digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Business

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

M/M/1 queue diagram
An M/M/1 queueing node

In queueing theory, a discipline within the mathematical theory of probability, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node. D. G. Kendall proposed describing queueing models using three factors written A/S/c in 1953[1] where A denotes the time between arrivals to the queue, S the size of jobs and c the number of servers at the node. It has since been extended to A/S/c/K/N/D where K is the capacity of the queue, D is the queueing discipline[2] and N is the size of the population of jobs to be served.[3][4]

When the final three parameters are not specified (e.g. M/M/1 queue), it is assumed K = ∞, N = ∞ and D = FIFO.[5]

A: The arrival process[edit]

A code describing the arrival process. The codes used are:

Symbol Name Description Examples
M Markovian or memoryless[6] Poisson process (or random) arrival process. M/M/1 queue
MX batch Markov Poisson process with a random variable X for the number of arrivals at one time. MX/MY/1 queue
MAP Markovian arrival process Generalisation of the Poisson process.
BMAP Batch Markovian arrival process Generalisation of the MAP with multiple arrivals
MMPP Markov modulated poisson process Poisson process where arrivals are in "clusters".
D Degenerate distribution A deterministic or fixed inter-arrival time. D/M/1 queue
Ek Erlang distribution An Erlang distribution with k as the shape parameter.
G General distribution Although G usually refers to independent arrivals, some authors prefer to use GI to be explicit.
PH Phase-type distribution Some of the above distributions are special cases of the phase-type, often used in place of a general distribution.

S: The service time distribution[edit]

This gives the distribution of time of the service of a customer. Some common notations are:

Symbol Name Description Examples
M Markovian or memoryless[6] Exponential service time. M/M/1 queue
MY bulk Markov Exponential service time with a random variable Y for the number of arrivals at one time. MX/MY/1 queue
D Degenerate distribution A deterministic or fixed service time. M/D/1 queue
Ek Erlang distribution An Erlang distribution with k as the shape parameter.
G General distribution Although G usually refers to independent service time, some authors prefer to use GI to be explicit. M/G/1 queue
PH Phase-type distribution Some of the above distributions are special cases of the phase-type, often used in place of a general distribution.
MMPP Markov modulated poisson process Exponential service time distributions, where the rate parameter is controlled by a Markov chain.[7]

c: The number of servers[edit]

The number of service channels (or servers). The M/M/1 queue has a single server and the M/M/c queue c servers.

K: The number of places in the system[edit]

The capacity of the system, or the maximum number of customers allowed in the system including those in service. When the number is at this maximum, further arrivals are turned away. If this number is omitted, the capacity is assumed to be unlimited, or infinite.

Note: This is sometimes denoted C + k where k is the buffer size, the number of places in the queue above the number of servers C.

N: The calling population[edit]

The size of calling source. The size of the population from which the customers come. A small population will significantly affect the effective arrival rate, because, as more jobs queue up, there are fewer left available to arrive into the system. If this number is omitted, the population is assumed to be unlimited, or infinite.

D: The queue's discipline[edit]

The Service Discipline or Priority order that jobs in the queue, or waiting line, are served:

Symbol Name Description
FIFO/FCFS First In First Out/First Come First Served The customers are served in the order they arrived in.
LIFO/LCFS Last in First Out/Last Come First Served The customers are served in the reverse order to the order they arrived in.
SIRO Service In Random Order The customers are served in a random order with no regard to arrival order.
PNPN Priority service Priority service, including preemptive and non-preemptive. (see Priority queue)
PS Processor Sharing
Note: An alternative notation practice is to record the queue discipline before the population and system capacity, with or without enclosing parenthesis. This does not normally cause confusion because the notation is different.

