Editing Queueing theory (section)

== History==
{{anchor|Overview of the development of the theory}}<!--anchored with previous section title-->

In 1909, [[Agner Krarup Erlang]], a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory.<ref>{{cite web |url=http://pass.maths.org.uk/issue2/erlang/index.html |title=Agner Krarup Erlang (1878-1929) &#124; plus.maths.org |publisher=Pass.maths.org.uk |access-date=2013-04-22 |date=1997-04-30 |archive-date=2008-10-07 |archive-url=https://web.archive.org/web/20081007225944/http://pass.maths.org.uk/issue2/erlang/index.html |url-status=live }}</ref><ref>{{Cite journal | last1 = Asmussen | first1 = S. R. | last2 = Boxma | first2 = O. J. | author-link2 = Onno Boxma| doi = 10.1007/s11134-009-9151-8 | title = Editorial introduction | journal = [[Queueing Systems]] | volume = 63 | issue = 1–4 | pages = 1–2 | year = 2009 | s2cid = 45664707 }}</ref><ref>{{cite journal | author-link = Agner Krarup Erlang | first = Agner Krarup | last = Erlang
| title = The theory of probabilities and telephone conversations | journal = Nyt Tidsskrift for Matematik B | volume = 20 | pages = 33–39 | archive-url = https://web.archive.org/web/20111001212934/http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf | archive-date = 2011-10-01 | url = http://oldwww.com.dtu.dk/teletraffic/erlangbook/pps131-137.pdf | year = 1909}}</ref> He modeled the number of telephone calls arriving at an exchange by a [[Poisson process]] and solved the [[M/D/1 queue]] in 1917 and [[M/D/k queue|M/D/''k'' queue]]ing model in 1920.<ref name="century">{{Cite journal | last1 = Kingman | first1 = J. F. C. | author-link1 = John Kingman | title = The first Erlang century—and the next | journal = [[Queueing Systems]] | volume = 63 | issue = 1–4 | pages = 3–4 | year = 2009 | doi = 10.1007/s11134-009-9147-4| s2cid = 38588726 }}</ref> In Kendall's notation:

* M stands for "Markov" or "memoryless", and means arrivals occur according to a Poisson process
* D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service
* ''k'' describes the number of servers at the queueing node (''k'' = 1, 2, 3, ...)

If the node has more jobs than servers, then jobs will queue and wait for service.

The [[M/G/1 |M/G/1 queue]] was solved by [[Felix Pollaczek]] in 1930,<ref>Pollaczek, F., Ueber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. Z. 1930</ref> a solution later recast in probabilistic terms by [[Aleksandr Khinchin]] and now known as the [[Pollaczek–Khinchine formula]].<ref name="century" /><ref name="century1" />

After the 1940s, queueing theory became an area of research interest to mathematicians.<ref name="century1">{{Cite journal | last1 = Whittle | first1 = P. | author-link1 = Peter Whittle (mathematician)| doi = 10.1287/opre.50.1.227.17792 | title = Applied Probability in Great Britain | journal = [[Operations Research (journal)|Operations Research]]| volume = 50 | issue = 1 | pages = 227–239| year = 2002 | jstor = 3088474| doi-access = free }}</ref> In 1953, [[David George Kendall]] solved the GI/M/''k'' queue<ref>Kendall, D.G.:Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain, Ann. Math. Stat. 1953</ref> and introduced the modern notation for queues, now known as [[Kendall's notation]]. In 1957, Pollaczek studied the GI/G/1 using an [[integral equation]].<ref>Pollaczek, F., Problèmes Stochastiques posés par le phénomène de formation d'une queue</ref> [[John Kingman]] gave a formula for the [[Mean sojourn time|mean waiting time]] in a [[G/G/1 queue]], now known as [[Kingman's formula]].<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | author-link = John Kingman| doi = 10.1017/S0305004100036094 | author2 = <!-- (exclude bad crossref data) --> | last2 = Atiyah | title = The single server queue in heavy traffic | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]| volume = 57 | issue = 4 | page = 902 | date=October 1961 | jstor = 2984229| bibcode = 1961PCPS...57..902K | s2cid = 62590290 }}</ref>

[[Leonard Kleinrock]] worked on the application of queueing theory to [[message switching]] in the early 1960s and [[packet switching]] in the early 1970s. His initial contribution to this field was his doctoral thesis at the [[Massachusetts Institute of Technology]] in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the [[ARPANET]], a forerunner to the Internet.

The [[matrix geometric method]] and [[matrix analytic method]]s have allowed queues with [[phase-type distribution|phase-type distributed]] inter-arrival and service time distributions to be considered.<ref>{{Cite journal | last1 = Ramaswami | first1 = V. | doi = 10.1080/15326348808807077 | title = A stable recursion for the steady state vector in markov chains of m/g/1 type | journal = Communications in Statistics. Stochastic Models | volume = 4 | pages = 183–188 | year = 1988 }}</ref>

Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.<ref>{{Cite book | last1 = Morozov | first1 = E. |chapter = Stability analysis of a multiclass retrial system withcoupled orbit queues | doi = 10.1007/978-3-319-66583-2_6 | title = Proceedings of 14th European Workshop| series = Lecture Notes in Computer Science | volume = 17| pages = 85–98 | year = 2017 | doi-access = free|isbn=978-3-319-66582-5 }}</ref> 

Modern day application of queueing theory concerns among other things [[product development]] where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.<ref>{{cite journal |title=Simulation and queueing network modeling of single-product production campaigns |date=1992 |url=https://www.sciencedirect.com/science/article/abs/pii/0098135492800185 |doi=10.1016/0098-1354(92)80018-5 |last1=Carlson |first1=E.C. |last2=Felder |first2=R.M. |journal=Computers & Chemical Engineering |volume=16 |issue=7 |pages=707–718 }}</ref>

Problems such as performance metrics for the [[M/G/k queue|M/G/''k'' queue]] remain an open problem.<ref name="century" /><ref name="century1" />