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Queueing Theory

Lines are a fact of life. They result from a company trying to keep the costs of providing its services in check in the face of fluctuations in customer demand. With enough resources to handle spikes, companies never make customers wait. Of course profit would drop significantly, because employees would be idle much of the time. On the other hand, if the company staffs for the average demand, they won’t have enough resources to cover the busy periods. During those times, customers end up waiting in long lines.

The expected waiting time of an operation can be calculated with some math-intensive calculations. The answer, though only tells you how long a customer would have to wait in a given situation. It doesn’t necessarily tell you anything about whether or not a customer actually will wait.

What does that mean? Sometimes, the length of a line does little to discourage people from waiting. Imagine a group of people come up to a trendy nightclub with a line around the corner. They may queue up without batting an eye. The same situation occurs at theme parks at the premiere attractions. People wait in line for extended times for the best rides.

But what happens when the payoff for waiting is nothing extraordinary? Imagine you walk into a coffee shop or a fast food restaurant, and the line is out the door. There is a good chance that you will abandon the line, and go somewhere else.

Most businesses are in this position. Long lines mean lost business. But when margins are slim, there is not much wiggle room to add personnel to handle the spikes. Understanding queueing theory becomes exceptionally important in this situation.

This term is not intended to present the math behind queueing theory. The calculations can be quite complicated, and take quite a bit of skill, especially for the more complicated equations. Instead, this term is presented to help you understand how queueing theory can help you improve your operation. Consult an expert in the field for assistance in crunching the numbers.

Factors Affecting Queue Length

There are six major factors that define how a line develops.[1]

  1. The population of potential customers. The size of the potential population of customers has a big impact on lines. A drive-through coffee stand along a remote road has a different customer base than a trendy downtown coffee shop. Keep in mind that these populations are not static. Some demographics shift slowly over time. In other instances, the change may be dramatic. A new factory complex may be built right across the road from a restaurant. Keep in mind that competition doesn’t change the size of the potential population, just your slice of it.
  2. Arrival characteristics. The manner in which customers arrive, both in quantity and in pattern, affect how long they will wait. Think of the difference between a tour bus showing up at a fast food restaurant versus a series of morning commuters. Different days of the week or times of day might have different arrival patterns. Night clubs don’t get many patrons at 10:30 am on a Wednesday. Regardless of the pattern, there is a statistical distribution that can describe the arrival rate. As mentioned earlier, some arrivals will abandon the line. If they depart after seeing how long the line is, but before getting in line, it is called balking. Some will wait in line for a period of time, and then leave, a process known as reneging.
  3. Line layout. The physical setup of the line affects service. Does the line have a fixed capacity? Some years back, during the company’s heyday, a Krispy Kreme donut shop opened near my home. It generated huge lines that went out of the parking lot, and into the street. Eventually, the police had to manage the traffic there. There was not capacity for the line during peak periods. Another factor that affects the mathematics whether there are multiple lines that feed into one area, such as at a ski lift.
  4. Who gets served first? Most lines follow the first come, first serve rule, but there are other methods.  Reservations change the sequence. Triage in an emergency room affects the wait, since more serious problems jump to the front of the line. Express lanes provide a situational advantage. ‘Gold’ programs, such as for rental cars can factor in, if they share the same service people with the rest of the line. (Note: If the ‘Gold’ program had a separate line, it would have independent queueing calculations.)
  5. The operation. The time it takes to process customers, and the process flow affect how people leave the line. Some operations consist of a single process, such as a checkout at a supermarket. Others, like fast food, may make you wait multiple times throughout the service. You may wait to place your order, and then to pick up your food. This is known as the phase in queueing theory. Another factor is the number of channels. This simply means whether the customers flow through a single line, such as the wandering queue at a fast food restaurant, or a multichannel operation as you would see in a supermarket checkout area. Batch processing also affects the line. Think of rides at an amusement park again. Many people jump on the ride at the same time. In general, operations are categorizes as single or multi-channels, meaning routes through the process, and single or multi-phase, meaning steps in the process. In some cases, though, the flow is of a mixed type[2], where, say a single phase, single channel line flows into a multichannel, single phase line. Finally, the overall speed of service defines the service rate—how many people can be served per time period.
  6. Disposition of finished customer. Does the customer go back to the population? That theme park ride is an example of this. The customer goes back out into the big park, and may eventually jump back into line. Flu vaccines, on the other hand, make the customer leave the population for a year.

Queueing theory is most relevant in service operations. Customers, in most cases, dislike waiting (also called queueing time). The length of time they will wait is directly related to the value they place in the service they are waiting for. The greater the perceived value, the more likely a person is to stand in line.

Managers who run operations that have customer queues (including queues of customer surrogates, like orders or service emails) should get a rudimentary understanding of queueing theory at a minimum. Developing a stronger understanding, though, tends to lead to more effective decisions about how to manage waiting time for customers. Managers who are not math wizard should find someone who is that can crunch the numbers for them. Having that working knowledge, though, will at least help them understand what data they need, and will help them know what levers that they can pull to improve their service levels.

