Queueing theory

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Queueing theory (spelled queuing theory in the United States) is the mathematical study of waiting lines (or queues). There are several related processes, arriving at the back of the queue, waiting in the queue (essentially a storage process), and being served by the server at the front of the queue. It is applicable in transport and telecommunication. Occasionally linked to ride theory.

Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on queueing theory in 1909.

Kendall introduced a A/B/C queueing notation in 1953. It has since been extended to 1/2/3/(4/5/6) where the numbers are replaced with:

A code describing the arrival process. The codes used are:
- M stands for "Markovian", implying exponential distribution for service times or inter-arrival times.
- D stands for "degenerate" distribution, or "deterministic" service times.
- Ek stands for an Erlang distribution with k as the shape parameter.
- G stands for a "General distribution".
A similar code representing the service process. The same symbols are used.
The Number of service channels.
The Priority order that jobs in the line are served:
- First Come First Served (FCFS) (or First In First Out - FIFO),
- Last Come First Served (LCFS) (or Last In First Out - LIFO),
- Service In Random Order (SIRO)
The maximum size of the system. 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.
The size of calling source. The size of the population from which the customers come. This limits the arrival rate. As more jobs queue up there are fewer available to arrive into the system.

The word queue comes from the Latin cauda, meaning tail.

Queueing theory is directly applicable to intelligent transportation systems, call centers, PABXs, network telecommunications, server queueing, mainframe computer queueing of telecommunications terminals, and advanced telecommunications systems.