Traffic equations

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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."[1]

Jackson network[edit]

In a Jackson network, the mean arrival rate \lambda_i at each node i in the network is given by the sum of external and internal arrivals. If external arrivals have rate \gamma_i and the routing matrix[clarification needed] is P, the traffic equations are,[2] (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.[1]

Gordon–Newell network[edit]

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[edit]

  1. ^ a b Mitrani, Isi (1998). Probabilistic modelling. Cambridge University Press. ISBN 0-521-58530-9. [page needed]
  2. ^ Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. ISBN 0-201-54419-9. [page needed]