“Kendall's notation”的意思、由来-开放百科全书

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 service time distribution 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, N is the size of the population of jobs to be served, and D is the queueing discipline^[2].^[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

Symbol

Name

Description

Examples

Markovian or memoryless^[6]

Poisson process (or random) arrival process (i.e., exponential inter-arrival times).

M/M/1 queue

M^X

batch Markov

Poisson process with a random variable X for the number of arrivals at one time.

M^X/M^Y/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".

Degenerate distribution

A deterministic or fixed inter-arrival time.

D/M/1 queue

E_k

Erlang distribution

An Erlang distribution with k as the shape parameter (i.e., sum of k i.i.d. exponential random variables).

General distribution

Although G usually refers to independent arrivals, some authors prefer to use GI to be explicit.

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

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

Symbol

Name

Description

Examples

Markovian or memoryless^[6]

Exponential service time.

M/M/1 queue

M^Y

bulk Markov

Exponential service time with a random variable Y for the size of the batch of entities serviced at one time.

M^X/M^Y/1 queue

Degenerate distribution

A deterministic or fixed service time.

M/D/1 queue

E_k

Erlang distribution

An Erlang distribution with k as the shape parameter (i.e., sum of k i.i.d. exponential random variables).

General distribution

Although G usually refers to independent service time, some authors prefer to use GI to be explicit.

M/G/1 queue

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

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

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.

N: The calling population

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

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)

Processor Sharing

References

1. ^{{Cite journal | last1 = Kendall | first1 = D. G. | authorlink1 = David George Kendall| title = Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain | doi = 10.1214/aoms/1177728975 | jstor = 2236285| journal = The Annals of Mathematical Statistics | volume = 24 | issue = 3 | pages = 338 | year = 1953| url = http://projecteuclid.org/euclid.aoms/1177728975 | pmid = | pmc = }}
2. ^{{cite book|chapter=A Problem of Standards of Service (Chapter 15)|first=Alec Miller|last=Lee|year=1966|title=Applied Queueing Theory|publisher=MacMillan|location=New York|isbn=0-333-04079-1}}
3. ^{{cite book|title=Operations research: an introduction|edition=Preliminary|year=1968|first=Hamdy A.|last=Taha}}
4. ^{{cite book|title=Operations Research: Algorithms And Applications|first=Rathindra P.|last=Sen|publisher=Prentice-Hall of India|isbn=81-203-3930-4|year=2010|page=518}}
5. ^{{Cite book | last1 = Gautam | first1 = N. | chapter = Queueing Theory | doi = 10.1201/9781420009712.ch9 | title = Operations Research and Management Science Handbook | series = Operations Research Series | volume = 20073432 | pages = 1–2 | year = 2007 | isbn = 978-0-8493-9721-9 | pmid = | pmc = }}
6. ^¹{{Cite book | last1 = Zonderland | first1 = M. E. | last2 = Boucherie | first2 = R. J. | chapter = Queuing Networks in Health Care Systems | doi = 10.1007/978-1-4614-1734-7_9 | title = Handbook of Healthcare System Scheduling | series = International Series in Operations Research & Management Science | volume = 168 | pages = 201 | year = 2012 | isbn = 978-1-4614-1733-0 | pmid = | pmc = }}
7. ^{{cite web|url=http://fic.wharton.upenn.edu/fic/papers/99/p9940.html|title=#99-40-B: A Single-Server Queue with Markov Modulated Service Times|first1=Yong-Ping|last1=Zhou|first2=Noah|last2=Gans|date=October 1999|publisher=Financial Institutions Center, Wharton, UPenn|accessdate=2011-01-11}}