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

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'''Jackson's theorem''' is a theorem by [[James R. Jackson]] in [[queueing theory]].<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 solutions in other networks has been the subject of much research, including 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's theorem''' is the first significant development in the theory of [[queueing theory|networks of queues]]. It assumes an open queueing network of single-server queues with the following characteristics:

==Theorem==

Assuming an open queueing network of single-server queues with the following notation and assumptions:

* M = # of queues in the system, not counting queue 0 which represents the outside world

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* Arrivals from the outside world are Poisson. All queues have exponential service time distributions.

then a unique equilibrium state probability distribution exists and is given, for some state vector <math>\scriptstyle{(k_1,k_2,\ldots,k_M)}</math>

==Product form of Jackson's network==

:<math>P(k_1,k_2,\dots,k_M)=\prod_{i=1}^{M}

\left[\left(\frac{\lambda_i}{\mu_i}\right)^{k_i}\left(1-\frac{\lambda_i}{\mu_i}\right)\right]

= \prod_{i=1}^{M}[(1-\rho_i)\rho_i^{k_i}]</math><br>

(where <math>\scriptstyle{\rho_i=\frac{\lambda_i}{\mu_i}}</math>).

==See also==

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*[[Little's law]]

==External linksReferences==

{{ref-list}}

*Sinclair, B. (2005, June 9). ''[http://cnx.rice.edu/content/m10888/latest/ Jackson's Theorem]''. Connexions

[[Category:Stochastic processes]]