In queueing theory

Queueing theory

Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...

, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify the queueing model

Queueing model

In queueing theory, a queueing model is used to approximate a real queueing situation or system, so the queueing behaviour can be analysed mathematically...

 that a queueing system corresponds to. First suggested by D. G. Kendall

David George Kendall

David George Kendall FRS was an English statistician, who spent much of his academic life in the University of Oxford and the University of Cambridge. He worked with M. S...

 in 1953 as a three-factor A/B/C notation system for characterising queues, it has since been extended to include K and D by Lee and N by Taha.

The notation now appears in most standard reference work about queueing theory, e.g. Algorithmic Analysis of Queues

Notation

A queue is described in shorthand notation by A/B/C/K/N/D or the more concise A/B/C. In this concise version, it is assumed K = ∞, N = ∞ and D = FIFO.

A: The arrival process

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

Symbol	Name	Description
M	Markovian	Poisson process Poisson process A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...  (or random) arrival process.
MX	batch Markov	Poisson process Poisson process A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...  with a random variable X for the number of arrivals at one time.
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.
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 Phase-type distribution A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random...	Some of the above distributions are special cases of the phase-type, often used in place of a general distribution.

B: The service time distribution

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

Symbol	Name	Description
M	Markovian	Exponential Exponential distribution In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...  service time.
D	Degenerate distribution	A deterministic or fixed service time.
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.
PH	Phase-type distribution Phase-type distribution A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random...	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 Exponential distribution In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...  service time distributions, where the rate parameter is controlled by a Markov chain.

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.

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

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)
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.

External links

• http://www.doc.ic.ac.uk/~nd/surprise_97/journal/vol4/wll1/main.htm
• http://www.everything2.com/index.pl?node_id=1055043