## 3 Detailed Problem Formulation
### Positioning Satisfaction  
When the user performs a VCR operation, he/she hopes that the player resumes playing the video exactly from the designated point. As mentioned in the anchor-prefetching strategy, the actual playback point of the player does not exactly coincide with the designated point by the user. Thus, the concept of positioning satisfaction is generated.  
As shown in Figure 4, \(\alpha\) is the middle of the anchor interval. Therefore, if the user jumps to point \(\alpha\) after a VCR operation, the worst case occurs, and the positioning satisfaction is minimal. However, if the user jumps to point \(\gamma\) or \(\beta\), because the anchors have been perfected, the best case occurs, and the positioning satisfaction is 1 (i.e., the maximum value). By closing from \(\alpha\) to \(\beta\) (or \(\gamma\)), the positioning satisfaction is increased.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Interaction** & **Video Type** & **Distribution** & **Parameters** \\ \hline \multirow{2}{*}{Forward jump length} & Short videos & \multirow{2}{*}{Weibull} & \(\alpha=0.13214\), \(\beta=0.58631\) \\  & Long videos & & \(\alpha=0.10916\), \(\beta=0.54129\) \\ \hline \multirow{2}{*}{Backward jump length} & Short videos & \multirow{2}{*}{Weibull} & \(\alpha=0.08108\), \(\beta=0.59850\) \\  & Long videos & & \(\alpha=0.05959\), \(\beta=0.59236\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Jump length characterization.  
Figure 3: Average playback delay with different anchor intervals.  
Satisfaction is a fuzzy concept of people psychology, so we use the membership function of fuzzy mathematics to define the positioning satisfaction. We define that the positioning satisfaction is 1 if the time gap of the actual playback point and the designated point is less than 5 s, and a greater difference implies lower positioning satisfaction. The positioning satisfaction with time gap is calculated according to Equation (2):  
\[S(x)=\begin{cases}\begin{aligned} 1,&& x\leq 5s\\ e^{-(\frac{x-5}{\lambda})^{2}},&& x>5s\end{aligned}\end{cases} \tag{2}\]  
where \(\lambda\) is a constant strength factor.  
We have studied how the anchor interval affects the positioning satisfaction using simulations. Since the user more likely jumps forward for a VCR operation [18; 19]. We mainly focus on the forward-search behavior. After the video was played for 1 min, the user randomly performed 20 VCR operations; then, the average positioning satisfaction was recorded. As shown in Figure 5, with the increase in anchor interval, the positioning satisfaction gradually decreases. From this perspective, a smaller anchor interval corresponds to higher positioning satisfaction and better system performance.  
It is not acceptable for the user to experience a long playback delay when they perform a VCR operation, i.e., a shorter playback delay corresponds to a better user experience. According to the analysis in Section 3.1, a larger anchor interval is better.  
The positioning satisfaction reflects the difference between the actual playback point and the designated point. A smaller difference indicates higher satisfaction of the user. According to the analysis in Section 3.2, a smaller anchor interval is better.  
Therefore, the playback delay and positioning satisfaction are two conflicting factors.  
Figure 4: Seeking in a video.  
Figure 5: Average satisfaction with different anchor intervals.