# Kendall’s Notation¶

Kendall’s notation is used as shorthand to denote single node queueing systems [WS09].

A queue is characterised by:

$A/B/C/X/Y/Z$

where:

• $$A$$ denotes the distribution of inter-arrival times

• $$B$$ denotes the distribution of service times

• $$C$$ denotes the number of servers

• $$X$$ denotes the queueing capacity

• $$Y$$ denotes the size of the population of customers

• $$Z$$ denotes the queueing discipline

For the parameters $$A$$ and $$B$$, a number of shorthand notation is available. For example:

• $$M$$: Markovian or Exponential distribution

• $$E$$: Erlang distribution (a special case of the Gamma distribution)

• $$C_k$$: Coxian distribution of order $$k$$

• $$D$$: Deterministic distribution

• $$G$$ / $$GI$$: General / General independent distribution

The parameters $$X$$, $$Y$$ and $$Z$$ are optional, and are assumed to be $$\infty$$, $$\infty$$, and First In First Out (FIFO) respectively. Other options for the queueing schedule $$Z$$ may be SIRO (Service In Random Order), LIFO (Last In First Out), and PS (Processor Sharing).

Some examples:

• $$M/M/1$$:
• Exponential inter-arrival times

• Exponential service times

• 1 server

• Infinite queueing capacity

• Infinite population

• First in first out

• $$M/D/\infty/\infty/1000$$:
• Exponential inter-arrival times

• Deterministic service times

• Infinite servers

• Infinite queueing capacity

• Population of 1000 customers

• First in first out

• $$G/G/1/\infty/\infty/\text{SIRO}$$:
• General distribution for inter-arrival times

• General distribution for service times

• 1 server

• Infinite queueing capacity

• Infinite population

• Service in random order

• $$M/M/4/5$$:
• Exponential inter-arrival times

• Exponential service times

• 4 servers

• Queueing capacity of 5

• Infinite population

• First in first out