In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes called a Jacksonian network) is a class of queueing networks where the equilibrium distribution is particularly simple to compute as the network has a product-form solution. 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, including ideas used in the development of the Internet. The networks were first identified by James R. Jackson and his paper was reprinted in the journal Management Science’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’ Jackson was inspired by the work of Burke and Reich, though Jean 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". An earlier product-form solution was found by R. R. P. Jackson for tandem queues (a finite chain of queues where each customer must visit each queue in order) and cyclic networks (a loop of queues where each customer must visit each queue in order). A Jackson network consists of a number of nodes, where each node represents a queue in which the service rate can be both node-dependent (different nodes have different service rates) and state-dependent (service rates change depending on queue lengths). Jobs travel among the nodes following a fixed routing matrix. All jobs at each node belong to a single "class", and jobs follow the same service-time distribution and the same routing mechanism. Consequently, there is no notion of priority in serving the jobs: all jobs at each node are served on a first-come, first-served basis. Jackson networks where a finite population of jobs travel around a closed network also have a product-form solution described by the Gordon–Newell theorem.
Necessary conditions for a Jackson network A network of m interconnected queues is known as a Jackson network or Jacksonian network if it meets the following conditions:
if the network is open, any external arrivals to node i form a Poisson process, All service times are exponentially distributed and the service discipline at all queues is first-come, first-served, a customer completing service at queue i will either move to some new queue j with probability P i j {\displaystyle P_{ij}} or leave the system with probability 1 − ∑ j = 1 m P i j {\displaystyle 1-\sum _{j=1}^{m}P_{ij}} , which, for an open network, 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 queues where the utilization ρ i {\displaystyle \rho _{i}} is less than 1 at every queue, the equilibrium state probability distribution exists and for state ( k 1 , k 2 , … , k m ) {\displaystyle \scriptstyle {(k_{1},k_{2},\ldots ,k_{m})}} is given by the product of the individual queue equilibrium distributions
π ( k 1 , k 2 , … , k m ) = ∏ i = 1 m π i ( k i ) = ∏ i = 1 m [ ρ i k i ( 1 − ρ i ) ] . {\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})].}
The result π ( k 1 , k 2 , … , k m ) = ∏ i = 1 m π i ( k i ) {\displaystyle \pi (k_{1},k_{2},\ldots ,k_{m})=\prod _{i=1}^{m}\pi _{i}(k_{i})} also holds for M/M/c model stations with ci servers at the i th {\displaystyle i^{\text{th}}} station, with utilization requirement ρ i < c i {\displaystyle \rho _{i}<c_{i}} .
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