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In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers where customers cannot leave the network.[1] Jackson's theorem cannot be applied to closed networks because the queue length at a node in the closed network is limited by the population of the network. The Gordon–Newell theorem calculates the open network solution and then eliminates the infeasible states by renormalizing the probabilities. Calculation of the normalizing constant makes the treatment more awkward as the whole state space must be enumerated. Buzen's algorithm or mean value analysis can be used to calculate the normalizing constant more efficiently.[2]

Definition of a Gordon–Newell network

A network of m interconnected queues is known as a Gordon–Newell network[3] or closed Jackson network[4] if it meets the following conditions:

the network is closed (no customers can enter or leave the network),
all service times are exponentially distributed and the service discipline at all queues is FCFS,
a customer completing service at queue i will move to queue j with probability \( P_{ij} \), with the \( P_{ij} \) such that \( {\textstyle \sum _{j=1}^{m}P_{ij}=1}, \)
the utilization of all of the queues is less than one.

Theorem

In a closed Gordon–Newell network of m queues, with a total population of K individuals, write \( {\displaystyle \scriptstyle {(k_{1},k_{2},\ldots ,k_{m})}} \) (where ki is the length of queue i) for the state of the network and S(K, m) for the state space

\( {\displaystyle S(K,m)=\left\{\mathbf {k} \in \mathbb {N} ^{m}{\text{ such that }}\sum _{i=1}^{m}k_{i}=K\right\}.} \)

Then the equilibrium state probability distribution exists and is given by

\( {\displaystyle \pi (k_{1},k_{2},\ldots ,k_{m})={\frac {1}{G(K)}}\prod _{i=1}^{m}\left({\frac {e_{i}}{\mu _{i}}}\right)^{k_{i}}} \)

where service times at queue i are exponentially distributed with parameter μi. The normalizing constant G(K) is given by

\( {\displaystyle G(K)=\sum _{\mathbf {k} \in S(K,m)}\prod _{i=1}^{m}\left({\frac {e_{i}}{\mu _{i}}}\right)^{k_{i}},} \)

and ei is the visit ratio, calculated by solving the simultaneous equations

\( } {\displaystyle e_{i}=\sum _{j=1}^{m}e_{j}p_{ji}{\text{ for }}1\leq i\leq m.} \)

See also

BCMP network

References

Gordon, W. J.; Newell, G. F. (1967). "Closed Queuing Systems with Exponential Servers". Operations Research. 15 (2): 254. doi:10.1287/opre.15.2.254. JSTOR 168557.
Buzen, J. P. (1973). "Computational algorithms for closed queueing networks with exponential servers" (PDF). Communications of the ACM. 16 (9): 527. doi:10.1145/362342.362345.
Daduna, H. (1982). "Passage Times for Overtake-Free Paths in Gordon-Newell Networks". Advances in Applied Probability. 14 (3): 672–686. doi:10.2307/1426680.

Gong, Q.; Lai, K. K.; Wang, S. (2008). "Supply chain networks: Closed Jackson network models and properties". International Journal of Production Economics. 113 (2): 567. doi:10.1016/j.ijpe.2007.10.013.

vte

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 Round-robin 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

Information systems

Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

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