Gordon–Newell theorem

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.[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] Contents 1 Definition of a Gordon–Newell network 2 Theorem 3 See also 4 References 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 {displaystyle P_{ij}} , with the {displaystyle 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}=Kright}.} 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 }}1leq ileq 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. hide vte Queueing theory Single queueing nodes D/M/1 queueM/D/1 queueM/D/c queueM/M/1 queue Burke's theoremM/M/c queueM/M/∞ queueM/G/1 queue Pollaczek–Khinchine formulaMatrix analytic methodM/G/k queueG/M/1 queueG/G/1 queue Kingman's formulaLindley equationFork–join queueBulk queue Arrival processes Poisson point processMarkovian arrival processRational arrival process Queueing networks Jackson network Traffic equationsGordon–Newell theorem Mean value analysisBuzen's algorithmKelly networkG-networkBCMP network Service policies FIFOLIFOProcessor sharingRound-robinShortest job nextShortest remaining time Key concepts Continuous-time Markov chainKendall's notationLittle's lawProduct-form solution Balance equationQuasireversibilityFlow-equivalent server methodArrival theoremDecomposition methodBeneš method Limit theorems Fluid limitMean-field theoryHeavy traffic approximation Reflected Brownian motion Extensions Fluid queueLayered queueing networkPolling systemAdversarial queueing networkLoss networkRetrial queue Information systems Data bufferErlang (unit)Erlang distributionFlow control (data)Message queueNetwork congestionNetwork schedulerPipeline (software)Quality of serviceScheduling (computing)Teletraffic engineering Category Categories: Probability theoremsQueueing theory

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