# Online Distributed User Association for Heterogeneous Radio Access Network
## 6 Performance Evaluation  
Here, we consider a three-tier HetNet environment consisting of macro, pico, and femto BSs. We develop the HetNet environment and run the simulation on the MATLAB platform. The BSs are uniformly and independently distributed in a 2000 \(\times\) 2000 m\({}^{2}\) space. The macro BS is placed at the central location of the space, and 4 pico and 20 femto BSs are deployed in the whole region. We also consider a uniform and non-uniform deployment of the 500 mobile UEs in the network scenario. For non-uniform deployment, we place the UEs around Femto and Pico BSs. Figure 5a,b depicts the UE and BS deployment for uniform and non-uniform UE deployment respectively. These UEs arrive to the system online--that is, there is one UE arrival per unit time. The spectrum bands of different tiers are orthogonal to each other, and hence do not interfere with each other, whereas BSs which belong to the same tier do interfere. The transmission powers of macro, pico, and femto BSs are 46 dBm, 35 dBm, and 24 dBm, respectively. The path-loss model for macro BS is taken as \(128+37.1^{\text{st}}\text{log}_{10}(D)\), where \(D\) is in kilometers. The path-loss model for pico and femto BS is taken as \(37+30^{\text{rd}}\text{log}_{10}(D)\), where \(D\) is in meters. We assume log-normal shadowing with a standard deviation \(\sigma_{5}=8\) dB and a bandwidth of 10 MHz. The thermal noise power is \(\sigma^{2}=-104\) dBm. For the different \(d\) values in Algorithm 3, we restrict the choices to only two, as the probability for higher \(d\) decreases in a real environment [54]. We perform the simulation 100 times and averaged out the results. The following are the performance matrices used for the evaluation of our results.  
(a) Cumulative distribution function (CDF) of spectrum efficiency  
(b) Normalized minimum user rate [19]  
The minimum user rate is normalized with the optimum value. The optimum value is the maximum among the minimum user rate on each scheme. Let \(R\) be the minimum data rate experienced by the UE(s) in the network. The value \(R\) will vary from one scheme to other. Let there be \(N\) schemes, so the normalized user rate, \(\widetilde{\beta}_{min}\), for each scheme, \(n\in N\), is given by \(\widetilde{\beta}_{min}=\frac{R_{n}}{max\{R_{1},R_{2},\ldots,R_{N}\}}\).  
(c) Normalized sum user rate [19]  
The sum user rate of the network is normalized with the optimum value. The optimum value is the maximum among the sum user rate on each scheme. Let \(S\) be the sum user rate of the network. The value \(S\) will vary from one scheme to other. The normalized sum user rate, \(\widetilde{\beta}_{sum}\), for each scheme, \(n\in N\), is given by \(\widetilde{\beta}_{sum}=\frac{S_{n}}{max\{S_{1},S_{2},\ldots,S_{N}\}}\).  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Algorithm** & **Association Rule** & **Time Complexity** & **Scheme** \\ \hline
1 & Centralized & \(O(U)\); \(U\) is the total number & Greedy based \\  & & of UEs in the network & \\ \hline
2 & Distributed & \(O(C)\); C is the average number & \\  & & of BSs a UE can access to & \\ \hline
3 & Distributed & \(O\) (\(d1\)+\(d2\)); \(d1\) and \(d2\) are the & \\  & & number of choices in the two levels & d-choices based \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of different algorithms.