Gordon–Newell theorem

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. 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.

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

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 k_i is the length of queue i) for the state of the network and S(K, m) for the state space

Then the equilibrium state probability distribution exists and is given by

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