In a video-on-Demand(VOD) service, a server has to return to he normal playback quickly at a certain new frame position after interactive operations such as jump or last playback. In this paper, we propose an efficient scheme to write a transmission schedule for a playback restart of a video stream at a new frame position after interactive operations. The proposed scheme is based on convergence characteristics, that is transmission schedules with different playback startup frame position in a video stream meet each other at some frame position. The scheme applies a bandwidth smoothing from a new frame position to a convergence position without considering all remaining frames of a video stream. And then the scheme transmits video dta according to the new schedule from the new frame position to the convergence position, and then transmits the remaining video data according to the reference schedule from the convergence position, and then transmits the remaining video data according to the reference schedule from the convergence position to the last frame position. In this paper, we showed that there existed the convergence position corresponding to nay frame position in a video stream through many experiments based on MPEG-1 bit trace data. With the convergence we reduced the computational overhead of a bandwidth smoothing, which was applied to find a new transmission schedule after interactive operations. Also, storage overhead is greatly reduced by storing pre-calculated schedule information up to the convergence position for each I frame position of a video stream with video data off-line. By saving information on a transmission schedule off-line along with the video data and searching the schedule corresponding to the specified restarting frame position, we expect the possibility of normal playback of a video stream with small tolerable playback startup delay.