# Kendall's notation

### From Wikipedia

In queueing theory, **Kendall's notation** (or sometimes **Kendall notation**) is the standard system used to describe and classify the queueing model that a queueing system corresponds to. First suggested by D. G. Kendall in 1953 as a three-factor *A/B/C* notation system for characterising queues, it has since been extended to include up to six different factors.

The notation now appears in most standard reference work about queueing theory. e.g. ^{[1]}

## Contents |

## 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 (or random) arrival process. |

M^{X} | batch Markov | Poisson process 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 | 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 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 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. |

### C: The number of servers

The number of service channels (or 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.

- 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.

## References

- ↑ Tijms, H.C,
*Algorithmic Analysis of Queues*, Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003.