# G-Networks with Adders
## 2 New Types of Customers: The Adders  
In the model we consider, \(N\) distinct servers have queues where ordinary customers can line up waiting for service, and the server handles them individually, in some order which does not depend on the individual customers' service needs. The service times of successive customers are independent and identically distributed exponential variables at each of the queues, and the service rates for the different queues are \(\mu_{i}\ \geq 0\) for \(i=1\,,...,\ N\). Denote by \((K_{1}(t),\...\,,K_{N}(t))\) the queue lengths where \(K_{i}(t)\) is the number of normal customers at queue \(i\) at time \(t\), and let \(p(\vec{k},t)=Pr[(K_{1}(t),\...\,,K_{N}(t))=\vec{k}]\). In addition to positive customers, negative customers, triggers, new types of customers that we call Adders can arrive to a queue from some node (possibly the same one). External arrivals of positive and negative arrivals occur according to mutually independent Poisson processes, with rates \(\lambda_{i}^{+},\ \lambda_{i}^{-}\), respectively, and:  
* 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. After a positive customer ends its service at time \(t\geq 0\) at queue \(i\), we will have \(K_{i}(t^{+})=K_{i}(t)-1\), and then the customer can then leave the network with probability \(d_{i}\), or it joins some queue \(j\) as:  
* A positive customer with probability \(P_{ij}^{+}\); hence \(K_{j}(t^{+})=K_{j}(t)+1\) and we can have \(P_{ii}\geq 0\),
* A negative customer with probability \(P_{ij}^{-}\), but we require that \(P_{ii}^{-}=0\), so that \(K_{j}(t^{+})=max[0,K_{j}(t)-1]\),
* 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 \(K_{j}(t^{+})=max[0,K_{j}(t)-1].1[K_{i}(t)>0]\), and \(K_{m}(t^{+})=K_{m}(t)+1[K_{i}(t)>0,K_{j}(t)>0]\),
* Finally as the new customer type, the Adder with probability \(P_{ij}^{A}\), and we have \(d_{i}+\sum_{j=1}^{N}\ [P_{ij}^{A}\ +\ P_{ij}^{+}+P_{ij}^{-}+\sum_{m=1}^{N}Q_{ijm}]=1\) for each queue \(i\). Let \(w_{ij}(k),k\in\{1,\ 2,\...\ \}\) be a (possibly non-honest) probability distribution with \(\sum_{k=1}^{\infty}\ w_{ij}(k)\leq 1\), and let its generating function be: \(W_{ij}(y)=\sum_{k=1}^{\infty}w_{ij}(k)y^{k},\ |y|\leq 1\). The Adder will replace the current number of customers at the queue where it arrives, by a random number of customers which depends on the probability \(R_{j}(t)\) that the queue was busy just before the Adder arrived, where \(R_{j}(t)\equiv Pr[K_{j}(t)>0]\). The queue length \(K_{j}(t^{+})\), just after the Adder arrives at \(j\), is a random variable whose probability distribution is:  
\[Pr[K_{j}(t^{+})=x]\equiv\gamma_{ij}(x,t) = [1-R_{j}(t)]W_{ij}(R_{j}(t))(R_{j}(t))^{x},\ x>0,\] \[= 1-R_{j}(t)W_{ij}(R_{j}(t))\ if\ x=0,\]  
so that:  
\[Pr[K_{j}(t^{+})>0]=R_{j}(t)W_{ij}(R_{j}(t)). \tag{1}\]  
We immediately see the potential application of Adders, as a way to reduce queue length and make the system more stable, since we have \(Pr[K_{j}(t^{+})>0]\ \leq\ Pr[K_{j}(t)>0]\) since \(W_{ij}(R_{j}(t))\leq 1\). _Chapman-Kolmogorov Equations_  
Let \(\vec{e}_{i}\) be the \(N\)-vector which is 0 everywhere, except in position \(i\) where it has the value 1: \(\vec{e}_{i}\ =\ (0,\..\,1,\..\,0)\).