# G-Networks with Adders
## 2 New Types of Customers: The Adders
The Chapman-Kolmogorov equations, or \"master equations\" [20,21] for the system we have defined are:  
\[\frac{dp(\vec{k},t)}{dt}=-\sum_{i=1}^{N}\left[\lambda_{i}^{+}+(\ \mu_{i}+ \lambda_{i}^{-}\ )1_{\{k_{i}>0\}}\right]\ p(\vec{k},t)\]  
\[+\sum_{i=1}^{N}\lambda_{i}^{+}p(\vec{k}-\vec{e}_{i},t)1_{\{k_{i}>0\}}+\sum_{i= 1}^{N}(\ \mu_{i}d_{i}+\lambda_{i}^{-}\ )p(\vec{k}+\vec{e}_{i},t)\]  
\[+\sum_{i=1}^{N}\sum_{j=1}^{N}\mu_{i}p(\vec{k}+\vec{e}_{i}-\vec{e}_{j},t)P_{ij} ^{+}1_{\{k_{j}>0\}}\]  
\[+\sum_{i=1}^{N}\sum_{j=1}^{N}[\mu_{i}p(\vec{k}+\vec{e}_{i}+\vec{e}_{j},t)P_{ij }^{-}+\mu_{i}p(\vec{k}+\vec{e}_{i},t)P_{ij}^{-}1_{\{k_{j}=0\}}]\]  
\[+\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{m=1}^{N}\ \mu_{i}\ p(\vec{k}+\vec{e}_{i}+ \vec{e}_{m}-\vec{e}_{j},t)Q_{imj}1_{\{k_{j}>0\}}\]  
\[+\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{m=1}^{N}\ \mu_{i}\ p(\vec{k}+\vec{e}_{i},t)Q_{ jim}1_{\{k_{j}=0\}}+\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{y=0}^{\infty}\ \mu_{i}\ P_{ij}^{A}\ \gamma_{ij}(k_{j},t)p(\vec{k}+\vec{e}_{i}+\vec{e}_{j}(y-k_{j}),t).\]  
**3. The Product Form**  
Consider the following system of non-linear equations:  
\[\Lambda_{i}^{+}=\lambda_{i}^{+}+\sum_{j=1}^{N}[\mu_{j}\rho_{j}P_{ji}^{+}+\mu_ {j}\rho_{j}P_{ji}^{A}\rho_{i}W_{ji}(\rho_{i})+\sum_{m=1}^{N}\mu_{j}\rho_{j} \rho_{m}Q_{jmi}], \tag{2}\]  
\[= 1-R_{j}(t)W_{ij}(R_{j}(t))\ if\ x=0,\]0  
\[= 1-R_{j}(t)W_{ij}(R_{j}(t))\ if\ x=0,\]1  
**Theorem 1. **_If the system of \(3N\) Equations (2)-(4) have a solution with \(0\leq\rho_{i}<\ 1\), for \(i=1,\...\,N\), then the equilibrium probability for the G-network with Adders is:_  
\[= 1-R_{j}(t)W_{ij}(R_{j}(t))\ if\ x=0,\]2  
_where the marginal queue length probabilities are \(p_{i}(k_{i})=lim_{t\rightarrow\infty}Pr[K_{i}(t)=k_{i}]=(\rho_{i})^{k_{i}}(1 \ -\ \rho_{i})\), \(k_{i}\geq\ 0\)._  
_A Simple Example_  
In the introduction we had mentioned that the G-network with Adders has a \"stabilising\" or load reducing property, and we would like to illustrate this with a simple example before we develop the proof of the Theorem. Consider a system with just a single queue, and assume that the arrival rate of positive customers to this queue is identical to the service rate, so that \(\lambda\ =\ \mu\). Assume that there are no negative customers arriving to the queue, nor any feedback of customers back to the queue after service, so that \(\lambda^{-}=0\) and \(P_{11}^{+}=P_{11}^{-}=0\). Also assume that there are no \"true\" departures so that each customer after a service turns into an Adder, and we have \(P_{11}^{A}=1\). Then (2)-(4) give us the utilisation factor for the queue that is expressed as:  
\[= 1-R_{j}(t)W_{ij}(R_{j}(t))\ if\ x=0,\]3  
while if their were no Adders, and if each customer departed after service, the utilisation factor would be larger at the value \(\frac{\lambda^{+}}{\mu}\). Clearly, the average queue length with Adders will also be smaller than the average queue length without Adders but where every customer departs the queue after a service. Thus quite obviously and surprisingly, the queue without Adders and with ordinary departures is less stable than a queue without any customer departures but with Adders.