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