What are the four main characteristics of waiting lines which determine the queuing model?

1. What is queuing theory?

Queuing theory (or queueing theory) refers to the mathematical study of the formation, function, and congestion of waiting lines, or queues.

At its core, a queuing situation involves two parts.

1. Someone or something that requests a service—usually referred to as the customer, job, or request.
2. Someone or something that completes or delivers the services—usually referred to as the server.

To illustrate, let’s take two examples. When looking at the queuing situation at a bank, the customers are people seeking to deposit or withdraw money, and the servers are the bank tellers. When looking at the queuing situation of a printer, the customers are the requests that have been sent to the printer, and the server is the printer.

Queuing theory scrutinizes the entire system of waiting in line, including elements like the customer arrival rate, number of servers, number of customers, capacity of the waiting area, average service completion time, and queuing discipline. Queuing discipline refers to the rules of the queue, for example whether it behaves based on a principle of first-in-first-out, last-in-first-out, prioritized, or serve-in-random-order.

2. How did queuing theory start?

Queuing theory was first introduced in the early 20th century by Danish mathematician and engineer Agner Krarup Erlang.

Erlang worked for the Copenhagen Telephone Exchange and wanted to analyze and optimize its operations. He sought to determine how many circuits were needed to provide an acceptable level of telephone service, for people not to be “on hold” (or in a telephone queue) for too long. He was also curious to find out how many telephone operators were needed to process a given volume of calls.

His mathematical analysis culminated in his 1920 paper “Telephone Waiting Times”, which served as the foundation of applied queuing theory. The international unit of telephone traffic is called the Erlang in his honor.

3. What are the different types of queuing systems?

Queuing theory uses the Kendall notation to classify the different types of queuing systems, or nodes. Queuing nodes are classified using the notation A/S/c/K/N/D where:

• A is the arrival process
• S is the mathematical distribution of the service time
• c is the number of servers
• K is the capacity of the queue, omitted if unlimited
• N is the number of possible customers, omitted if unlimited
• D is the queuing discipline, assumed first-in-first-out if omitted

For example, think of an ATM.

It can serve: one customer at a time; in a first-in-first-out order; with a randomly-distributed arrival process and service distribution time; unlimited queue capacity; and unlimited number of possible customers.

Queuing theory would describe this system as a M/M/1 queue (“M” here stands for Markovian, a statistical process to describe randomness).

Queuing theory calculators out there often require choosing a queuing system from the Kendall notation before calculating inputs.

4. Why is queuing theory important?

Waiting in line is a part of everyday life because as a process it has several important functions. Queues are a fair and essential way of dealing with the flow of customers when there are limited resources. Negative outcomes arise if a queue process isn’t established to deal with overcapacity.

For example, when too many visitors navigate to a website, the website will slow and crash if it doesn’t have a way to change the speed at which it processes requests or a way to queue visitors.

Related: What is an Online Queue System?

Or, imagine planes waiting for a runway to land. When there is an excess of planes, the absence of a queue would have real safety implications as planes all tried to land at the same time.

Queuing theory is important because it helps describe features of the queue, like average wait time, and provides the tools for optimizing queues. From a business sense, queuing theory informs the construction of efficient and cost-effective workflow systems.

5. What are the applications of queuing theory?

Queuing theory is powerful because the ubiquity of queue situations means there are countless and diverse applications of queuing theory.   

Queuing theory has been applied, just to name a few, to:

• telecommunications
• transportation
• logistics
• finance
• emergency services
• computing
• industrial engineering
• project management

Before we look at some specific applications, it’s helpful to understand Little’s Law, a formula that helps to operationalize queuing theory in many of these applications.

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