Research on video-on-demand transmissions is essentially divided into periodic broadcasting methods and on-demand methods. Periodic broadcasting is aimed to schedule transmissions off-line, so that an optimized time schedule is achieved. On the other hand video-ondemand has to deal with constraints at requesting times. Thus, studies on these areas have been quite isolated. Obviously, in periodic broadcasting all parameters are known in advance, so timetables can be accurately adjusted and it is assumed transmissions can be arranged to use less bandwidth than video-on-demand. In this paper, we analyze the convergence of both paradigms, showing that the claims that argue that VoD schemes use more bandwidth than PB ones are not necessarily true. We state this argument by proving how to convert any periodic broadcasting method into an on-demand one, which will use equal or less bandwidth. Moreover, we show that this converted on-demand method can also offer shorter serving times. Ó 200...