Traffic equations

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]^:125

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 and internal arrivals. If external arrivals have rate $\gamma_i$ and the routing matrix 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

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

↑ 1.0 1.1 Mitrani, I. (1997). "Queueing networks". Probabilistic Modelling. p. 122. doi:10.1017/CBO9781139173087.005. ISBN 9781139173087.
↑ Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. ISBN 0-201-54419-9.

Queueing theory

Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue

Arrival processes	Poisson process Markovian arrival process Rational arrival process

Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network

Service policies	FIFO LIFO Processor sharing Shortest job first Shortest remaining time

Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method

Limit theorems	Fluid limit Mean field theory Heavy traffic approximation Reflected Brownian motion

Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue

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