# G-Networks with Adders

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. New Types of Customers: The Adders

- Only positive customers can remain in a queue to wait for service. Negative customers, triggers and adders disappear once they have accomplished their mission.
- Negative customers are a special case of triggers, since the customer they remove from a queue will then directly leave the network rather than joining another queue.
- Positive customers, negative customers (including batch removal) and triggers, are covered in early work [18] and lead to a “ product form” solution for the joint probability of network state in equilibrium, provided external arrivals are Poisson processes, service times are independent and identically distributed, and the movement of customers is described by a Markov chain.

- A positive customer with probability ${P}_{ij}^{+}$; hence ${K}_{j}\left({t}^{+}\right)={K}_{j}\left(t\right)+1$ and we can have ${P}_{ii}\ge 0$,
- A negative customer with probability ${P}_{ij}^{-}$, but we require that ${P}_{ii}^{-}=0$, so that ${K}_{j}\left({t}^{+}\right)=max[0,{K}_{j}\left(t\right)-1]$,
- A trigger that moves one positive customer from some other node j to node m, if there is at least one such customer, with probability ${Q}_{ijm}$; we require that ${Q}_{ijm}=0$ if $i=j$ or $i=m$ or $j=m$. As a result ${K}_{j}\left({t}^{+}\right)=max[0,{K}_{j}\left(t\right)-1].1[{K}_{i}\left(t\right)>0]$, and ${K}_{m}\left({t}^{+}\right)={K}_{m}\left(t\right)+1[{K}_{i}\left(t\right)>0,{K}_{j}\left(t\right)>0]$,
- Finally as the new customer type, the Adder with probability ${P}_{ij}^{A}$, and we have ${d}_{i}+{\sum}_{j=1}^{N}\phantom{\rule{3.33333pt}{0ex}}[{P}_{ij}^{A}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{P}_{ij}^{+}+{P}_{ij}^{-}+{\sum}_{m=1}^{N}{Q}_{ijm}]=1$ for each queue i.

#### Chapman-Kolmogorov Equations

## 3. The Product Form

**Theorem**

**1.**

#### A Simple Example

## 4. Proof of the Theorem

**Proof.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Fourneau, J.-M.; Gelenbe, E.
G-Networks with Adders. *Future Internet* **2017**, *9*, 34.
https://doi.org/10.3390/fi9030034

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Fourneau J-M, Gelenbe E.
G-Networks with Adders. *Future Internet*. 2017; 9(3):34.
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Fourneau, Jean-Michel, and Erol Gelenbe.
2017. "G-Networks with Adders" *Future Internet* 9, no. 3: 34.
https://doi.org/10.3390/fi9030034