Queueing theory

Queueing theory is the study of waiting lines, or queues. It helps us understand how long people might wait and how to make lines more efficient. By creating models, scientists can predict how long a line will be and how much time people will spend waiting.

This idea started with a man named Agner Krarup Erlang, who worked for a telephone company. He wanted to figure out how to handle calls, so he made models to describe how calls flow. His work started a field called teletraffic engineering.

Today, queueing theory is used in many places, like telecommunications, traffic planning, computing, project management, and industrial engineering. It helps design better systems in factories, shops, offices, and hospitals, making sure services run smoothly and people don’t wait too long.

Description

Queueing theory is a part of management science. It helps businesses solve problems with math. It looks at waiting lines, or queues, to guess how long people might wait. By using chances — or probabilities — instead of exact times, it can find average wait times, how many people might be waiting, and if servers (like cashiers) are busy or free.

The main goal is to compare the current system with possible changes. This helps find the best way to save money, cut down waiting time, and work better. Two common models are single-server systems (one person serving) and multiple-server systems (several people serving). They can change based on things like service time length and the number of people waiting.

Single queueing nodes

A queue or queueing node is like a black box. Jobs, also called customers or requests, come to the queue. They might wait a little, get processed, and then leave.

The queue has one or more servers that handle the jobs. For example, at a supermarket, customers arrive and are helped by a cashier. The cashier can only help one customer at a time. If the cashier is busy, customers might wait in line, or they might leave if there is no place to wait.

Queueing networks

Queue networks are systems with several lines of waiting, called queues, connected together. After a person or item is helped at one spot, they might move to another spot to wait again, or they might leave the whole system.

These networks can be described using numbers. For a network with m spots, we can use an m–dimensional vector (x₁, x₂, ..., x_m) where each x_i shows how many people or items are waiting at each spot. Some special kinds of queue networks, like tandem queues and Jackson networks, help us understand how these systems work on average.

Queueing theory is used in many places, like computers and networks, to help make things run better. It helps us understand how to make waiting lines shorter and systems faster, whether it's at a supermarket, on the internet, or in many other places.