Continuous Improvement and Queueing Theory

As you may imagine with all that goes into determining how lines affect customers, you have a great many opportunities to make improvements. It all comes down to finding the balance between service levels and cost. Continuous improvement efforts reduce the costs to make lines flow more quickly.

CI efforts can affect operations with queues in a number of ways.

  1. Improve data collection. Data collection is a process that takes time and effort. Streamline it, and make sure you are getting the data you need. It is worth a kaizen to improve this necessary, but non-value added process.
  2. Determine service levels needed. It is not just a random guess. Companies need to know where they should be to meet their goals. Just grabbing an industry standard might not be right for you.
  3. Split your time. This means that all time periods are not created equally. Airports near port cities get demand surges when cruise ships come in. Lunch periods spike demand for fast food places. The first hour a call center is opened might have a surge in demand. Break up your day, and manage the individual time buckets.
  4. Manage staffing. Staffing should be balanced to the need. With good data, you can come up with a flexible staffing plan. This means scheduling breaks away from peak demand times, hiring part-timers to bolster peak demand periods, and flexing people in and out of the team as demand dictates.
  5. Speeding up service. Find ways to serve customers better and quicker. Manage and share best practices among the team. Automate steps, if possible.
  6. Eliminate misrouted customers. Don’t let customers get misrouted—it wastes capacity and it irritates the customers.
  7. Reduce the need for service. In some cases, you want lots of customers, like at a restaurant or ordering line. In others, such as in the ‘Complaint Department’, you don’t. Figure out what the complaints are, and eliminate them. This entails a lot of cooperation throughout the company.
  8. Fill the voids. This is most applicable in call centers. Front-load calls (least operator concept) to make bigger time chunks available for the service people at the back of the phone queue. Then give them something else to do while they are waiting to help the next customer. It lets you keep those people productive, while still keeping high service levels. Nothing says that a technical support person cannot also enter customer orders during their down time.
  9. Manage the queue time. When customers are in line, give them something to do to make them feel that their time is being used. Disney does a good job of this, putting displays up along the winding lines. Many hotels also put mirrors on the wall by their elevators. It makes customers feel like the wait is shorter, because they have something to do during the brief wait for the elevator doors to open. Calming music also makes waits feel shorter. An even better option is if the activity while waiting can help productivity. Posting a menu on the wall lets customers decide what they want to eat while waiting in line.
  10. Avoid the line. Find ways to give customers an alternative to waiting in line. Send them to a website, for example, if that is appropriate in your situation. Self-checkouts and kiosks are becoming ever more popular alternatives to lines. Be wary, though, that some people want the sound of a human voice to solve their problem.
  11. Outsource. Be wary of this option. There are lots of financial benefits from outsourcing, but most people view this type of customer service as inferior to in-house service. Make sure you full understand the needs of your business before doing this.
  12. Inform the customer. Customers like to know how long they have to wait. They feel better knowing that there will be a ten-minute wait than if they are in the dark.
  13. Inform your team. Employees need the same information that customers do. If they know how long the queue is, or how long customers have been waiting, they can adjust their process, call people back off break, or get help.
  14. Alter demand. There is more control to demand than people realize. Cloudy, dark days on golf courses likely brings attendance way down. Discounts on those days can bring it back up. Happy hour specials are designed to bring people into restaurants early or to get them to sit in the bar area.
  15. Use a phone tree. This can help route calls properly, and give customers some information. Phone trees (automated answering systems) have become more mainstream, and have less of the negative emotion surrounding them than in the early days of their adoption. They still lag having a live person answer a call, though. If you choose to use one, make sure of two things. They must correctly route a person, and the messages should be short. Every additional layer is a chance for a customer to abandon the tree and shop elsewhere.
  16. Use a queue calculator. Many queueing theory tools are available, either free or for a price. They can help you with many of the steps listed above—crunching your data, finding appropriate staffing levels, etc. They do the math for you. All you need to do is punch in the numbers and interpret and act on the results.

[1] Fundamentals of Operations Management. Davis, Aquilano, & Chase, 1999, p. 291

[2] Queueing theory can get complicated in a hurry.  A single line might feed three service reps at a medical facility, which each feed one of several doctors with their own wait.  The doctors share an X-ray machine, which then feeds back to the same doctor, and possible goes back to the X-ray, after waiting for a casting room.  The patient then waits in a pharmacy line.  Lots of math involved in figuring this one out!

  • The spelling of this theory can be confusing. You will see it as both “queueing” and “queuing”. Both are listed in the dictionary, and are correct spellings. The former, and the one used throughout this term, appear to be in more common use. Note, though, that the spell-check function in Microsoft Office prefers the latter.
  • The math behind queueing theory can be daunting. Because the output is used to make important, expensive decisions, make sure the answers are correct. Consult an expert if you are not confident in our skills as a mathematician.

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