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Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the system of Copenhagen Telephone Exchange company, a Danish company. The ideas have since seen applications including telecommunication, traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management.

Virtual queuing is a concept used in inbound call centers. Call centers use an Automatic Call Distributor (ACD) to distribute incoming calls to specific resources (agents) in the center. ACDs hold queued calls in First In, First Out order until agents become available. From the caller’s perspective, without virtual queuing they have only two choices: wait until an agent resource becomes available, or abandon (hang up) and try again later. From the call center’s perspective, a long queue results in many abandoned calls, repeat attempts, and customer dissatisfaction.

Virtual queuing systems allow customers to receive callbacks instead of waiting in an ACD queue. This solution is analogous to the “fast lane” option (e.g. Disney's FASTPASS) used at amusement parks, which often have long queues to ride the various coasters and attractions. A computerized system allows park visitors to secure their place in a “virtual queue” rather than waiting in a physical queue.

In the brick-and-mortar retail and business world, virtual queuing for large organizations similar to the FASTPASS and Six Flags' Flash Pass, have been in use successfully since 1999 and 2001 respectively. For small businesses, the virtual queue management solutions come in two types: (a) based on SMS text notification and (b) apps on smartphones and tablet devices, with in-app notification and remote queue status views.

The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively.

The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The name "knapsack problem" dates back to the early works of mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage.