## 4 Results
### Statistical Error Propagation in QoE considering Correlation between PLR and Jitter  
We carried out an analysis to estimate the correlation between QoS parameters as presented in Table 4. We found that on average the UK network service providers have around 20-40% linear correlation (PCC) between their measured PLR and jitter. This level of correlation is significant and hence we considered it to evaluate the propagated asymptotic standard error of QoE for VoD applications. We took the sample sizes of 400, 800 and 1200 from the population of the commercially measured data. The measurements were anonymised by following the privacy policy and they were distributed over a period of almost 50 h. Using the evaluated PCC and Equation (5), we calculated the asymptotic standard error of the QoE at different operating points of PLR and jitter. This was used to evaluate the confidence interval width around mean QoE at operating points of PLR and jitter. The resulting confidence interval width of the QoE at a certain operating point of jitter was plotted against PLR and are presented in Figures 2 and 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**PLR** & **Absolute Standard** & **Relative Standard** & **Propagated Standard** & **Propagated CI** \\
**Configuration (\%)** & **Error in PLR (\%)** & **Error in PLR** & **Error in QoE (MOS)** & **Width of QoE (MOS)** \\ \hline
0.05 & 0.016 & 0.320 & 0.10 & 0.40 \\
0.10 & 0.015 & 0.150 & 0.10 & 0.40 \\
0.50 & 0.036 & 0.072 & 0.12 & 0.45 \\
1.0 & 0.074 & 0.074 & 0.23 & 0.50 \\
5 & 0.160 & 0.032 & 0.02 & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: NetEm PLR uncertainty and propagated error in QoE. \begin{table}
\begin{tabular}{c c} \hline \hline
**PLR Operating Points** & **\(\mathbf{\partial}\)QoE/\(\mathbf{\partial}\)PLR(MOS)** \\ \hline
0.0005 & \(-\)562.7 \\
0.001 & \(-\)519.8 \\
0.005 & \(-\)293.4 \\
0.01 & \(-\)155.2 \\
0.05 & \(-\)1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First derivative of QoE w.r.t PLR at operating points. \begin{table}
\begin{tabular}{c c} \hline \hline
**Jitter Operating Points (ms)** & **\(\mathbf{\partial}\)QoE/\(\mathbf{\partial}\)jitter (MOS/ms)** \\ \hline
10 & \(-\)0.06 \\
20 & \(-\)0.05 \\
40 & \(-\)0.03 \\
80 & \(-\)0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First derivative of QoE w.r.t jitter at operating points. Based on further analysis we found that the confidence interval width is not affected by the jitter magnitude for the model presented in Equation (4). Hence, we only used the lower and upper limiting two values of jitter (10 ms and 80 ms) in our results for this model. The trend in confidence interval width of QoE vs. PLR at all operating points of jitter is the same. We observe that:  
* Uncertainty (measured as CI width) in QoE has a peak somewhere between PLR = 0.001 and PLR = 0.01: this has great significance for network and service operators, as the mean PLR written into most SLAs is around the value of PLR > 0.001 and PLR < 0.01 [35]. * Uncertainty (measured as CI width) in QoE rises to a peak between PLR = 0.001 and PLR = 0.01, and then rapidly diminishes as PLR either increases or decreases. * This shape is constant regardless of the jitter value, from very small (10 ms) to very large (80 ms) jitter values. * The shape is also consistent regardless of the values of PCC, from very small (10%) to relatively larger (30%). Figure 3: Propagated uncertainty in QoE vs. PLR at jitter = 80 ms with correlation. Figure 2: Propagated uncertainty in QoE vs. PLR at jitter = 10 ms with correlation. * The absolute predicted values of 95% CI width in QoE decreases for increasing sample size. However, for sample size 800 (UK sampling guideline) CI width in QoE is large and peaks at around 3 units of MOS. The CI width of the QoE is large between PLR > 0.001 and PLR < 0.01. PLR = 0.001 to PLR = 0.01 is the most common operational magnitude of PLR in real network operation and are recorded in the SLAs for VoD applications [35].