==Generalized #REDIRECT[[Jackson network ==]]▼In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]] '''Jackson's theorem''' is a theorem by [[James R. Jackson]].<ref>''Jobshop-like Queueing Systems'' [[James R. Jackson]] in Management Science, Vol. 10, No. 1 (Oct., 1963), pp. 131-142</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>''Networks of Queues'' [[F. P. Kelly]] in Advances in Applied Probability, Vol. 8, No. 2 (Jun., 1976), pp. 416-432</ref> including ideas used in the development of the Internet.<ref>''Comments on "Jobshop-Like Queueing Systems": The Background'' [[James R. Jackson]] in Management Science, Vol. 50, No. 12, Ten Most Influential Titles of Management Sciences First Fifty Years (Dec., 2004), p. 1803</ref> The paper was printed in the journal [[Management Science]]’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’<ref>''Jobshop-like Queueing Systems'' [[James R. Jackson]] in Management Science, Vol. 50, No. 12, Ten Most Influential Titles of Management Sciences First Fifty Years</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}}</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>
==Definition of a Jackson network==
A network of ''m'' interconnected queues is known as a '''Jackson network''' if it meets the following conditions:
# the network is open and any external arrivals to node ''i'' form a [[Poisson process]],
# all service times are exponentially distributed and the service discipline at all queues is [[FCFS]],
# a customer completing service at queue ''i'' will either move to some new queue ''j'' with probability <math>P_{ij}</math> or leave the system with probability <math>1-\sum_{j=1}^{m}P_{ij}</math>, which is non-zero for some subset of the queues,
# the [[utilization]] of all of the queues is less than one.
==Theorem==
In an open Jackson network of ''m'' [[M/M/1 model|M/M/1 queues]] where the [[utilization]] <math>\rho_i</math> is less than 1 at every queue, the equilibrium state probability distribution exists and for state <math>\scriptstyle{(k_1,k_2,\ldots,k_m)}</math> is given by the product of the individual queue equilibrium distributions
:<math>\pi (k_1,k_2,\ldots,k_m) = \prod_{i=1}^{m} \pi_i(k_i) = \prod_{i=1}^{m} [\rho_i^{k_i} (1-\rho_i)].</math>
The result <math>\pi (k_1,k_2,\ldots,k_m) = \prod_{i=1}^{m} \pi_i(k_i)</math> also holds for [[M/M/c model]] stations with ''c<sub>i</sub>'' servers at the <math>i^{th}</math> station, with utilization requirement <math>\rho_i < c_i</math>.
▲==Generalized Jackson network==
A '''generalized Jackson network''' allows [[renewal process|renewal arrival processes]] that need not be Poisson processes, and independent, identically distributed non-exponential service times. In general, this network does not have a [[product form solution|product form stationary distribution]], so approximations are sought.<ref>''Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization'' by Hong Chen, David D. Yao Published by Springer, 2001 ISBN 0387951660, 9780387951669</ref>
==See also==
*[[Gordon–Newell network]]
*[[BCMP network]]
*[[G-network]]
*[[Little's law]]
==Notes==
{{Reflist}}
[[Category:Stochastic processes]]
[[Category:Mathematical theorems]]
[[Category:Queueing theory]]
[[nl:Jackson-netwerk]]
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