.. _kendall-notation: ================== Kendall's Notation ================== Kendall's notation is used as shorthand to denote single node queueing systems [WS09]_. A queue is characterised by: .. math:: A/B/C/X/Y/Z where: + :math:`A` denotes the distribution of inter-arrival times + :math:`B` denotes the distribution of service times + :math:`C` denotes the number of servers + :math:`X` denotes the queueing capacity + :math:`Y` denotes the size of the population of customers + :math:`Z` denotes the queueing discipline For the parameters :math:`A` and :math:`B`, a number of shorthand notation is available. For example: + :math:`M`: Markovian or Exponential distribution + :math:`E`: Erlang distribution (a special case of the Gamma distribution) + :math:`C_k`: Coxian distribution of order :math:`k` + :math:`D`: Deterministic distribution + :math:`G` / :math:`GI`: General / General independent distribution The parameters :math:`X`, :math:`Y` and :math:`Z` are optional, and are assumed to be :math:`\infty`, :math:`\infty`, and First In First Out (FIFO) respectively. Other options for the queueing schedule :math:`Z` may be SIRO (Service In Random Order), LIFO (Last In First Out), and PS (Processor Sharing). Some examples: + :math:`M/M/1`: + Exponential inter-arrival times + Exponential service times + 1 server + Infinite queueing capacity + Infinite population + First in first out + :math:`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 + :math:`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 + :math:`M/M/4/5`: + Exponential inter-arrival times + Exponential service times + 4 servers + Queueing capacity of 5 + Infinite population + First in first out