References[edit]

  1. ^ Kendall, D. G. (1953). "Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain". The Annals of Mathematical Statistics 24 (3): 338. doi:10.1214/aoms/1177728975. JSTOR 2236285.  edit
  2. ^ Lee, Alec Miller (1966). "A Problem of Standards of Service (Chapter 15)". Applied Queueing Theory. New York: MacMillan. ISBN 0-333-04079-1. 
  3. ^ Taha, Hamdy A. (1968). Operations research: an introduction (Preliminary ed.). 
  4. ^ Sen, Rathindra P. (2010). Operations Research: Algorithms And Applications. Prentice-Hall of India. p. 518. ISBN 81-203-3930-4. 
  5. ^ Gautam, N. (2007). "Queueing Theory". Operations Research and Management Science Handbook. Operations Research Series. 20073432. pp. 1–2. doi:10.1201/9781420009712.ch9. ISBN 978-0-8493-9721-9.  edit
  6. ^ a b Zonderland, M. E.; Boucherie, R. J. (2012). "Queuing Networks in Health Care Systems". Handbook of Healthcare System Scheduling. International Series in Operations Research & Management Science 168. p. 201. doi:10.1007/978-1-4614-1734-7_9. ISBN 978-1-4614-1733-0.  edit
  7. ^ Zhou, Yong-Ping; Gans, Noah (October 1999). "#99-40-B: A Single-Server Queue with Markov Modulated Service Times". Financial Institutions Center, Wharton, UPenn. Retrieved 2011-01-11. 

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Kendall's_notation — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
304 videos foundNext > 

CSE567-13-30A: Introduction to Queueing Theory for Computer System Performance Evaluation

Audio recording of a class lecture by Prof. Raj Jain on Introduction to Queueing Theory. The talk covers Basic Components of a Queue, Kendall Notation A/S/m/...

CSE567-13-30B: Introduction to Queueing Theory for Computer System Performance Evaluation

Second part of audio recording of a class lecture by Prof. Raj Jain on Introduction to Queueing Theory. The talk covers Basic Components of a Queue, Kendall ...

Lesson 3: Ukulele First Chords, Notation, and Strumming

Link for the text: http://www.amazon.com/dp/B005CRQ30W/ref=cm_sw_r_tw_dp_KXS3sb1HATK79&tag=kytc-20 (aff) Thank you so much if you've already joined my Patreo...

Kendall's tau - Explained Simply + Examples (part 1)

I describe what Kendall's tau is and provide 2 examples with step by step calculations and explanations.

Exponents and Scientific Notation "Quick" Lesson

AQA FP1 9.01 Series: Introducing Sigma Notation

I introduce Sigma notation, how it is used and what is in the formula booklet.

Composition, Notations and Score for Live Audiovisual Preformance

Benton-C Bainbridge, Adam Kendall, Katherine Liberovskaya and Andrea Parkins, moderated by Ana Carvalho. A series of four presentations around composition wi...

Ran Dank, piano, Nick Kendall, violin, Thomas Carroll, cello, aka The Tosslers - Tokyo, summer 2011

Ran Dank, piano, Thomas Carroll, cello, Nick Kendall, violin. Intense rehearsing in Tokyo Japan.

ADVANCED SLAP BASS by Scott Whitley Lesson 3 - Victor Wooten style lick 1

Buy the TAB (with standard notation) here.... http://www.scott-whitley.com/downloads/327/ Web - http://www.scott-whitley.com/ Facebook - https://www.facebook...

Physics Wood, Kendall, Morgan in order of magnitude)

304 videos foundNext > 

1 news items

 
中文IT技术网站
Fri, 08 Aug 2014 04:30:57 -0700

对于开发者、技术管理者、架构师、系统分析师和项目经理来说,创建具备高性能特征的复杂软件都是一件极其困难的事。 然而,通过了解一些基本原理、性能问题的解决和预防可以更简单、可靠。 本文讲述了这些基本原理,涵盖了 ...
Loading

Oops, we seem to be having trouble contacting Twitter

Talk About Kendall's notation

You can talk about Kendall's notation with people all over the world in our discussions.

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!