The Azimuth Project
Queueing theory

Idea

The simplest sort of queuing model describes (for example) customers entering a line, being dealt with by one or more ‘servers’, and then leaving. Such queuing models can be represented using Kendall’s notation:

A/B/S/K/N/D A/B/S/K/N/D

where:

  • AA is the interarrival time distribution: the probability distribution of times for the next customer to enter the queue.
  • BB is the service time distribution: the probability distribution of times for a customer being service to have their service completed.
  • SS is the number of servers
  • KK is the system capacity, meaning the maximum number of customers who can be ‘waiting in line’.
  • NN is the calling population, meaning the maximum number of customers who enter.
  • D is the service discipline assumed: for example, ‘first in first out’ (FIFOFIFO), ‘last in first out’ (LIFOLIFO), etcetera.

Often the last members are omitted, so the notation becomes A/B/SA/B/S and it is assumed that K=K = \infty, N=N = \infty and D=FIFOD = FIFO.

Some of the most popular probability distributions for AA and BB are:

  • MM: Markovian, meaning a Poisson process.

  • E kE^k: an Erlang kk-process, which is the convolution of kk identical Poisson processes.

  • DD: a degenerate distribution, meaning a delta function at some fixed waiting time.