G-network

Definition

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A network of m interconnected queues is a G-network if

1. each queue has one server, who serves at rate μ_i,
2. external arrivals of positive customers or of triggers or resets form
   Poisson processes
   of rate ${\displaystyle \scriptstyle {\Lambda _{i}}}$ for positive customers, while triggers and resets, including negative customers, form a Poisson process of rate ${\displaystyle \scriptstyle {\lambda _{i}}}$,
3. on completing service a customer moves from queue i to queue j as a positive customer with probability ${\displaystyle \scriptstyle {p_{ij}^{+}}}$, as a trigger or reset with probability ${\displaystyle \scriptstyle {p_{ij}^{-}}}$ and departs the network with probability ${\displaystyle \scriptstyle {d_{i}}}$,
4. on arrival to a queue, a positive customer acts as usual and increases the queue length by 1,
5. 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 disappear after they have taken their action, so that they are in fact "control" signals in the network,

• 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 G-queue.

Stationary distribution

Define the utilization at each node,

${\displaystyle \rho _{i}={\frac {\lambda _{i}^{+}}{\mu _{i}+\lambda _{i}^{-}}}}$

where the ${\displaystyle \scriptstyle {\lambda _{i}^{+},\lambda _{i}^{-}}}$ for ${\displaystyle \scriptstyle {i=1,\ldots ,m}}$ satisfy

${\displaystyle \lambda _{i}^{+}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{+}+\Lambda _{i}\,}$

(1)

${\displaystyle \lambda _{i}^{-}=\sum _{j}\rho _{j}\mu _{j}p_{ji}^{-}+\lambda _{i}.\,}$

(2)

${\displaystyle W^{\ast }(s)={\frac {\mu (1-\rho )}{\lambda ^{+}}}{\frac {s+\lambda +\mu (1-\rho )-{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}{\lambda ^{-}-\lambda ^{+}-\mu (1-\rho )-s+{\sqrt {[s+\lambda +\mu (1-\rho )]^{2}-4\lambda ^{+}\lambda ^{-}}}}}}$

where λ = λ⁺ + λ⁻ and ρ = λ⁺/(λ⁻ + μ), requiring ρ < 1 for stability.