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# Queuing Theory

Updated on April 16, 2024
Article byWallstreetmojo Team
Edited byRaisa Ali
Reviewed byDheeraj Vaidya, CFA, FRM

## What is Queuing Theory?

Queuing theory refers to the study comprising a queue’s features, functions, and imperfections. This mathematical study is very relevant in operations research since its appropriate application helps in eliminating operational bottlenecks and service failures.

The concept was introduced by Danish mathematician Agner Krarup Erlang. Generally, a queue is associated with limited resource delivery, and modeling such processes requires queuing theory. Its application enables the managers to explore the optimal supply of finite resources necessary to meet consumer demands in various instances.

### Key Takeaways

• Queuing theory is primarily the analysis of various aspects of a queue or waiting line.
• Its analysis helps the businesses handle a queue more productively without hurting the customers’ interest, optimizing cost and customer satisfaction.
• The theory involves multiple factors. It starts with the customer or entity entry, then their movement, services rendered to them, and finally, the expression about the service.
• It helps in developing various queuing models. Examples of its application include a virtual queue management system.

### How does Queuing Theory work?

Queuing theory in operations research contributes to designing an efficient queuing system for a business. The theory guides the professionals to systematically explore the finest method and arrange the setup. It gives primary importance to balancing efficient service and the system’s economic viability. An efficient queuing system in place enhances customer service and competitive advantage.

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The queuing system requires incessant improvisations to keep pace with technological advancement and innovative business practices. Over the years, while designing a queuing system, the following basic factors are integral.

• Arrival: The process starts with the arrival of a single individual or a group of individuals. They may come in different intervals, and it may influence the operations. Check on the formation of the queue and note down any variation in the arrival. Track every aspect of the process at this stage.
• Movement: This part mainly focuses on the movement of the queue and the individual’s behavior. It’s like monitoring their activities and looking at whether the customer is impatient or is habituated with the situation. Take feedback and see how they react to it. Please note down where they want any changes. Many times, it is observed huge gatherings in a small place tend to develop negative attributes towards a business. In such a case, the customer might choose a different option. Adapt necessary arrangements or alternative procedures to keep the customer and increase efficiency.
• Service: It is one of the vital parts of the process. If more time is taken to solve the query, it will increase the line. In addition, it may cause boredom and frustration in customers. A better understanding and application of the theory is important in reducing the negative impacts of the long waiting line and long response time.
• Expression: It is the final step of the process. It is important to note that the person leaving the queue makes an impression on the people standing next to him. An individual’s negative feedback is bound to affect the business. Therefore, preference should be given to every person and worked with full diligence. An ideal expression speaks a lot of the services offered to him.

Let’s look into the basic queuing theory formula for a queuing system explained by Little’s Law.

L= λ*W

Or

Number of items in the queue = Arrival rate × Average time spent in the queue

• L:  Average number of items or customers in the system,
• λ: Average arrival rate,
• W: Average time an item spends in the system

Another formula based on the queuing system model by Erlang derived from Little’s Law is the following:

L = (λ – σ )/ μ

• L:  Average number of items or customers in the system (Length of the queue)
• λ: Arrival rate
• σ: Dropout rate
• μ: Departure rate

### Queuing Theory Example

Let’s look into a queuing theory example:

Mr. A went to a food joint and wanted to grab a tasty snack. So, he stands in a line and waits for his turn to order the food. After ordering the food, he has to wait to receive it, as they require a little more time to prepare it. Sometimes, a long queue and a further wait for the food might negatively affect the customers.

In this restaurant formation of a queue is common, and customers have to wait for the service. So, by applying proper queuing theory, the restaurant can create an optimal solution. Based on the theory’s analysis, the restaurant operation model increases efficiency and reduces cycle time.

Some of the analyses that can be derived using the theory in this scenario include the expected waiting time in the queue, the average time in the system, average service time, average waiting time at the cash counter, the expected queue length, the expected number of customers served at one time, as well as the probability of the system to be in certain states, such as empty or full.

### Application of Queuing Theory

The application of queuing theory is not inherent to any specific sector. It is predominantly applied in industries like retail, logistics, and hospitality. The relevance hit its peak during the Covid 19 pandemic period. Dealing with long lines and resolving issues during the emergence of the COVID-19 pandemic is very hectic. Businesses have introduced different queue management systems keeping public safety as a priority. Its use is evident in the following cases where changes are made, keeping in mind the customers’ safety.

There is software on the market that promotes virtual waiting. A virtual queue management system organizes clients in a virtual waiting line or queue not visibly waiting in line to receive a product or service. Customers can wait virtually using a virtual queue management system since they are not bound to a specific waiting area.

Queueing theory helps in the process of hassle-free checkout at the store. Many retail outlets have come with new rules and no customer contact service. As a result, retail stores have made slots so that a mass gathering can be waived off. The serpentine line with multiple cashiers or registers is one of the efficient queuing systems used in fast food places since the occurrence of an impediment does not stop the entire line; only one cashier must tackle that perplexing issue while the others plow through the remaining effort.

What is the queue theory example?

There are many instances involving queues that require the application of queue theory to streamline the process by removing the events causing inefficiencies. Examples of its applications are a virtual queue management system enabling virtual waiting lines and a serpentine line with multiple cashiers.

What is queuing theory in Operations Research?

This mathematical study has immense relevance in Operations Research. Businesses can utilize it to remove operational bottlenecks and enhance product and service delivery. Furthermore, businesses can gain a competitive advantage by continuously improvising the queuing system to keep pace with technological advancements. However, businesses should balance service efficiency and the system’s economic viability.

What is queueing theory problem?

It is the problems or situations involving queuing or waiting and describing a failure situation. Its rectification depends on creating an efficient queuing system model based on the theories analysis.

This has been a guide to What is Queuing Theory & its Definition. Here we explain how Queuing Theory Works in Operations Research along with examples and applications. You may also have a look at the following articles to learn more –