Traffic equations

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title: "Traffic equations" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["queueing-theory"] description: "Equations describing traffic rate" topic_path: "general/queueing-theory" source: "https://en.wikipedia.org/wiki/Traffic_equations" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Equations describing traffic rate ::

In queueing theory, a discipline within the mathematical theory of probability, traffic equations are equations that describe the mean arrival rate of traffic, allowing the arrival rates at individual nodes to be determined. Mitrani notes "if the network is stable, the traffic equations are valid and can be solved."

Jackson network

In a Jackson network, the mean arrival rate \lambda_i at each node i in the network is given by the sum of external arrivals (that is, arrivals from outside the network directly placed onto node i, if any), and internal arrivals from each of the other nodes on the network. If external arrivals at node i have rate \gamma_i, and the routing matrix is P, the traffic equations are, (for i = 1, 2, ..., m)

::\lambda_i = \gamma_i + \sum_{j=1}^m p_{ji}\lambda_j.

This can be written in matrix form as

::\lambda(I-P)=\gamma , ,

and there is a unique solution of unknowns \lambda_i to this equation, so the mean arrival rates at each of the nodes can be determined given knowledge of the external arrival rates \gamma_i and the matrix P. The matrix I − P is surely non-singular as otherwise in the long run the network would become empty.

Gordon–Newell network

In a Gordon–Newell network there are no external arrivals, so the traffic equations take the form (for i = 1, 2, ..., m)

::\lambda_i = \sum_{j=1}^m p_{ji} \lambda_j.

Notes

References

1. (1997). "Probabilistic Modelling".
2. As explained in the [[Jackson network]] article, jobs travel among the nodes following a fixed routing matrix.
3. (December 2010). ["Performance Modelling of Communication Networks and Computer Architectures"](https://archive.org/details/performancemodel0000harr }}{{Page needed). Addison-Wesley.

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