## 5 Numerical Results
### Maximum Bit Error Rate and Maximum Interference  
We study the effect of changing the maximum Bit Error Rate, BER\({}_{\max}\) and maximum interference \(I_{max}\) on the optimal values of total rate, total power and total transmission delay for the entire network.  
In order to study the effect of varying the maximum interference, \(I_{max}\) on the total rate, total delay and total power, we fix the packet size to \(Ls=300\) Bytes and use all the other parameters as provided in Table 1. The results of the simulations are given in Figure 1.  
We also set the packet size to \(Ls=300\) Bytes and vary the maximum BER (BER\({}_{\max}\)), in order to study the effect of BER\({}_{\max}\) on the total power, total rate and total delay. The results of this are summarized in Figure 2.  
From Figure 0(a),b, it can be observed that both the total rate and total power are decreasing functions of \(I_{max}\) for values of \(I_{max}\) less than 0.18 and 0.1, respectively. The curves of total rate and total power start to get flat as \(I_{max}\) increases beyond 0.18 and 0.1, respectively. The total delay curves also start to become relatively flat after a maximum interference value of 0.18. Thus, we can conclude that for a packet size of 300 Bytes and the parameters in Table 1, an interference of 0.18 is a good threshold value for PUs. This is because both graphs are generally flat for the values \(I_{max}\geq 0.18\). An increase in PU interference generally leads to a decrease in the rate of transmission and the power of the SUs. However, an increase in PU interference will generally lead to an increase in the time taken by an SU to transmit on the channel, hence the increase in transmission delay. These behaviors are shown in the simulation results in Figure 1. The behavior can be attributed to the fact that the SUs are not the first priority users of the channels and have to vacate the channels every time a PU comes on the channel.  
It should also be noted that case 3 (more weight on power) leads to the least amount of power and rate, and also the highest amount of delay. On the other hand, case 4 (more weight on power) leads to the lowest transmission delay and the highest power and rate. The case of equal weights for all the objective functions, case 1, gives optimal values between case 3 and case 4.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{\lambda_{3}}\) \\ \hline case 1 & \(\frac{1}{3}\) & \(\frac{1}{3}\) & \(\frac{1}{3}\) \\ \hline case 2 & 0.2 & 0.2 & 0.6 \\ \hline case 3 & 0.2 & 0.6 & 0.2 \\ \hline case 4 & 0.6 & 0.2 & 0.2 \\ \hline \end{tabular}
\end{table}
Table 2: Different cases used in simulations.  
Figure 1: Effect of changing maximum interference \(I_{max}\), on the optimal rate, power and delay.  
From Figure 2a,b, the total rate and total power are an increasing function of maximum BER, until a threshold value of \(\text{BER}_{\text{max}}=0.01\) is reached. On the other hand, the total delay is a decreasing function of maximum BER until a threshold value of \(\text{BER}_{\text{max}}=0.01\) is reached, as seen from Figure 2c. After the threshold value of \(\text{BER}_{\text{max}}=0.01\), is reached, the curves for total delay, total rate and total power become flat. This shows that the threshold value for \(\text{BER}_{\text{max}}\) for our channel is 0.01. Before reaching the threshold \(\text{BER}_{\text{max}}=0.01\), the total power and total rate are an increasing function of maximum BER, and the total transmission delay is a decreasing function of maximum BER. This is because of the fact that allowing for a higher percentage of corrupted bits leads to an increase in total transmission rate and power, and at the same time leads to less transmission time. Using case 1 gives results that are generally in-between (cases 2, 4) and case 3. Optimizing using case 3 gives the least power and rate and also the highest delay, whereas using (cases 2, 4) gives the highest power and rate, and the lowest transmission delay.