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In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], a '''
| url=https://www.researchgate.net/profile/Erol-Gelenbe-2/publication/239294946_Random_Neural_Networks_with_Negative_and_Positive_Signals_and_Product_Form_Solution/links/60a3b1eb458515952dd4b706/Random-Neural-Networks-with-Negative-and-Positive-Signals-and-Product-Form-Solution.pdf}}</ref><ref>{{cite journal | title = Turning Back Time – What Impact on Performance? | first = Peter | last = Harrison | author-link = Peter G. Harrison | journal = [[The Computer Journal]] | volume = 53 | doi = 10.1093/comjnl/bxp021 | issue = 6 | pages = 860–868 | year = 2009 | citeseerx = 10.1.1.574.9535 }}</ref> 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
*"triggers", which arrive from other queues or from outside the network, and which displace customers and move them to other queues
A [[product
==Definition==
{{no footnotes|section|date=February 2012}}
A network of ''m'' interconnected queues is a ''
# each queue has one server, who serves at rate ''μ<sub>i</sub>'',
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# on completing service a customer moves from queue ''i'' to queue ''j'' as a positive customer with probability <math>\scriptstyle{p_{ij}^{+}}</math>, as a trigger or reset with probability <math>\scriptstyle{p_{ij}^{-}}</math> and departs the network with probability <math>\scriptstyle{d_i}</math>,
# on arrival to a queue, a positive customer acts as usual and increases the queue length by 1,
# on arrival to a queue, the negative customer reduces the length of the queue by some random number (if there is at least one positive customer present at the queue), while a trigger moves a customer probabilistically to another queue and a reset sets the state of the queue to its steady-state if the queue is empty when the reset arrives. All triggers, negative customers and resets
* note that normal customers leaving a queue can become triggers or resets and negative customers when they visit the next queue.
A queue in such a network is known as a '''
==Stationary distribution==
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:<math>\rho_i = \frac{\lambda^+_i}{\mu_i + \lambda^-_i}</math>
where the <math>\scriptstyle{\lambda^+_i, \lambda^-_i}</math> for <math>\scriptstyle{i=1,\ldots,m}</math> satisfy
{{NumBlk|:|<math>\lambda^+_i = \sum_j \rho_j \mu_j p^+_{ji} + \Lambda_i \,</math>|{{EquationRef|1}}}}
{{NumBlk|:|<math>\lambda^-_i = \sum_j \rho_j \mu_j p^-_{ji} + \lambda_i. \,</math>|{{EquationRef|2}}}}
Then writing (''n''<sub>1</sub>,
:<math>\pi(n_1,n_2,\ldots,n_m) = \prod_{i=1}^m (1 - \rho_i)\rho_i^{n_i}.</math>
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It is sufficient to show <math>\pi</math> satisfies the [[balance equation|global balance equations]] which, quite differently from Jackson networks are non-linear. We note that the model also allows for multiple classes.
==Response time distribution==
The response time is the length of time a customer spends in the system. The response time distribution for a single
:<math>W^\ast(s) = \frac{\mu(1-\rho)}{\lambda^+}\frac{s+\lambda+\mu(1-\rho)-\sqrt{[s+\lambda+\mu(1-\rho)]^2-4\lambda^+\lambda^-}}{\lambda^--\lambda^+-\mu(1-\rho)-s+\sqrt{[s+\lambda+\mu(1-\rho)]^2-4\lambda^+\lambda^-}}</math>
where ''λ'' = ''λ''<sup>+</sup> + ''λ''<sup>−</sup> and ''ρ'' = ''λ''<sup>+</sup>/(''λ''<sup>−</sup> + ''μ''), requiring ''ρ'' < 1 for stability.
The response time for a tandem pair of
==References==
{{Reflist}}
{{Queueing theory}}
[[Category:Queueing theory]]
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