The origin of queueing theory and its application traces back to Erlang’s historical
work for telephony networks as recently celebrated by the Erlang Centennial, 100
Years of Queueing, Copenhagen, recalling his first paper in 1909. Ever since, the
simplicity and fundamental flavour of Erlang’s famous expressions, such as his loss
formula for an incoming call in a circuit switched system to be lost, has remained
intriguing. It has motivated the development of results with similar elegance and
expression power for various systems modeling congestion and competition over
resources.
A second milestone was the step of queueing theory into queueing networks as
motivated by the first so-called product form results for assembly type networks
in manufacturing in the nineteen fifties (R.R.P. Jackson 1954, J.R. Jackson 1957,
and E. Koenigsberg 1958, 1959). These results revealed that the queue lengths at
nodes of a network, where customers route among the nodes upon service completion
in equilibrium can be regarded as independent random variables, that is, the
equilibrium distribution of the network of nodes factorizes over (is a product of) the
marginal equilibrium distributions of the individual nodes as if in isolation. These
networks are nowadays referred to as Jackson networks.
A third milestone was inspired by the rapid development of computer systems
and brought the attention for service disciplines such as the Processor Sharing discipline
introduced by Kleinrock in 1967. More complicated multi server nodes and
service disciplines such as First-Come-First-Served, Last-Come-First-Served and
Processor Sharing, and their mixing within a network have led to a surge in theoretical
developments and a wide applicability of queuing theory.
Queueing networks have obtained their place in both theory and practice. New
technological developments such as Internet and wireless communications, but also
advancements in existing applications such as manufacturing and production systems,
public transportation, logistics, and health care, have triggered many theoretical
and practical results.
