Jackson's theorem (queueing theory): Difference between revisions

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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]] '''Jackson's theorem''' is a theorem by [[James R. Jackson]].<ref name="jackson">{{cite journal|title=Jobshop-like Queueing Systems|first=James R.|last=Jackson|journal=Management Science|volume=10|number=1|month=Oct.|year=1963|pages=131-142|doi=10.1287/mnsc.1040.0268|jstor=300461492627213}}</ref> It was the first significant development in the theory of [[queueing theory|networks of queues]], and generalising and applying the ideas of the theorem to search for similar [[product form solution]]s in other networks has been the subject of much research,<ref>''{{cite journal|title=Networks of Queues'' [[|authorlink=F. P. Kelly]] in|first=F. P.|last=Kelly|journal=Advances in Applied Probability, Vol. |volume=8, No. |number=2 (|month=Jun., |year=1976), pp. |pages=416-432|jstor=1425912}}</ref> including ideas used in the development of the Internet.<ref>''{{cite journal|title=Comments on "Jobshop-Like Queueing Systems": The Background'' [[|first=James R. |last=Jackson]] in |journal=|Management Science, Vol. |volume=50, No. |number=12, Ten Most Influential Titles of Management Sciences First Fifty Years (Dec., |month=December|year=2004), p. 1803|pages=1796-1802|jstor=30046150}}</ref> The paper was printed in the journal [[Management Science]]’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’<ref>''{{cite journal|title=Jobshop-likeLike Queueing Systems'' [[|first=James R. |last=Jackson]] in |journal=|Management Science, Vol. |volume=50, No. |number=12, Ten Most Influential Titles of Management Sciences First Fifty Years|month=December|year=2004|pages=1796-1802|jstor=30046149}}</ref>

Jackson was inspired by the work of [[Burke's theorem|Burke]] and Reich,<ref>{{cite journal|title=Waiting Times When Queues are in Tandem|journal=[[Annals of Mathematical Statistics]]|volume=28|number=3|month=September|year=1957|first=Edgar|last=Reich|doi=10.1214/aoms/1177706889|jstor=2237237}}</ref> though Walrand notes "product form results … [are] a much less immediate result of the output theorem than Jackson himself appeared to believe in his fundamental paper".<ref>{{cite journal|title=A Probabilistic Look at Networks of Quasi-Reversible Queues|journal=[[IEEE Transactions on Information Theory]]|volume=29|number=6|month=November|year=1983|first=Jean|last=Walrand|doi=10.1109/TIT.1983.1056762}}</ref>