# Game Theory-Based Load-Balancing Algorithms for Small Cells Wireless Backhaul Connections +
## 2 Materials and Methods  
### Problem Formulation and System Model  
The considered networking scenario is presented in Figure 1. The AP of the small cell offers high bit rate communications to its active users, using some access technology such as LTE, WiFi, etc., and connects to the core network using the wireless links of several macro Radio Access Networks (RANs) (LTE, 3G+, WiMAX, 5G, etc.) which cover the area of the small cell.  
A particular case is represented by the deployment of a small cell inside a public transportation vehicle (bus, train, etc.) to provide ubiquitous and high bit-rate communications to the passengers and the equipment of the vehicles (video surveillance system, infotainment system, etc.). In this situation, depicted in Figure 2, one of the most cost-effective solutions for providing backhaul connectivity for the AP is to use the wireless links of the macro cells which cover the route of the vehicle as backhaul links. The AP is represented in this case by an Advanced Mobile Router which works together with a dedicated advanced gateway, the Service Continuity Gateway (see Figure 2), to provide ubiquitous connectivity.  
A single shared macro cell link might not be able to provide the needed backhaul capacity, at least not in circumstances when the number of small cell users is large and bandwidth-demanding services are accessed. The solution is to use several macro cell links established in different macro cells covering the small cell and aggregate the bandwidth offered by these macro cell links. The problem which must be solved is how to distribute/route in an efficient way the service flows of the users connected to the small cell AP over the macro cells shared links while fulfilling the QoS requirements of the user's service flows.  
The considered system model for the uplink transmission is depicted in Figure 3 and includes the AP node, which receives the services flows, \(f_{i}\), generated by the users, and is connected to an Advanced Gateway node (AG) by several parallel virtual tunnels. Both nodes have LB capabilities over the virtual tunnels connecting them. Such an advanced gateway node could be part of a virtualized infrastructure layer [1; 2], and its functionalities can be defined and implemented using SDN technologies [9]. As in most communication systems, e.g., LTE and WiMAX, each duplex communication channel is formed of two independent simplex communication channels. The modeling of the downlink transmission is like that of the uplink transmission, only the direction of the flows is different. Each flow \(f_{i}\), \(i=1,\ldots,K\), is characterized by a distribution function \(\Gamma_{i}(t)\) which describes the packets'  
Figure 1: The considered networking scenario.  
Figure 2: Heterogeneous network architecture that provides ubiquitous connectivity to public transportation vehicles.  
arrival rate, and a function \(\Theta_{i}(x)\) which describes the distribution of the packets' size, \(x\). The average bit rate \(R_{i}^{av}\) of the \(i\)-th flow is given by:  
\[R_{i}^{av}=\int\int\Gamma_{i}(t)\Theta_{i}(x)dxdt \tag{1}\]  
It is assumed that there are also several output flows and there is a one-to-one correspondence between the \(i\)-th input flow and \(i\)-th output flow.