page_title: Queueing theory
section_title: Single queueing nodes
hierarchical_section_title: Queueing theory / Single queueing nodes
caption_attribution_description: English: a simple diagram showing the behaviour of a queueing node, with arrivals, servers in various states, and a departure
context_page_description: 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 Copenhagen telephone exchange. 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.
context_section_description: Simple description and analogy
  A queue, or queueing node can be thought of as nearly a black box. Jobs or "customers" arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue (see Fig. 1).
  The queueing node is not quite a pure black box, however, since there is some information we need to specify about the inside of the queuing node. The queue has one or more "servers" which can each be paired with an arriving job until it departs, after which that server will be free to be paired with another arriving job (see Fig. 2).
  An analogy often used is that of the cashier at a supermarket. There are other models, but this is one commonly encountered in the literature. Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting where a customer will leave immediately if the cashier is busy when the customer arrives, is referred to as a queue with no buffer (or no "waiting area", or similar terms). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n.
caption: Fig. 2. A queueing node with 3 servers. Server a is idle, and thus an arrival is given to it to process. Server b is currently busy and will take some time before it can complete service of its job. Server c has just completed service of a job and thus will be next to receive an arriving job.
exif: {}