Kendall's notation

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 service channels open 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]

Waiting queue at Ottawa station.

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

First example: M/M/1 queue
M/M/1 queue diagram
An M/M/1 queueing node.

A M/M/1 queue means that the time between arrivals is Markovian (M), i.e. the inter-arrival time follows an exponential distribution of parameter λ. The second M means that the service time is Markovian: it follows an exponential distribution of parameter μ. The last parameter is the number of service channel which one (1).

Description of the parameters

In this section, we describe the parameters A/S/c/K/N/D from left to right.

A: The arrival process

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

SymbolNameDescriptionExamples
MMarkovian or memoryless[6]Poisson process (or random) arrival process (i.e., exponential inter-arrival times).M/M/1 queue
MXbatch MarkovPoisson process with a random variable X for the number of arrivals at one time.MX/MY/1 queue
MAPMarkovian arrival processGeneralisation of the Poisson process.
BMAPBatch Markovian arrival processGeneralisation of the MAP with multiple arrivals
MMPPMarkov modulated poisson processPoisson process where arrivals are in "clusters".
DDegenerate distributionA deterministic or fixed inter-arrival time.D/M/1 queue
EkErlang distributionAn Erlang distribution with k as the shape parameter (i.e., sum of k i.i.d. exponential random variables).
GGeneral distributionAlthough G usually refers to independent arrivals, some authors prefer to use GI to be explicit.
PHPhase-type distributionSome 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:

SymbolNameDescriptionExamples
MMarkovian or memoryless[6]Exponential service time.M/M/1 queue
MYbulk MarkovExponential service time with a random variable Y for the size of the batch of entities serviced at one time.MX/MY/1 queue
DDegenerate distributionA deterministic or fixed service time.M/D/1 queue
EkErlang distributionAn Erlang distribution with k as the shape parameter (i.e., sum of k i.i.d. exponential random variables).
GGeneral distributionAlthough G usually refers to independent service time, some authors prefer to use GI to be explicit.M/G/1 queue
PHPhase-type distributionSome of the above distributions are special cases of the phase-type, often used in place of a general distribution.
MMPPMarkov modulated poisson processExponential 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 queue

The capacity of queue, or the maximum number of customers allowed in the queue. 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 customers are in system, there are fewer free customers 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 (used by default).
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.
PQ Priority Queuing There are several options: Preemptive Priority Queuing, Non Preemptive Queuing, Class Based Weighted Fair Queuing, Weighted Fair Queuing.
PS Processor Sharing The customers are served in the determine order with no regard of arrival order.
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
  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): 338–354. doi:10.1214/aoms/1177728975. JSTOR 2236285.
  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 978-81-203-3930-9.
  5. Gautam, N. (2007). "Queueing Theory". Operations Research and Management Science Handbook. Operations Research Series. Vol. 20073432. pp. 1–2. doi:10.1201/9781420009712.ch9. ISBN 978-0-8493-9721-9.
  6. 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. Vol. 168. p. 201. doi:10.1007/978-1-4614-1734-7_9. ISBN 978-1-4614-1733-0.
  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. Archived from the original on 2010-06-21. Retrieved 2011-01-11.
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