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Queuing Theory

Last updated date: 04th Mar 2024
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What is Queuing Theory?

The study of all the varied dynamics of lines or queues, as well as how they might be altered to run more efficiently, is the focus of queueing theory. Queuing theory is an area of mathematics that analyses and models the act of queueing. It is essentially the study of how people act when they have to wait in line to make a purchase or receive a service, as well as what sorts of queue structure move people through lines the most efficiently, and how many people can a specific queuing arrangement process through the line in a particular time frame.

The queuing models have two aspects at its core.

  • The customer, job, or request are all terms used to describe someone or something who demands a service.

  • The server refers to the person or thing that completes or provides the services.

Let's look at queuing theory in operation research examples. 

Consumers trying to deposit or withdraw money are the customers, and bank tellers are the servers in a bank queuing situation. The customers in a printer's queue scenario are the requests that have been made to the printer, and the server is the printer.

Queuing theory in operation research examines the entire system of standing in line, including factors such as customer arrival rate, number of servers, number of customers, waiting room capacity, average service completion time, and queuing discipline. The rules of the queue, such as whether it operates on a first-in-first-out, last-in-first-out, prioritised, or serve-in-random-order basis, are referred to as queuing discipline.

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History of Queuing Theory

  • Agner Krarup Erlang, a Danish mathematician and engineer, was the first to establish queueing theory in the early twentieth century.

  • Erlang was employed by the Copenhagen Telephone Exchange, and he intended to examine and improve the company's processes. He wanted to know how many circuits were required to offer an acceptable level of telephone service, with people not having to wait too long on hold or in a phone queue. He also wanted to know how many telephone operators were required to handle a certain volume of calls.

  • His mathematical research culminated in the paper Telephone Waiting Times, published in 1920, which became the foundation for the applied queuing theory. In his honour, the Erlang is the international telephone traffic unit.

Basics of Probability and Queuing theory

Queuing theory is primarily a tool for calculating costs. Most firms would find it excessively expensive, or symptomatic of a lack of customers, to run in such a way that none of their customers or clients ever had to wait in line.

To give a simple example, a movie theatre would need to add fifty to one hundred ticket booths to avoid the situation of people needing to wait in line to purchase a movie ticket. The theatre, on the other hand, could clearly not afford to compensate a hundred ticket salespeople.

As a result, businesses employ queuing theory information to set up their operational operations in order to strike a balance between the cost of supplying clients and the difficulty caused by having to wait in line.

The individuals in line and the performance of the service they are waiting for are the fundamentals of queuing models.

Queuing theory problems are commonly divided into four groups in studies on the subject:

  • Arrival: The procedure for getting customers to the front of the line or a queue.

  • Queue: That is, the character or operation of the queue itself. How does the line advance?

  • Service: The process of providing a customer with the service they have requested. For example, when being seated and then served in a restaurant, the restaurant must consider the dynamics of two independent queues: the line of people waiting to be seated and the line of people who have already been seated and are waiting to be served. The latter can be divided into two lines: the line to have your order taken and the line to have your food delivered to your table.

  • Leaving: The act of departing from a queue position. Businesses that provide a drive-through service, for example, must consider how customers exiting the drive-through may affect customers entering the parking lot.

Now let us discuss in details the four groups of queuing theory in operation research.


  • The average number of individuals who arrive during a specific time frame, such as one hour, is one factor to consider when it comes to the arrival of individuals at the queuing location.

  • Significant variations in the volume of traffic or arrivals that occur at different times of the day, or on different days of the week or month, are a related factor.

  • For example, grocery stores know that on Friday mornings between 10 a.m. and noon, they require more personnel than on Wednesday mornings between 10 a.m. and noon to avoid queues getting backed up.


  • How does the line progress? Is it better for a bank to have a single line of customers waiting for the next available teller or cashier, or is it better to have distinct queues for each teller? When presenting such a question, human behaviour characteristics become an integral aspect of queuing theory. 

  • While feeding one line of clients to four different teller stations rather than four separate lines at each teller station may not have a substantial impact on the speed or efficiency with which customers are served, it may have a major impact on customer satisfaction.

  • While ultimately, regardless of the line arrangement, the wait time to be served can be approximately the same, customers may feel or perceive that they are served more quickly if only two or three people waiting in line behind each distributor station having their own queue rather than having to stand behind 10 or 12 individuals where one customer line is fed into all four teller stations.

  • There are also some basic considerations to make: Will using only one line result in a line that stretches all the way out the door if the business office is small? Many individuals may be put off from conducting business there if they witness a situation like that. They might instead go to a competitor who appears to have a shorter wait time.

