---
title: "Queueing theory"
chunk: 4/4
source: "https://en.wikipedia.org/wiki/Queueing_theory"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:56:55.531499+00:00"
instance: "kb-cron"
---

=== Mean-field limits ===
Mean-field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues m approaches infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.

=== Heavy traffic/diffusion approximations ===

In a system with high occupancy rates (utilisation near 1), a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion, Ornstein–Uhlenbeck process, or more general diffusion process. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative orthant.

=== Fluid limits ===

Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable but have an unstable fluid limit.

=== Queueing applications ===
Queueing theory finds widespread application in computer science and information technology. In networking, for instance, queues are integral to routers and switches, where packets queue up for transmission. By applying queueing theory principles, designers can optimize these systems, ensuring responsive performance and efficient resource utilization.
Beyond the technological realm, queueing theory is relevant to everyday experiences. Whether waiting in line at a supermarket or for public transportation, understanding the principles of queueing theory provides valuable insights into optimizing these systems for enhanced user satisfaction. At some point, everyone will be involved in an aspect of queuing. What some may view to be an inconvenience could possibly be the most effective method.
Queueing theory, a discipline rooted in applied mathematics and computer science, is a field dedicated to the study and analysis of queues, or waiting lines, and their implications across a diverse range of applications. This theoretical framework has proven instrumental in understanding and optimizing the efficiency of systems characterized by the presence of queues. The study of queues is essential in contexts such as traffic systems, computer networks, telecommunications, and service operations.
Queueing theory delves into various foundational concepts, with the arrival process and service process being central. The arrival process describes the manner in which entities join the queue over time, often modeled using stochastic processes like Poisson processes. The efficiency of queueing systems is gauged through key performance metrics. These include the average queue length, average wait time, and system throughput. These metrics provide insights into the system's functionality, guiding decisions aimed at enhancing performance and reducing wait times.

== See also ==

== References ==

== Further reading ==
Gross, Donald; Carl M. Harris (1998). Fundamentals of Queueing Theory. Wiley. ISBN 978-0-471-32812-4. Online
Zukerman, Moshe (2013). Introduction to Queueing Theory and Stochastic Teletraffic Models (PDF). arXiv:1307.2968.
Deitel, Harvey M. (1984) [1982]. An introduction to operating systems (revisited first ed.). Addison-Wesley. p. 673. ISBN 978-0-201-14502-1. chap.15, pp. 380–412
Gelenbe, Erol; Isi Mitrani (2010). Analysis and Synthesis of Computer Systems. World Scientific 2nd Edition. ISBN 978-1-908978-42-4.
Newell, Gordron F. (1 June 1971). Applications of Queueing Theory. Chapman and Hall.
Leonard Kleinrock, Information Flow in Large Communication Nets, (MIT, Cambridge, May 31, 1961) Proposal for a Ph.D. Thesis
Leonard Kleinrock. Information Flow in Large Communication Nets (RLE Quarterly Progress Report, July 1961)
Leonard Kleinrock. Communication Nets: Stochastic Message Flow and Delay (McGraw-Hill, New York, 1964)
Kleinrock, Leonard (2 January 1975). Queueing Systems: Volume I – Theory. New York: Wiley Interscience. pp. 417. ISBN 978-0-471-49110-1.
Kleinrock, Leonard (22 April 1976). Queueing Systems: Volume II – Computer Applications. New York: Wiley Interscience. pp. 576. ISBN 978-0-471-49111-8.
Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984). Quantitative System Performance: Computer System Analysis Using Queueing Network Models. Prentice-Hall, Inc. ISBN 978-0-13-746975-8.
Jon Kleinberg; Éva Tardos (30 June 2013). Algorithm Design. Pearson. ISBN 978-1-292-02394-6.

== External links ==

Teknomo's Queueing theory tutorial and calculators
Virtamo's Queueing Theory Course
Myron Hlynka's Queueing Theory Page
LINE: a general-purpose engine to solve queueing models