G-network

This article is about the mathematical model. For the wireless standard, see 802.11g.

In queueing theory, a discipline within the mathematical theory of probability, a G-network (generalized queueing network,[1][2] often called a Gelenbe network[3]) is an open network of G-queues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic re-routing or traffic destruction, as well as a model for neural networks.[4][5] A G-queue is a network of queues with several types of novel and useful customers:

  • positive customers, which arrive from other queues or arrive externally as Poisson arrivals, and obey standard service and routing disciplines as in conventional network models,
  • negative customers, which arrive from another queue, or which arrive externally as Poisson arrivals, and remove (or 'kill') customers in a non-empty queue, representing the need to remove traffic when the network is congested, including the removal of "batches" of customers</ref>[6][7]
  • "triggers", which arrive from other queues or from outside the network, and which displace customers and move them to other queues

A product-form solution superficially similar in form to Jackson's theorem, but which requires the solution of a system of non-linear equations for the traffic flows, exists for the stationary distribution of G-networks while the traffic equations of a G-network are in fact surprisingly non-linear, and the model does not obey partial balance. This broke previous assumptions that partial balance was a necessary condition for a product-form solution. A powerful property of G-networks is that they are universal approximators for continuous and bounded functions, so that they can be used to approximate quite general input-output behaviours.[8]

  1. ^ Gelenbe, Erol (1991). "Product-form queueing networks with negative and positive customers" (PDF). Journal of Applied Probability. 28 (3): 656–663. doi:10.2307/3214499.
  2. ^ Gelenbe, Erol (Sep 1993). "G-Networks with Triggered Customer Movement". Journal of Applied Probability. 30 (3): 742–748. doi:10.2307/3214781. JSTOR 3214781.
  3. ^ Gelenbe, Erol; Fourneau, Jean-Michel (2002). "G-networks with resets". Performance Evaluation. 49 (1/4): 179–191. doi:10.1016/S0166-5316(02)00127-X.
  4. ^ Gelenbe, Erol (1989). "Random neural networks with negative and positive signals and product form solution" (PDF). Neural Computation. 1 (4): 502–510. doi:10.1162/neco.1989.1.4.502.
  5. ^ Harrison, Peter (2009). "Turning Back Time – What Impact on Performance?". The Computer Journal. 53 (6): 860–868. CiteSeerX 10.1.1.574.9535. doi:10.1093/comjnl/bxp021.
  6. ^ Gelenbe, Erol (1993). "G-Networks with signals and batch removal". Probability in the Engineering and Informational Sciences. 7 (3): 335–342. doi:10.1017/s0269964800002953.
  7. ^ Artalejo, J.R. (Oct 2000). "G-networks: A versatile approach for work removal in queueing networks". European Journal of Operational Research. 126 (2): 233–249. doi:10.1016/S0377-2217(99)00476-2.
  8. ^ Gelenbe, Erol; Mao, Zhi-Hong; Da Li, Yan (1999). "Function approximation with spiked random networks". IEEE Transactions on Neural Networks. 10 (1): 3–9. CiteSeerX 10.1.1.46.7710. doi:10.1109/72.737488. PMID 18252498.