# Game Theory-Based Load-Balancing Algorithms for Small Cells Wireless Backhaul Connections +
## 2 Materials and Methods
### Modeling the Load Balancing on a Set of Tunnels as a Multiplayer Game
#### 2.2.3 Combinatorial Auction-Based Load Balancing
We define the potential function associated with the delay tolerant traffic as:  
\[\phi_{i}=\tanh\biggl{(}\frac{R_{i}}{R_{i}^{ap}}\biggr{)}=\frac{e^{\frac{2R_{i }}{R_{i}^{ap}}}-1}{e^{\frac{2R_{i}}{R_{i}^{ap}}}+1} \tag{26}\]  
where \(R_{i}^{ap}\) is the expected value of the average bit rate and can be computed according to (1). This parameter allows the differentiation between different non-real-time flows. The maximum value of this potential function is also 1. The potential function was defined in such a way as to express user satisfaction with the increasing values of the throughput. Figure 5b shows the variation of the proposed utility function with the instantaneous bit rate, \(L_{j}\), when the expected average bit rate \(R_{i}^{av}\) has different values. The hyperbolic  
Figure 5: Potential functions for delay sensitive and best effort traffic types: (**a**) Variation of the potential function for delay sensitive traffic with latency \(\tau_{i}\) when coefficient \(c_{i}\) has different values; (**b**) Variation of the potential function for best effort traffic with the instantaneous bit rate of the tunnel, \(L_{i}\), for different values of the expected average bit rate \(R_{i}^{ap}\). tangent function is a good choice for representing the progressive transition between two extreme states in our situation: not satisfied at all and totally satisfied with the throughput provided by the system. It also allows a good separation between non-real-time services with different average rate constraints. The operations involved by the Combinatorial Auction-Based LB mechanism are described in Algorithm 3, presented below. ```
Each new traffic initially is associated with default tunnel \(j=1\). At each arriving packet update the flows' potential function \(\phi_{f_{i}}\). In each moment \(mT;\ m=0,\ 1,\ 2,\ldots;T\)-interpacket delay or timer period, perform the following steps: 1:for\(j=1\) to N do
2: compute \(\Phi_{j}^{m}=\frac{\sum\limits_{f\in\mathbf{K}_{j}}\phi_{f_{i}}}{|\mathbf{K} _{j}|}\)
3: compute \(\Delta\Phi_{j}^{m}=\Phi_{j}^{m}-\Phi_{j}^{m-1}\)
4:while\(\Delta\Phi_{j}^{m}<imposed\)treshold do// the QoS decreases considerably Select the flow with the lowest potential. 5:\(i=\arg\min\limits_{f_{i}\in\mathbf{K}_{j}}\left(\phi_{f_{i}}\right)\) Create a new arrangement: 6:\(\mathbf{s}=\left(\mathbf{K}_{1}\cup\{f_{i}\},\mathbf{K}_{2}\cup\{f_{i}\}, \ldots,\mathbf{K}_{j}\cap\{f_{i}\},\ldots,\mathbf{K}_{\mathbf{N}}\cup\{f_{i} \}\right)\)
7: each bidder \(z\
eq j\) submits a bid \(b_{z}(\mathbf{K}_{z})\) for the new subset \(\mathbf{K}_{z}\) Route flow \(f_{i}\) through the tunnel which submitted the lowest bid. 8:\(t=\arg\min\limits_{ \begin{array}{c}z=1,\ldots,N\\ z\
eq j\end{array} }\left(b_{z}(\mathbf{K}_{z})\right)\) Associate flow \(i\) to tunnel \(t\). 9:\(\mathbf{K}_{t}=\mathbf{K}_{t}\cup\{f_{i}\}\)
10: recompute \(\Phi_{j}^{m}=\frac{\sum\limits_{f\in\mathbf{K}_{j}}\phi_{f_{i}}}{|\mathbf{K} _{j}|}\)
11: recompute \(\Delta\Phi_{j}^{m}=\Phi_{j}^{m}-\Phi_{j}^{m-1}\)
12:endwhile
13:endfor
```  
**Algorithm 3** Combinatorial Auction-based load-balancing algorithm