  • In reality, doing business with a competitor may entail roughly the same amount of time spent in line. The only difference could be that the competition picked distinct lines for each service station rather than a single line for all of them, avoiding a wait that stretches all the way out the door. You can see that, in addition to any operational efficiency concerns, queue aesthetics must be considered.


  • Variables exist in regard to the actual providing of service as well. 

  • The express lane in grocery shops, which is allocated for people buying a modest number of items, is a good example. Customers satisfaction is improved by allowing customers who are only buying a few items to check out more quickly, rather than having to wait in line behind other customers with full carts of food, according to grocery businesses that use queuing theory.

  • The average time it takes to serve each customer or client, the number of servers required for maximum operational and cost-effectiveness, and the rules dictating the order in which clients are served are all elements that have an impact on actually providing service. While most queues work on a first-come, first-served basis, this isn't always the best option for some firms.

  • A classic example is the emergency room waiting area at a hospital. Patients are served according to the severity of their illness or damage, rather than on a first-come, first-served basis. It demands the addition of a service step called triage, in which a nurse assesses each patient's emergency severity to determine where that patient should be placed in the service line.


  • Basic logistical issues are frequently related to clients leaving a queue area.

  • Businesses having drive-through operations, for example, must consider how customers exiting the drive-through may affect incoming traffic to the facility, as mentioned above.

  • Another departure-related factor is a restaurant's decision on whether to have servers present bills and collect payment at a customer's table or have customers pay their bill to a cashier as they leave.

Importance of Queuing Theory

  • Waiting in line is a common occurrence in everyday life because it serves various key functions as a process. When there are limited resources, queues are a fair and necessary manner of dealing with the flow of clients. If there isn't a queuing process in place to deal with overcapacity, bad things happen.

  • For example, if a website has too many visitors, it will slow down and fail if it does not include a mechanism to adjust the speed at which requests are processed or a mechanism to queue visitors. Consider planes waiting to land on a runway. When there are too many planes to land at once, the lack of a queue has actual safety issues when jets try to land at the same time.

  • Queuing theory is significant because it helps to describe queue characteristics such as average wait time and gives tools for queue optimization. 

  • Queuing theory influences the design of efficient and cost-effective workflow systems from a commercial standpoint.

Applications of Queuing Theory

Queuing theory finds its application in various sectors. Few are listed below:

  • Telecommunications

  • Transportation

  • Logistics

  • Finance

  • Emergency services

  • Computing

  • Industrial engineering

  • Project management

Little’s Law

Little's law developed by John Little asserts that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system in queueing theory. Little’s law is a topic within the mathematical probability and queuing theory.

L = λ W


L is the average number of customers in the system.

λ is the average arrival rate into the system.

W is the average amount of time spent in the system.

Although it appears to be a simple result, it is rather surprising because the link is unaffected by the arrival process distribution, service distribution, service order, or virtually everything else.

Little's law holds true for every system, but it is especially true for systems inside systems. So, in a bank, the client queue could be one subsystem, and each of the tellers could be another, and Little's result could be applied to both. The sole requirements are for the system to be stable and non-preemptive; this eliminates transition states like initial setup and shutdown.

In some circumstances, not only can the average number in the system be mathematically related to the average wait, but the complete probability distribution of the number in the system can also be mathematically related to the wait.

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Queuing models can assist explain the math behind how queues work. When it comes to human lineups, however, queue psychology is crucial to comprehend the experience. According to queue psychology studies, it's not how long individuals wait that affects whether they have a great or negative line experience, but how they feel while waiting. Little's Law is useful because it allows us to solve important variables such as the average wait time in a queue or the number of customers in a queue using only two other inputs.

FAQs on Queuing Theory

1. What is Queuing Theory?

Ans: The mathematical study of waiting lines, or queues, is known as queueing theory. A queueing model is created in order to anticipate queue lengths and waiting times. Because the results are frequently employed for making business choices regarding the resources needed to offer a service, queuing theory is widely considered a branch of operations research.

2. What are the Four Groups of Queuing Theory?

Ans: The four groups of queuing models are:

  • Arrival

  • Queue

  • Service 

  • Leaving

3. What is Little’s Law and Its Importance in Probability And Queuing Theory?

Ans: Without knowing any other properties of the line, Little's Law relates the capacity of a queuing system, the average time spent in the system, and the average arrival rate into the system. Without Little's Law, project management techniques like Lean and Kanban would not exist. Little's Law is useful because it allows us to solve for crucial variables such as the average wait time in a queue or the number of consumers in a queue using only two additional inputs.