## 5 Experiments
### Evaluate Performance  
#### 5.2.1 The Accuracy of Estimation  
The simulation results for Scenario-1 and Scenario-2 are displayed in Figures 5 and 6, respectively. In MC, conventional IS, and the proposed IS simulation, there are \(10^{6}\) [cycles] of regenerative cycles. Figures 5 and 6 demonstrate that the performance of the proposed method is comparable to that of the MC and traditional IS. Otherwise, the estimated value of the IS approach can be produced at a value smaller than \(10^{-25}\), whereas the MC technique only estimates the probability \(P(Q>q)\) with a minimum value of roughly \(10^{-6}\). When \(q\) is sufficiently large, the event \(\{Q>q\}\) does not happen frequently, making it impossible to estimate the value using the MC approach, or requiring a significant amount of time to acquire the estimated value. Consequently, in these situations, the IS approach is helpful.  
Simulation time: We now want to evaluate the simulation times of the MC method, conventional IS, and proposed IS simulation. In Case 4, we will simulate to estimate the tail probability \(P(Q>q)\) for \(q=20[packets]\) for all three methods. The number of RCs in this experiment is \(10^{7}\) for MC simulation and \(10^{2}\) for both conventional IS and the proposed IS simulation. To calculate the sample mean and sample variance, we ran the simulation 30 times to obtain the estimated value of the tail probability. The outcome is presented in Table 8, where the estimation variances are nearly the same. The simulation time using Monte Carlo (MC) is 1136.81 s, whereas it is 0.39 s using conventional Importance Sampling (IS) and 0.35 s using the proposed IS simulation. Therefore, the proposed IS simulation has a simulation time that is approximately 3200 times faster than the MC simulation and is slightly shorter than traditional IS. The reason can be explained as follows: First, it is the difference between the calculation of the weight function of conventional IS and the proposed IS. Conventional IS simulation uses Equation (18) to calculate the weighting function. This equation is more complex than Equation (19) used in the proposed IS. In  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(\mathbf{\lambda_{0}}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{r_{01}}\) & \(\mathbf{r_{10}}\) & \(\mathbf{\mu}\) & \(\mathbf{\lambda_{mean}}\) \\ \hline
4 & 0.8 & 0.5 & 0.1 & 0.4 & 1.0 & 0.74 \\
5 & 0.6 & 0.3 & 0.1 & 0.4 & 1.0 & 0.54 \\
6 & 0.4 & 0.3 & 0.7 & 0.3 & 1.0 & 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Scenario-2: parameters for MC simulation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & \(\mathbf{\lambda_{0}}\) & \(\mathbf{\lambda_{1}}\) & \(\mathbf{r_{01}}\) & \(\mathbf{r_{10}}\) & \(\mathbf{\theta_{0}^{*}}\) & \(\mathbf{\mu^{*}}\) \\ \hline
4 & 0.8 & 0.5 & 0.1 & 0.4 & 0.54 & 0.58 \\
5 & 0.6 & 0.3 & 0.1 & 0.4 & 1.05 & 0.35 \\
6 & 0.4 & 0.3 & 0.7 & 0.3 & 1.89 & 0.15 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Scenario-2: IS simulation’s parameters.  
Equation (18), it takes time to determine the states \(\{i,j\}\) of the MMPP. Moreover, the second part \(e^{-\theta_{1}^{*}(j-i)}\) of this equation also takes more time to calculate. Hence, the simulation time of conventional IS is longer than that of the proposed IS simulation.  
Figure 5: Scenario-1: performance comparison of estimates by MC, conventional IS, and the proposed method (\(\lambda_{1}=\lambda_{2}\)).  
Figure 6: Scenario-2: performance comparison of estimates by MC, conventional IS, and the proposed (\(\lambda_{1}\
eq\lambda_{2}\)).