Jackson network

In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes Jacksonian network[1]) is a class of queueing network 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,[2] including ideas used in the development of the Internet.

[3] The networks were first identified by James R. Jackson[4][5] and his paper was re-printed in the journal Management Science’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’[6] Jackson was inspired by the work of Burke and Reich,[7] 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".

[9] 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.

is given by the product of the individual queue equilibrium distributions The result

also holds for M/M/c model stations with ci servers at the

In an open network, jobs arrive from outside following a Poisson process with rate

Each arrival is independently routed to node j with probability

, including both external arrivals and internal transitions: (Since the utilisation at each node is less than 1, and we are looking at the equilibrium distribution i.e. the long-run-average behaviour, the rate of jobs transitioning from j to i is bounded by a fraction of the arrival rate at j and we ignore the service rate

All jobs leave each node also following Poisson process, and define

denote the number of jobs at node i at time t, and

is determined by the following system of balance equations: where

Suppose a vector of independent random variables

is well defined, then the equilibrium distribution of the open Jackson network has the following product form: for all

By the product form and formula (3), we have: Substituting these into the right side of

are equal.⟨ This theorem extends the one shown above by allowing state-dependent service rate of each node.

Suppose we have a three-node Jackson network shown in the graph, the coefficients are: Then by the theorem, we can calculate: According to the definition of

, we have: Hence the probability that there is one job at each node is: Since the service rate here does not depend on state, the

s simply follow a geometric distribution.

A generalized Jackson network allows 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 stationary distribution, so approximations are sought.

[13] Under some mild conditions the queue-length process[clarification needed]

of an open generalized Jackson network can be approximated by a reflected Brownian motion defined as

This is a two-order approximation obtained by relation between general Jackson network with homogeneous fluid network and reflected Brownian motion.

The parameters of the reflected Brownian process is specified as follows: where the symbols are defined as: They are defined in this way: Let

is a driftless Brownian process with covariate matrix