Queueing theory

The queuing theory ( or operation theory ) is a branch of probability theory and operations research and thus an example of applied mathematics. It deals with the mathematical analysis of systems in which jobs are processed by service stations, and answers the questions of the characteristic quantities such as the stability of the waiting system, the number of customers in the system, their waiting time, etc., it supports management decisions through the deployment of personnel and the handling process and helps a performance measurement system to expand. Their application ranges from computers, telecommunications systems, transportation systems to logistics to manufacturing systems.

System

Basically, there is a queuing system of an operating range in which to edit one or more service units orders, and a waiting room, waiting in the incoming orders at currently not free, but available service units to the operation. Processed orders leave the system.

A queuing system will be described with six parameters (here in order of the Kendall notation):

Using these assumptions, the queuing theory provides statements about performance parameters such as the mean queue length, the number of customers in the queuing system, the average waiting time or something similar. By David George Kendall a uniform notation has been developed to describe the response systems, the Kendall notation. Queuing Systems without waiting area are referred to as loss systems. Key messages are the law by Little, Erlang B and Erlang C as well as the set of Gordon Newell.

Applications

The queuing theory is employed in the analysis of computers, telecommunications systems (call centers ), transport systems ( traffic flow ), logistics, and manufacturing systems. Depending on the application the abstract concepts of order and service station have very different meanings.

Several of these (simple ) response systems can be assembled to form the so-called queuing networks. For the mathematical analysis of response systems, various approaches have been developed. These include Markov chains, Petri nets, and discrete-event simulation.

History

The first application of queuing theory was carried out by the mathematician Agner Krarup Erlang in 1909 for the dimensioning of telephone switching systems ( The Theory of Probabilities and Telephone Conversations ). In the 1930s, the Pollaczek - Khinchin formula allowed further simplification of the theory. Later, significant contributions came from David George Kendall, Dennis Victor Lindley, James R. Jackson, Gordon, Gordon F. Newell, Felix Pollaczek, Carl Adam Petri, Leonard Kleinrock and Paul Ehrenfest. With the development of computers and computer networks, the research gained in this area also important.