# Game Theory-Based Load-Balancing Algorithms for Small Cells Wireless Backhaul Connections +
## 2 Materials and Methods
### Problem Formulation and System Model
#### 2.1.1 Virtual Tunnel Modeling  
Each tunnel between the AP and AGN nodes is modeled as a First-In-First-Out (FIFO) queue with a single server whose service time depends on the instantaneous physical channel conditions in the wireless network where the tunnel is passing. To each tunnel, \(j\), is associated a convex function \(T_{j}(n_{j},C_{j}(t))\), which returns the expected waiting time for a newly arrived packet, if, after queuing the packet there are \(n_{j}\) packets waiting in the queue, and the instantaneous capacity of the physical link is \(C_{j}(t)\). Because the data flow fed to the input of a tunnel is a mixture of different types of traffic, real-time and non-real-time traffic with different packet sizes and packet interarrival times, it can be considered that the packets arrive according to a Poisson process [33]. It should be mentioned that the Poisson traffic model is not the most realistic model for many traffic types, but it is widely accepted and provides mathematical tractability. Some more complex modeling involving a mixture of Poisson distributions is given in [33]. The data packets' interarrival times on different tunnels are independent random variables exponentially distributed with parameter \(\lambda_{Aj}\)[34]. The interarrival time, \(\
u_{Aj}\), with parameter \(\lambda_{Aj}\) has the probability distribution function:  
\[P\big{(}\
u_{Aj}\big{)}=\lambda_{Aj}\cdot e^{-\lambda_{Aj}\cdot\
u_{Aj}} \tag{2}\]  
The expectation of the number of packets fed to the input of tunnel \(j\) is:  
\[\mathbb{E}\big{(}\
u_{Aj}\big{)}=\lambda_{Aj} \tag{3}\]  
It is considered that the times necessary to transmit the packets, i.e., the service times on different tunnels, are independent and identically distributed with probability distribution function \(F_{Bj}(\cdot)\) and probability density \(f_{Bj}(\cdot)\). To ensure the stability of the queue the users' service flows must be combined, i.e., distributed on the available tunnels, so that the occupation rate \(\rho_{j}\) is less than one [34] on each tunnel. \[\rho_{j}=\frac{\lambda_{Aj}}{\mathbb{E}\big{(}B_{j}\big{)}^{-1}}<1 \tag{4}\]  
where \(\mathbb{E}\big{(}B_{j}\big{)}^{-1}\) represents the expectation of the packet rate on tunnel \(j\). Figure 3: The system model of the considered networking scenario. The state of the \(j\)-th tunnel can be described at a time moment by the tuple \((n_{j},\,\tau_{j})\) where \(n_{j}\) is the number of packets stored in the queue of the tunnel in discussion and \(\tau_{j}\) is the delay experienced by a packet in the transmission chain. The first parameter of the tunnel's state is discrete, while the second one is continuous, and this could complicate the analysis. However, if we consider only the queuing process of the tunnel, then the tunnel's state description can be simplified to \(n_{j}\) only, \(\tau_{j}\) being 0 for the new packets stored in the queue. We denote by \(L_{j,k}^{d}\) the number of packets remaining in the queue after the transmission of the \(k\)-th packet, and by \(d_{n^{j}}\) the fraction of packets that leave in the queue \(n_{j}\) packets after their transmission [34]. \[d_{n_{j}}=\lim\limits_{k\rightarrow\infty}P\left(L_{j,k}^{d}=n_{j}\right) \tag{5}\]  
The number of packets remaining in the queue after the \(k\)+1-th packet is sent, \(L_{j,k+1}^{d}\), is equal to the number of packets present when the \(k\)-th packet was sent minus one plus the number of packets that are stored in the queue during the transmission time of the \(k\)+1-th packet, i.e., during the service time of packet \(k\)+1. This last number of packets is denoted by \(A_{j,k+1}\) and we have [34]:  
\[ \begin{array}{l}L_{j,k+1}^{d}=L_{j,k}^{d}-1+A_{j,k+1};\,if\,L_{j,k}^{d}>0\\ L_{j,k+1}^{d}=A_{j,k+1};\,if\,L_{j,k}^{d}=0\end{array}