G-network

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.

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)

Then writing (n₁, ... ,n_m) for the state of the network (with queue length n_i at node i), if a unique non-negative solution ${\displaystyle \scriptstyle {(\lambda _{i}^{+},\lambda _{i}^{-})}}$ exists to the above equations (1) and (2) such that ρ_i for all i then the stationary probability distribution π exists and is given by

${\displaystyle \pi (n_{1},n_{2},\ldots ,n_{m})=\prod _{i=1}^{m}(1-\rho _{i})\rho _{i}^{n_{i}}.}$

The response time is the length of time a customer spends in the system. The response time distribution for a single G-queue is known[9] where customers are served using a FCFS discipline at rate μ, with positive arrivals at rate λ⁺ and negative arrivals at rate λ⁻ which kill customers from the end of the queue. The Laplace transform of response time distribution in this situation is[9][10]

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

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

The response time for a tandem pair of G-queues (where customers who finish service at the first node immediately move to the second, then leave the network) is also known, and it is thought extensions to larger networks will be intractable.[10]