the customer service that was delivered, and business might drop off before the manager decided that he should consider increasing the number of sales associates.

So, wouldn’t it be advantageous if the manager could learn about the effects of staffing (capacity) on customer service before customers’ impressions were permanently affected? How can a manager make better decisions about staffing the sales counter without experiencing the poor customer service that intuitive policies cause? Computer simulation is one very good approach to such problems, and this is a topic that is addressed later in this chapter. If computer simulation was used, then the manager could know ahead of time what customer service would be with one or two sales associates at the counter and could make a better decision right from the beginning without needing to experiment on real customers.

Situations with variation, like the fast food restaurant, are difficult to analyze, and interesting, because the results cannot be predicted exactly in advance and they vary day-to-day. Customers receive better service some days and poor service on other days, and all this depends on how many customers actually show up for lunch, whether they arrive steadily or in clumps, and the consistency with which the sales associate serves customers.

Lesson Summary

  • Something varies, or exhibits variation, when it is not always the same value. The rate of customer demand and the rate of customer service are variable from minute to minute and day to day.
  • Variation may be described using a probability distribution, which is a list of all the possible outcomes and the probability of each outcome for a given event.
  • Variation in demand rates and service rates affect the quality and speed of customer service, which in turn influences an operation's profitability.
  • Variation in demand rates and service rates cannot always, if ever, be predicted intuitively by managers.

Review Questions

  1. Which of the following four sequences of numbers exhibit variation, and which do not? Assume that these numbers are the number of customers that appear at a drug store checkout counter in successive hours.
  1. What are some of the reasons why the sales associate in the subsection entitled Variation Makes Decisions Difficult sometimes averaged serving fewer than 2.6 customers per minute and sometimes more than 2.6 customers per minute?
  2. What caused the fast food sales associate to be idle in the example in the section entitled Variation Makes Decisions Difficult.

Points to Consider

  • How can the effect of variation in customer demand and service times on customer service be accurately measured?
  • How exactly does variation in arrival times and service times cause lines to form and service times for customers to be much longer than anticipated?
  • If making business decisions is difficult when capacity and demand rates vary and using intuition results in poor customer service, then how can we make good decisions about how much capacity (how many sales associates) to have and be sure that we give customers prompt service?

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Date Created:
Aug 06, 2012
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Jan 30, 2016
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