Jackson's theorem (queueing theory)

Jackson's theorem is a theorem by James R. Jackson in queueing theory.^[1] It was the first significant development in the theory of 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,^[2] including ideas used in the development of the Internet.^[3] The paper was printed in the journal Management Science’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’^[4]

Theorem

In an open Jackson network of m queues where the utilization ${\displaystyle \rho _{i}}$  is less than 1 at every queue, the equilibrium state probability distribution exists and for state ${\displaystyle \scriptstyle {(k_{1},k_{2},\ldots ,k_{m})}}$  is given by the product of the individual queue equilibrium distributions

${\displaystyle \pi (k_{1},k_{2},\ldots ,k_{m})=\prod _{i=1}^{m}\pi _{i}(k_{i})=\prod _{i=1}^{m}[\rho _{i}^{k_{i}}(1-\rho _{i})].}$ 

See also

• Jackson network
• BCMP network
• Gordon–Newell network
• G-network
• Little's law

References

1. Jobshop-like Queueing Systems James R. Jackson in Management Science, Vol. 10, No. 1 (Oct., 1963), pp. 131-142
2. Networks of Queues F. P. Kelly in Advances in Applied Probability, Vol. 8, No. 2 (Jun., 1976), pp. 416-432
3. 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
4. Jobshop-like Queueing Systems James R. Jackson in Management Science, Vol. 50, No. 12, Ten Most Influential Titles of Management Sciences First Fifty Years